The document describes the rectangular coordinate system. It consists of two perpendicular axes (x-axis and y-axis) that intersect at the origin (0,0). Each point in the plane is assigned a unique coordinate pair (x,y) where x represents the distance from the y-axis and y represents the distance from the x-axis. The system divides the plane into four quadrants based on whether x and y are positive or negative.
The rectangular coordinate system represents points in a plane using perpendicular axes (x-axis and y-axis) that intersect at the origin (0,0). Each point is assigned an ordered pair (x,y) where x is the distance from the origin on the x-axis and y is the distance from the origin on the y-axis. The system divides the plane into four quadrants based on whether the x and y values are positive or negative. The rectangular coordinate system allows any point in the plane to be uniquely addressed using its x and y coordinates.
The document describes the rectangular coordinate system. It defines the x-axis and y-axis which intersect at the origin point (0,0). Each point in the plane is assigned an ordered pair (x,y) where x is the distance from the y-axis and y is the distance from the x-axis. The plane is divided into four quadrants based on whether x and y are positive or negative. Reflections of points across the axes are also described. Examples are provided to demonstrate labeling points and finding point coordinates.
The document describes the rectangular coordinate system. It establishes that a coordinate system assigns positions in a plane using ordered pairs of numbers (x,y). It defines the x-axis, y-axis, and origin at their intersection. Any point is addressed by its coordinates (x,y) where x represents horizontal distance from the origin and y represents vertical distance. The four quadrants divided by the axes are also defined based on positive and negative coordinate values. Reflections of points across the axes and origin are discussed. Finally, it introduces the concept of graphing mathematical relations between x and y coordinates to represent collections of points.
The document describes the rectangular coordinate system. It defines a coordinate system as assigning positions in a plane or space with addresses. The rectangular coordinate system uses a grid with two perpendicular axes (x and y) intersecting at the origin (0,0). Any point in the plane is located by its coordinates (x,y), where x is the distance right or left of the origin and y is the distance up or down. The four quadrants divided by the axes are labeled based on the signs of the x and y coordinates.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance from the y-axis and y is the distance from the x-axis. The four quadrants (I, II, III, IV) are defined by the intersection of the positive and negative sides of the x and y axes. Examples are given of labeling points and finding coordinates on the grid.
The document describes the rectangular coordinate system. It defines the system as consisting of a rectangular grid where each point in the plane is addressed by an ordered pair of numbers (x, y). The horizontal axis is called the x-axis and the vertical axis is called the y-axis. The point where the axes meet is called the origin. Each point's coordinates are defined by its distance from the origin on the x-axis and y-axis.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance right or left from the origin and y is the distance up or down. This divides the plane into four quadrants, with the signs of x and y determining which quadrant a point falls into. Examples are given of labeling points and finding coordinates on the grid.
The document describes the rectangular coordinate system. Each point in the plane can be located using an ordered pair (x, y) representing horizontal and vertical distances from the origin. The x-axis represents horizontal distance and the y-axis represents vertical distance. Changing the x-value moves a point right or left, and changing the y-value moves a point up or down. The four quadrants are defined by the positive and negative x and y axes. Reflecting a point across an axis results in another point with the same distances but opposite sign for the axis coordinate.
The rectangular coordinate system represents points in a plane using perpendicular axes (x-axis and y-axis) that intersect at the origin (0,0). Each point is assigned an ordered pair (x,y) where x is the distance from the origin on the x-axis and y is the distance from the origin on the y-axis. The system divides the plane into four quadrants based on whether the x and y values are positive or negative. The rectangular coordinate system allows any point in the plane to be uniquely addressed using its x and y coordinates.
The document describes the rectangular coordinate system. It defines the x-axis and y-axis which intersect at the origin point (0,0). Each point in the plane is assigned an ordered pair (x,y) where x is the distance from the y-axis and y is the distance from the x-axis. The plane is divided into four quadrants based on whether x and y are positive or negative. Reflections of points across the axes are also described. Examples are provided to demonstrate labeling points and finding point coordinates.
The document describes the rectangular coordinate system. It establishes that a coordinate system assigns positions in a plane using ordered pairs of numbers (x,y). It defines the x-axis, y-axis, and origin at their intersection. Any point is addressed by its coordinates (x,y) where x represents horizontal distance from the origin and y represents vertical distance. The four quadrants divided by the axes are also defined based on positive and negative coordinate values. Reflections of points across the axes and origin are discussed. Finally, it introduces the concept of graphing mathematical relations between x and y coordinates to represent collections of points.
The document describes the rectangular coordinate system. It defines a coordinate system as assigning positions in a plane or space with addresses. The rectangular coordinate system uses a grid with two perpendicular axes (x and y) intersecting at the origin (0,0). Any point in the plane is located by its coordinates (x,y), where x is the distance right or left of the origin and y is the distance up or down. The four quadrants divided by the axes are labeled based on the signs of the x and y coordinates.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance from the y-axis and y is the distance from the x-axis. The four quadrants (I, II, III, IV) are defined by the intersection of the positive and negative sides of the x and y axes. Examples are given of labeling points and finding coordinates on the grid.
The document describes the rectangular coordinate system. It defines the system as consisting of a rectangular grid where each point in the plane is addressed by an ordered pair of numbers (x, y). The horizontal axis is called the x-axis and the vertical axis is called the y-axis. The point where the axes meet is called the origin. Each point's coordinates are defined by its distance from the origin on the x-axis and y-axis.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance right or left from the origin and y is the distance up or down. This divides the plane into four quadrants, with the signs of x and y determining which quadrant a point falls into. Examples are given of labeling points and finding coordinates on the grid.
The document describes the rectangular coordinate system. Each point in the plane can be located using an ordered pair (x, y) representing horizontal and vertical distances from the origin. The x-axis represents horizontal distance and the y-axis represents vertical distance. Changing the x-value moves a point right or left, and changing the y-value moves a point up or down. The four quadrants are defined by the positive and negative x and y axes. Reflecting a point across an axis results in another point with the same distances but opposite sign for the axis coordinate.
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined, and tilted lines. Horizontal lines have the equation y=c, vertical lines x=c, and tilted lines are found using the point-slope formula y=m(x-x1)+y1, where m is the slope and (x1,y1) is a point on the line. Examples are given to demonstrate finding equations of lines given information about them.
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses sign charts of factorable polynomials. A polynomial is factorable if it can be written as the product of linear factors. The sign chart of a factorable polynomial follows an important rule: if a root has an even order, the signs are the same on both sides; if a root has an odd order, the signs are different on both sides. This is called the even/odd-order sign rule. An example demonstrates finding the sign chart of a polynomial by identifying the roots and their orders, and then applying the sign rule.
The document describes polar coordinates, which specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. Conversion formulas between polar (r, θ) and rectangular (x, y) coordinates are provided. An example problem illustrates plotting points from their polar coordinates and finding the corresponding rectangular coordinates.
The document discusses linear equations and how to graph them. It explains that linear equations relate the x-coordinate and y-coordinate of points in a straight line. To graph a linear equation, one finds ordered pairs that satisfy the equation by choosing values for x and solving for y, then plots the points. An example demonstrates graphing the linear equation y = 2x - 5 by making a table of x and y values and plotting the line.
The document discusses parabolas and their key properties:
- A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- The vertex is the point where the axis of symmetry intersects the parabola. The focus and directrix are a fixed distance (p) from the vertex.
- The latus rectum is the line segment from the focus to the parabola, perpendicular to the axis of symmetry. Its length is determined by the equation of the parabola.
The document discusses sign charts and inequalities. It explains how to determine if an expression is positive or negative when evaluated with different values of x. It provides examples of factoring expressions like x^2 - 2x - 3 to determine the signs. The key steps to create a sign chart are: 1) solve for f=0, 2) mark solutions on a number line, 3) sample points in each segment to determine the sign. A sign chart graphically shows the regions where an expression is positive, negative or zero.
This document discusses graphing parabolas using squares. It explains that parabolas can be graphed by comparing their equations to standard forms and using squares of length 2p, where p is determined by the equation. Examples are given of graphing parabolas from equations x^2=6y and y^2=8x. The focus, vertex, directrix, and latus rectum are identified for each parabola by comparing the equations to the standard forms.
The document discusses the properties of parabolas that open up or down and parabolas that open left or right. It provides the general form and standard form of each, noting that a positive value of a in the standard form indicates the direction the parabola opens (up if discussing vertical parabolas, right if discussing horizontal parabolas) while a negative a indicates the opposite direction. It also identifies the axis of symmetry formula for each. Examples are given to demonstrate finding the direction a parabola opens, its vertex, and axis of symmetry.
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
The document defines a parabola geometrically as the set of all points in a plane that are the same distance from a fixed point, called the focus, as they are from a fixed line, called the directrix. It provides examples of parabolas in standard form with the vertex at various points and opening in different directions. It also discusses how to write the standard form equation of a parabola given its focus and vertex, as well as how to find the focus and directrix of a parabola given its equation. Finally, it demonstrates how to use the method of completing the square to convert a general quadratic equation into the standard form needed for graphing a parabola.
This summary combines slides from Melanie Tomlinson and Morrobea on the topic of parabolas. The key points covered include:
- The geometric definition of a parabola as the set of all points equidistant from a fixed point (the focus) and fixed line (the directrix).
- Parabolas can be represented using various equation forms including vertex form, standard form, and general form.
- Methods for graphing parabolas by identifying features like the vertex, axis of symmetry, x-intercepts, focus, and directrix.
- Applications of parabolas to model real-world situations like searchlights and radio telescopes.
The document defines and describes ellipses. It states that an ellipse is the set of points whose sum of the distances to two fixed foci is a constant. An ellipse has a center, major axis, and minor axis. The standard form of the ellipse equation is given as (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center, a is the x-radius, and b is the y-radius. An example problem demonstrates how to identify these properties from a given ellipse equation and sketch the ellipse.
1. The document discusses parabolas and their key characteristics including focus, directrix, and standard equation forms.
2. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
3. The standard equation forms for parabolas are provided depending on the orientation and location of the vertex.
This document discusses partial fraction decompositions, which are used to integrate rational functions. It explains that a rational function P(x)/Q(x), where P and Q are polynomials, can be broken down into a sum of simpler rational formulas where the denominators are the factors of Q(x) according to the partial fraction decomposition theorem. Two methods are used to find the exact decomposition: evaluating at the roots of the least common denominator, and matching coefficients after expanding. Examples are provided to illustrate decomposing different types of rational functions.
The document discusses parabolas and their key properties. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex, axis of symmetry, and focus-directrix distance determine the shape and position of the parabola. Examples are provided to demonstrate how to find the equation of a parabola given properties like the vertex and focus.
1) The document discusses formulas for calculating area, volume, fluid pressure, and work based on the cross-sectional lengths and areas of regions and solids.
2) It provides examples of calculating the area of regions bounded by functions, the volume of solids of revolution, fluid pressure on a plate, and work needed to pump water.
3) The key concepts are using integrals to calculate quantities by summing cross-sectional lengths or areas and defining these lengths and areas based on the geometry of regions and solids.
To summarize the key steps for factoring polynomials:
1. Determine possible integer roots by finding the divisors of the constant term.
2. Use the remainder theorem or Ruffini's rule to check if an integer is a root by dividing the polynomial by (x - a) and checking if the remainder is zero.
3. Integer roots that produce a zero remainder are factors of the polynomial. Repeating this process allows one to fully factor the polynomial into linear terms.
The rectangular coordinate system (also known as Cartesian plane) was developed by René Descartes and uses two perpendicular number lines (x-axis and y-axis) that intersect at the origin (0,0) to locate points in a plane. Any point is identified with an ordered pair (x,y) denoting its distances from the x-axis and y-axis. The axes divide the plane into four quadrants, and points can be plotted or identified from their coordinates.
The rectangular coordinate system (also known as the Cartesian plane) was developed by René Descartes and uses two perpendicular number lines (the x-axis and y-axis) that intersect at the origin (0,0) to locate points in a plane. Any point is identified with an ordered pair (x,y) denoting its distances from the x-axis and y-axis. The axes divide the plane into four quadrants, and points can be plotted or identified based on their coordinates.
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined, and tilted lines. Horizontal lines have the equation y=c, vertical lines x=c, and tilted lines are found using the point-slope formula y=m(x-x1)+y1, where m is the slope and (x1,y1) is a point on the line. Examples are given to demonstrate finding equations of lines given information about them.
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses sign charts of factorable polynomials. A polynomial is factorable if it can be written as the product of linear factors. The sign chart of a factorable polynomial follows an important rule: if a root has an even order, the signs are the same on both sides; if a root has an odd order, the signs are different on both sides. This is called the even/odd-order sign rule. An example demonstrates finding the sign chart of a polynomial by identifying the roots and their orders, and then applying the sign rule.
The document describes polar coordinates, which specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. Conversion formulas between polar (r, θ) and rectangular (x, y) coordinates are provided. An example problem illustrates plotting points from their polar coordinates and finding the corresponding rectangular coordinates.
The document discusses linear equations and how to graph them. It explains that linear equations relate the x-coordinate and y-coordinate of points in a straight line. To graph a linear equation, one finds ordered pairs that satisfy the equation by choosing values for x and solving for y, then plots the points. An example demonstrates graphing the linear equation y = 2x - 5 by making a table of x and y values and plotting the line.
The document discusses parabolas and their key properties:
- A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- The vertex is the point where the axis of symmetry intersects the parabola. The focus and directrix are a fixed distance (p) from the vertex.
- The latus rectum is the line segment from the focus to the parabola, perpendicular to the axis of symmetry. Its length is determined by the equation of the parabola.
The document discusses sign charts and inequalities. It explains how to determine if an expression is positive or negative when evaluated with different values of x. It provides examples of factoring expressions like x^2 - 2x - 3 to determine the signs. The key steps to create a sign chart are: 1) solve for f=0, 2) mark solutions on a number line, 3) sample points in each segment to determine the sign. A sign chart graphically shows the regions where an expression is positive, negative or zero.
This document discusses graphing parabolas using squares. It explains that parabolas can be graphed by comparing their equations to standard forms and using squares of length 2p, where p is determined by the equation. Examples are given of graphing parabolas from equations x^2=6y and y^2=8x. The focus, vertex, directrix, and latus rectum are identified for each parabola by comparing the equations to the standard forms.
The document discusses the properties of parabolas that open up or down and parabolas that open left or right. It provides the general form and standard form of each, noting that a positive value of a in the standard form indicates the direction the parabola opens (up if discussing vertical parabolas, right if discussing horizontal parabolas) while a negative a indicates the opposite direction. It also identifies the axis of symmetry formula for each. Examples are given to demonstrate finding the direction a parabola opens, its vertex, and axis of symmetry.
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
The document defines a parabola geometrically as the set of all points in a plane that are the same distance from a fixed point, called the focus, as they are from a fixed line, called the directrix. It provides examples of parabolas in standard form with the vertex at various points and opening in different directions. It also discusses how to write the standard form equation of a parabola given its focus and vertex, as well as how to find the focus and directrix of a parabola given its equation. Finally, it demonstrates how to use the method of completing the square to convert a general quadratic equation into the standard form needed for graphing a parabola.
This summary combines slides from Melanie Tomlinson and Morrobea on the topic of parabolas. The key points covered include:
- The geometric definition of a parabola as the set of all points equidistant from a fixed point (the focus) and fixed line (the directrix).
- Parabolas can be represented using various equation forms including vertex form, standard form, and general form.
- Methods for graphing parabolas by identifying features like the vertex, axis of symmetry, x-intercepts, focus, and directrix.
- Applications of parabolas to model real-world situations like searchlights and radio telescopes.
The document defines and describes ellipses. It states that an ellipse is the set of points whose sum of the distances to two fixed foci is a constant. An ellipse has a center, major axis, and minor axis. The standard form of the ellipse equation is given as (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center, a is the x-radius, and b is the y-radius. An example problem demonstrates how to identify these properties from a given ellipse equation and sketch the ellipse.
1. The document discusses parabolas and their key characteristics including focus, directrix, and standard equation forms.
2. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
3. The standard equation forms for parabolas are provided depending on the orientation and location of the vertex.
This document discusses partial fraction decompositions, which are used to integrate rational functions. It explains that a rational function P(x)/Q(x), where P and Q are polynomials, can be broken down into a sum of simpler rational formulas where the denominators are the factors of Q(x) according to the partial fraction decomposition theorem. Two methods are used to find the exact decomposition: evaluating at the roots of the least common denominator, and matching coefficients after expanding. Examples are provided to illustrate decomposing different types of rational functions.
The document discusses parabolas and their key properties. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex, axis of symmetry, and focus-directrix distance determine the shape and position of the parabola. Examples are provided to demonstrate how to find the equation of a parabola given properties like the vertex and focus.
1) The document discusses formulas for calculating area, volume, fluid pressure, and work based on the cross-sectional lengths and areas of regions and solids.
2) It provides examples of calculating the area of regions bounded by functions, the volume of solids of revolution, fluid pressure on a plate, and work needed to pump water.
3) The key concepts are using integrals to calculate quantities by summing cross-sectional lengths or areas and defining these lengths and areas based on the geometry of regions and solids.
To summarize the key steps for factoring polynomials:
1. Determine possible integer roots by finding the divisors of the constant term.
2. Use the remainder theorem or Ruffini's rule to check if an integer is a root by dividing the polynomial by (x - a) and checking if the remainder is zero.
3. Integer roots that produce a zero remainder are factors of the polynomial. Repeating this process allows one to fully factor the polynomial into linear terms.
The rectangular coordinate system (also known as Cartesian plane) was developed by René Descartes and uses two perpendicular number lines (x-axis and y-axis) that intersect at the origin (0,0) to locate points in a plane. Any point is identified with an ordered pair (x,y) denoting its distances from the x-axis and y-axis. The axes divide the plane into four quadrants, and points can be plotted or identified from their coordinates.
The rectangular coordinate system (also known as the Cartesian plane) was developed by René Descartes and uses two perpendicular number lines (the x-axis and y-axis) that intersect at the origin (0,0) to locate points in a plane. Any point is identified with an ordered pair (x,y) denoting its distances from the x-axis and y-axis. The axes divide the plane into four quadrants, and points can be plotted or identified based on their coordinates.
This document provides an introduction to coordinate geometry and the Cartesian coordinate system. It defines key terms like coordinates, quadrants, and plotting points. The Cartesian plane is formed by the intersection of the x and y axes, with the origin at (0,0). Any point can be uniquely identified using an ordered pair (x,y) representing the distances from the x and y axes. Examples are given of plotting points and calculating distances between points on the plane using their coordinates. In summary, the document outlines the basic concepts of the Cartesian coordinate system used in coordinate geometry.
This document discusses linear equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It provides examples of linear equations and discusses how to graph them by plotting the x and y intercepts. It also explains how to determine if a given ordered pair is a solution to a linear equation by substituting the x and y values into the equation. Finally, it discusses different methods for solving systems of linear equations, including substitution and elimination.
1) A linear equation in two variables can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0.
2) The solution to a linear equation is the ordered pair that satisfies the equation when substituted into it.
3) Linear equations can be graphed on a Cartesian plane by plotting the solutions as points and connecting them.
This document provides information about coordinate grids, ordered pairs, and formulas in coordinate geometry. It defines key terms like coordinates, quadrants, and distance and section formulas. The distance formula calculates the distance between two points with coordinates (x1, y1) and (x2, y2). The section formula finds the coordinates of a point that divides a line segment between (x1, y1) and (x2, y2) in a given ratio. It also discusses finding the midpoint and calculating the area of a triangle using coordinates.
Cordinate geometry for class VIII and IXMD. G R Ahmed
Coordinate geometry uses the coordinate system of intersecting x and y axes to locate points in a plane. The x-axis is called the abscissa and the y-axis is called the ordinate. The point where they intersect is the origin, with coordinates (0,0). To find the coordinates of a point, its distance from the x-axis is measured as the abscissa and its distance from the y-axis as the ordinate, written as an ordered pair (x,y). The document provides examples of finding and writing the coordinates of points on a graph.
The document discusses the Cartesian coordinate system. It explains that the system uses a horizontal and vertical number line that intersect at the origin (0,0) to locate points in space. Any point has coordinates written as an ordered pair (x,y) representing its distance from the x-axis and y-axis. The plane is divided into four quadrants based on whether the x and y values are positive or negative. Examples are given to demonstrate plotting points in the quadrants and identifying the axes.
27 triple integrals in spherical and cylindrical coordinatesmath267
The document discusses cylindrical and spherical coordinate systems. It defines cylindrical coordinates as using polar coordinates in the xy-plane with z as the third coordinate. It provides an example of converting between rectangular and cylindrical coordinates. Spherical coordinates represent a point as (ρ, θ, φ) where ρ is the distance from the origin and θ and φ specify the direction. Conversion rules between the different systems are given.
The document discusses 3D coordinate systems. It explains that a 3D coordinate system adds a z-axis perpendicular to the x- and y-axes. There are two ways to orient the z-axis, resulting in right-hand and left-hand systems. Points in 3D space are identified by ordered triples (x,y,z). Graphs of equations in 3D are surfaces. Constant equations like x=k form planes parallel to coordinate planes.
The document discusses 3D coordinate systems. It explains that a 3D coordinate system adds a z-axis perpendicular to the x- and y-axes. There are two ways to orient the z-axis, resulting in right-hand and left-hand systems. Points in 3D space are identified by ordered triples (x, y, z). Linear equations in 3D define planes, and planes parallel to the coordinate planes satisfy equations of the form x=k, y=k, or z=k.
1. The document introduces analytic geometry and its use of Cartesian coordinate systems to determine properties of geometric figures algebraically.
2. It defines key concepts like directed lines and rectangular coordinates, and explains how to find the distance between two points and the area of polygons using their coordinates.
3. Formulas are provided to calculate distances between horizontal, vertical and slanted line segments, as well as the area of triangles and general polygons from the coordinates of their vertices. Sample problems demonstrate applying these formulas.
This document provides information about linear equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It discusses using the rectangular coordinate system to graph linear equations by plotting the x- and y-intercepts. It also describes how to determine if an ordered pair is a solution to a linear equation by substituting the x- and y-values into the equation. Finally, it briefly outlines common methods for solving systems of linear equations, including elimination, substitution, and cross-multiplication.
The document discusses coordinate geometry and the Cartesian coordinate system. It describes how René Descartes proposed using an ordered pair of numbers to describe the position of points on a plane. This allows curves and lines to be described through algebraic equations, linking algebra and geometry. The coordinate plane is defined by perpendicular x and y axes that intersect at the origin. Points on the plane are located using their coordinates (x, y), marking their distance from the two axes. The plane is divided into four quadrants by the intersecting axes.
The document discusses coordinate systems, ordered pairs, relations, and graphing ordered pairs on a coordinate plane. It defines a coordinate system as using two number lines that meet at right angles to locate points, with the x-axis and y-axis intersecting at the origin. Ordered pairs use the x-coordinate as the first number and y-coordinate as the second number to specify the location of a point. Relations show the connections between inputs and outputs and can be represented by ordered pairs, tables, or graphs.
The document discusses 3D coordinate systems. It explains that a z-axis is added perpendicular to the x- and y-axes to form a 3D coordinate system. There are two ways to orient the z-axis, known as the right-hand system and left-hand system. Every point in 3D space can be located using an ordered triple (x, y, z). The document also discusses the three coordinate planes and provides an example of sketching the graph of an equation in 3D space.
The document discusses coordinate geometry and plotting points on a Cartesian plane. It defines the x and y axes as perpendicular lines intersecting at the origin point. It then explains how to plot a point on the plane by locating its coordinates along the x and y axes. Several examples are given of plotting points with coordinates like (3,4), (-4,2), (-2,-5), and (2,-4).
The document provides information about coordinate planes and ordered pairs:
- A coordinate plane is formed by two number lines (x-axis and y-axis) intersecting at right angles at the origin point (0,0).
- Ordered pairs (x,y) are used to identify points in the coordinate plane, with x being the distance from the origin on the x-axis and y being the distance from the origin on the y-axis.
- Examples are given for naming quadrants, writing coordinates of points, and graphing points on a coordinate plane.
The document provides information about coordinate planes and ordered pairs:
- A coordinate plane is formed by two number lines (x-axis and y-axis) intersecting at right angles at the origin point (0,0).
- Ordered pairs (x,y) are used to identify points in the coordinate plane, with x being the distance from the y-axis and y being the distance from the x-axis.
- Examples are given of naming quadrants, writing coordinates of points, and graphing points on a coordinate plane.
Analytic geometry introduced in the 1630s by Descartes and Fermat uses algebraic equations to describe geometric figures on a coordinate system. It connects algebra and geometry by plotting points using a coordinate system with real number coordinates. This allows geometric shapes to be represented by algebraic equations which can be graphed. Key concepts include the Cartesian plane, slope, distance and midpoint formulas, and relationships between lines such as parallel, perpendicular and angles between lines based on their slopes.
The document discusses the concept of slope as it relates to functions. It introduces function notation and defines a function's output f(x). It explains that the slope of the line connecting two points (x, f(x)) and (x+h, f(x+h)) on a function graph is given by the difference quotient formula: m = (f(x+h) - f(x))/h. An example calculates the slope of the cord between points on the graph of f(x) = x^2 - 2x + 2.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Functions are typically represented by mathematical formulas using notation like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input, and the formula defines the output. The input box (parentheses) holds the input to be evaluated by the formula. New functions can be formed using addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate simplifying expressions involving function notation and evaluating functions for given inputs.
4 graphs of equations conic sections-circlesTzenma
There are two types of x-y formulas for graphing: functions and non-functions. Functions have y as a single-valued function of x, while non-functions cannot separate y and x. Many graphs of second-degree equations (Ax2 + By2 + Cx + Dy = E) are conic sections, including circles, ellipses, parabolas, and hyperbolas. These conic section shapes result from slicing a cone at different angles. Circles consist of all points at a fixed distance from a center point.
The document discusses quadratic functions and parabolas. It begins by defining quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It then provides an example of graphing the quadratic function y = x2 - 4x - 12. To do this, it finds the vertex by setting x = -b/2a, and uses the vertex and other points like the y-intercept to sketch the parabolic shape. It also discusses general properties of parabolas, such as being symmetric around a center line and having a highest/lowest point called the vertex that sits on this line.
The document discusses first degree (linear) functions. It explains that most real-world mathematical functions can be composed of formulas from three groups: algebraic, trigonometric, and exponential-log. Linear functions of the form f(x)=mx+b are especially important, where m is the slope and b is the y-intercept. The graphs of equations of the form Ax+By=C are straight lines. The slope formula for calculating the slope between two points (x1,y1) and (x2,y2) on a line is given as m=(y2-y1)/(x2-x1).
The document discusses the basic language of functions. A function assigns each input exactly one output. Functions can be defined through written instructions, tables, or mathematical formulas. The domain is the set of all inputs, and the range is the set of all outputs. Functions are widely used in mathematics to model real-world relationships.
The document discusses rational equations word problems involving multiplication-division operations and rate-time-distance problems. It provides an example of people sharing a taxi cost and forms a rational equation to determine the number of people. It also shows how to set up rate, time, and distance relationships using a table for problems involving hiking a trail with different rates of travel for the outward and return journeys.
The document discusses using rational equations to solve word problems involving costs shared among groups of people. It provides an example where a taxi costs $20 to rent for a group of x people, with the cost shared equally. If one person leaves the group, the remaining people each pay $1 more. Setting up the cost equations and subtracting them allows solving for x as 5, the number of original people in the group. A table is shown to organize the calculations for different inputs.
The document discusses ratios and proportions. It defines a ratio as two related quantities stated side by side, and gives an example of a 3:4 ratio of eggs to flour in a recipe. It explains how to write ratios as fractions and set up proportion equations. Proportions are equal ratios, like 3:4 being proportional to 6:8. The document shows how to solve proportion equations by cross-multiplying to obtain a regular equation that can then be solved for the unknown value.
The document discusses methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method is to combine the numerator and denominator into single fractions using cross multiplication. The second method is to multiply the lowest common denominator of all terms to both the numerator and denominator. Examples are provided to demonstrate both methods.
The document discusses techniques for combining fractions with opposite denominators. It explains that we can multiply the numerator and denominator by -1 to change the denominator to its opposite. It provides examples of switching fractions to their opposite denominators and combining fractions with opposite denominators by first switching one denominator so they are the same. It also discusses an alternative approach of pulling out a "-" from the denominator and passing it to the numerator when switching denominators, ensuring the leading term is positive for polynomial denominators.
The document discusses addition and subtraction of rational expressions. It states that fractions with the same denominator can be directly added or subtracted, while those with different denominators must first be converted to have a common denominator. The document provides an addition/subtraction rule and examples demonstrating how to perform these operations on rational expressions, including converting fractions to equivalent forms with a specified common denominator.
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required for each subject across different college requirements.
3 multiplication and division of rational expressions xTzenma
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which is that the product of two rational expressions is equal to the product of the numerators divided by the product of the denominators. It then gives examples of simplifying products and quotients of rational expressions by factoring and canceling like terms.
The document discusses terms, factors, and cancellation in mathematics expressions. It provides examples of identifying terms and factors in expressions, and using common factors to simplify fractions. Key points include:
- A mathematics expression contains one or more quantities called terms.
- A quantity multiplied to other quantities is a factor.
- To simplify a fraction, factorize it and cancel any common factors between the numerator and denominator.
The document discusses rational expressions, which are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They can be written in expanded or factored form. The factored form is useful for solving equations, determining the domain of a rational expression, evaluating inputs, and determining the sign of outputs. The domain excludes values that make the denominator equal to 0.
The document provides examples of how to translate word problems into mathematical equations using variables. It introduces using a system of two equations to solve problems involving two unknown quantities, labeled as x and y. An example word problem is provided where a rope is cut into two pieces, and the lengths of the pieces are defined using the variables x and y. The equations are set up and solved to find the length of each piece. The document also discusses organizing multiple sets of data into tables to solve word problems involving multiple entities.
The document provides an example of solving a system of linear equations using the substitution method. It begins with the system 2x + y = 7 and x + y = 5. It solves the second equation for x in terms of y, getting x = 5 - y. This expression for x is then substituted into the first equation, giving 10 - 2y + y = 7, which can be solved to find the value of y, and then substituted back into the original equation to find the value of x. The solution is presented as (2, 3). The document then provides two additional examples demonstrating how to set up and solve systems of equations using the substitution method.
The document discusses systems of linear equations. It provides examples to illustrate that we need as many equations as unknowns to solve for the unknown variables. For a system with two unknowns, we need two equations; for three unknowns, we need three equations. The document also gives examples of setting up and solving systems of linear equations to find unknown costs given information about total costs.
The document discusses equations of lines. It separates lines into two cases - horizontal and vertical lines which have a slope of 0 or undefined, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their characteristics like slope and intercept points.
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
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إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
💀💀💀💀💀💀💀💀💀💀
تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
How to Manage Reception Report in Odoo 17Celine George
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
How to Download & Install Module From the Odoo App Store in Odoo 17Celine George
Custom modules offer the flexibility to extend Odoo's capabilities, address unique requirements, and optimize workflows to align seamlessly with your organization's processes. By leveraging custom modules, businesses can unlock greater efficiency, productivity, and innovation, empowering them to stay competitive in today's dynamic market landscape. In this tutorial, we'll guide you step by step on how to easily download and install modules from the Odoo App Store.
2. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
Rectangular Coordinate System
3. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
4. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
5. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis.
6. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis.
7. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
8. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
Starting from the origin, each
point is addressed by its
ordered pair (x, y) where:
9. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
Starting from the origin, each
point is addressed by its
ordered pair (x, y) where:
x = amount to move
right (+) or left (–).
10. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
Starting from the origin, each
point is addressed by its
ordered pair (x, y) where:
x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
11. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
Rectangular Coordinate System
12. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3)
Rectangular Coordinate System
13. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right,
Rectangular Coordinate System
14. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
(4, –3)
P
15. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
16. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
17. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
B
18. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
B
C
19. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) corresponds to a point is called the
coordinate of the point, x is the x-coordinate and y is the
y-coordinate.
A
B
C
20. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) corresponds to a point is called the
coordinate of the point, x is the x-coordinate and y is the
y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R as shown.
P
Q
R
21. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) corresponds to a point is called the
coordinate of the point, x is the x-coordinate and y is the
y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R as shown.
P(4, 5),
P
Q
R
22. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) corresponds to a point is called the
coordinate of the point, x is the x-coordinate and y is the
y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R as shown.
P(4, 5), Q(3, -5),
P
Q
R
23. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) corresponds to a point is called the
coordinate of the point, x is the x-coordinate and y is the
y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R as shown.
P(4, 5), Q(3, -5), R(-6, 0)
P
Q
R
24. The coordinate of the
origin is (0, 0).
(0,0)
Rectangular Coordinate System
25. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(0,0)
Rectangular Coordinate System
26. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)
(0,0)
Rectangular Coordinate System
27. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
(0,0)
Rectangular Coordinate System
28. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
Any point on the y-axis
has coordinate of the
form (0, y).(0,0)
Rectangular Coordinate System
29. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
Any point on the y-axis
has coordinate of the
form (0, y).
(0, 6)
(0,0)
Rectangular Coordinate System
30. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
Any point on the y-axis
has coordinate of the
form (0, y).
(0, -4)
(0, 6)
(0,0)
Rectangular Coordinate System
31. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
Rectangular Coordinate System
The axes divide the plane
into four parts. Counter
clockwise, they are denoted
as quadrants I, II, III, and IV.
QIQII
QIII QIV
32. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
Rectangular Coordinate System
The axes divide the plane
into four parts. Counter
clockwise, they are denoted
as quadrants I, II, III, and IV.
QIQII
QIII QIV
(+,+)
33. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
Rectangular Coordinate System
The axes divide the plane
into four parts. Counter
clockwise, they are denoted
as quadrants I, II, III, and IV.
QIQII
QIII QIV
(+,+)(–,+)
34. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
Rectangular Coordinate System
Q1Q2
Q3 Q4
(+,+)(–,+)
(–,–) (+,–)
The axes divide the plane
into four parts. Counter
clockwise, they are denoted
as quadrants I, II, III, and IV.
Respectively, the signs of
the coordinates of each
quadrant are shown.
35. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
(5,4)
Rectangular Coordinate System
36. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
(5,4)(–5,4)
Rectangular Coordinate System
37. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
When the y-coordinate of
the a point (x, y) is changed
to its opposite as (x , –y),
the new point is the
reflection across the x-axis.
(5,4)(–5,4)
Rectangular Coordinate System
38. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
When the y-coordinate of
the a point (x, y) is changed
to its opposite as (x , –y),
the new point is the
reflection across the x-axis.
(5,4)(–5,4)
(5, –4)
Rectangular Coordinate System
39. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
When the y-coordinate of
the a point (x, y) is changed
to its opposite as (x , –y),
the new point is the
reflection across the x-axis.
(5,4)(–5,4)
(5, –4) (–x, –y) is the reflection of
(x, y) across the origin.
Rectangular Coordinate System
40. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
When the y-coordinate of
the a point (x, y) is changed
to its opposite as (x , –y),
the new point is the
reflection across the x-axis.
(5,4)(–5,4)
(5, –4) (–x, –y) is the reflection of
(x, y) across the origin.
(–5, –4)
Rectangular Coordinate System
43. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3)
Rectangular Coordinate System
A
(2, 3)
44. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
Rectangular Coordinate System
A B
(2, 3) (6, 3)
45. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
x–coord.
increased
by 4
(2, 3) (6, 3)
46. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3)
x–coord.
increased
by 4
(2, 3) (6, 3)
47. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3) - to the point C,
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)
48. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3) - to the point C,
this corresponds to moving A to the
left by 4.
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)
49. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3) - to the point C,
this corresponds to moving A to the
left by 4.
Hence we conclude that changes in the x–coordinates of a point
move the point right and left.
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)
50. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3) - to the point C,
this corresponds to moving A to the
left by 4.
Hence we conclude that changes in the x–coordinates of a point
move the point right and left.
If the x–change is +, the point moves to the right.
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)
51. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3) - to the point C,
this corresponds to moving A to the
left by 4.
Hence we conclude that changes in the x–coordinates of a point
move the point right and left.
If the x–change is +, the point moves to the right.
If the x–change is – , the point moves to the left.
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)
52. Again let A be the point (2, 3).
Rectangular Coordinate System
A
(2, 3)
53. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7)
Rectangular Coordinate System
A
(2, 3)
54. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
Rectangular Coordinate System
A
D
y–coord.
increased
by 4
(2, 3)
(2, 7)
55. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
y–coord.
increased
by 4
(2, 3)
(2, 7)
56. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
Similarly if the y–coordinate of
(2, 3) is decreased by 4 to
(2, 3 – 4) = (2, –1) - to the point E,
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)
57. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
Similarly if the y–coordinate of
(2, 3) is decreased by 4 to
(2, 3 – 4) = (2, –1) - to the point E,
this corresponds to
moving A down by 4.
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)
58. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
Similarly if the y–coordinate of
(2, 3) is decreased by 4 to
(2, 3 – 4) = (2, –1) - to the point E,
this corresponds to
moving A down by 4.
Hence we conclude that changes in the y–coordinates of a point
move the point right and left.
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)
59. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
Similarly if the y–coordinate of
(2, 3) is decreased by 4 to
(2, 3 – 4) = (2, –1) - to the point E,
this corresponds to
moving A down by 4.
Hence we conclude that changes in the y–coordinates of a point
move the point right and left.
If the y–change is +, the point moves up.
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)
60. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
Similarly if the y–coordinate of
(2, 3) is decreased by 4 to
(2, 3 – 4) = (2, –1) - to the point E,
this corresponds to
moving A down by 4.
Hence we conclude that changes in the y–coordinates of a point
move the point right and left.
If the y–change is +, the point moves up.
If the y–change is – , the point moves down.
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)
62. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
63. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4)
64. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
65. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
66. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
67. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100)
68. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
69. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D
that is 50 to the right and 30 below A?
70. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D
that is 50 to the right and 30 below A?
Here is the vertical format for the calculation:
(–2, 4)
point A
71. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D
that is 50 to the right and 30 below A?
Here is the vertical format for the calculation:
adding 50 to the x–coordinate to move right,
and –30 to the y–coordinate to move down.
(–2, 4)
point A
72. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D
that is 50 to the right and 30 below A?
Here is the vertical format for the calculation:
adding 50 to the x–coordinate to move right,
and –30 to the y–coordinate to move down.
(–2, 4)
+ (50, –30)
+ the “moves”
point A
73. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D
that is 50 to the right and 30 below A?
Here is the vertical format for the calculation:
adding 50 to the x–coordinate to move right,
and –30 to the y–coordinate to move down.
(–2, 4)
+ (50, –30)
(48, –26)
point A
+ the “moves”
74. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D
that is 50 to the right and 30 below A?
Here is the vertical format for the calculation:
adding 50 to the x–coordinate to move right,
and –30 to the y–coordinate to move down.
Hence D has the coordinate (–2 + 50, 4 – 30) = (48, –26).
(–2, 4)
+ (50, –30)
(48, –26)
point A
+ the “moves”
75. Rectangular Coordinate System
d. The point A(–2, 4) is 50 to the right and 30 below the point E
What’s the coordinate of the point E?
76. Rectangular Coordinate System
d. The point A(–2, 4) is 50 to the right and 30 below the point E
What’s the coordinate of the point E?
Let the coordinate of E be (a, b). (a , b) point E
77. Rectangular Coordinate System
d. The point A(–2, 4) is 50 to the right and 30 below the point E
What’s the coordinate of the point E?
Let the coordinate of E be (a, b).
In the vertical format we have:
(a , b)
+ (50, –30)
(–2, 4)
the “moves”
point A
point E
78. Rectangular Coordinate System
d. The point A(–2, 4) is 50 to the right and 30 below the point E
What’s the coordinate of the point E?
Let the coordinate of E be (a, b).
In the vertical format we have
Hence a + 50 = –2 so a = –52
and that b + (–30) = 4 so b = 34.
(a , b)
+ (50, –30)
(–2, 4)
the “moves”
point A
point E
79. Rectangular Coordinate System
d. The point A(–2, 4) is 50 to the right and 30 below the point E
What’s the coordinate of the point E?
Let the coordinate of E be (a, b).
In the vertical format we have
Hence a + 50 = –2 so a = –52
and that b + (–30) = 4 so b = 34.
Hence E is (–52 , 34).
(a , b)
+ (50, –30)
(–2, 4)
the “moves”
point A
point E
80. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D that is 50 to the right
and 30 below A?
We need to add 50 to the x–coordinate (to the right)
and subtract 30 from the y–coordinate (to go down).
Hence D has coordinate (–2 + 50, 4 – 30) = (48, –26).
81. Exercise. A.
a. Write down the coordinates of the following points.
Rectangular Coordinate System
AB
C
D
E
F
G
H
82. Ex. B. Plot the following points on the graph paper.
Rectangular Coordinate System
2. a. (2, 0) b. (–2, 0) c. (5, 0) d. (–8, 0) e. (–10, 0)
All these points are on which axis?
3. a. (0, 2) b. (0, –2) c. (0, 5) d. (0, –6) e. (0, 7)
All these points are on which quadrant?
4. a. (5, 2) b. (2, 5) c. (1, 7) d. (7, 1) e. (6, 6)
All these points are in which quadrant?
5. a. (–5, –2) b. (–2, –5) c. (–1, –7) d. (–7, –1) e. (–6, –6)
All these points are in which quadrant?
6. List three coordinates whose locations are in the 2nd
quadrant and plot them.
7. List three coordinates whose locations are in the 4th
quadrant and plot them.
83. C. Find the coordinates of the following points. Draw both
points for each problem.
Rectangular Coordinate System
The point that’s
8. 5 units to the right of (3, –2).
10. 4 units to the left of (–1, –5).
9. 6 units to the right of (–4, 2).
11. 6 units to the left of (2, –6).
12. 3 units to the left and 6 units down from (–2, 5).
13. 1 unit to the right and 5 units up from (–3, 1).
14. 3 units to the right and 3 units down from (–3, 4).
15. 2 units to the left and 6 units up from (4, –1).