The document discusses the point slope formula and provides examples of its use. Specifically, it contains:
1) The point slope formula: y - y1 = m(x - x1)
2) An example that finds the equation of the line passing through (-3,4) and (2,-6).
3) A second example that finds the equation (3x + 4y + 6 = 0) of the line passing through (2,-3) and parallel to the given line (3x + 4y - 5 = 0).
This document provides 10 practice problems for solving linear systems by graphing. The problems involve setting up systems of linear equations from given equations and points, then finding equations of lines parallel or perpendicular to given lines that pass through specified points.
1. The document discusses functions and relations through examples and questions.
2. It covers finding the value of functions, solving equations involving functions, and evaluating composite functions.
3. Key concepts covered include domain, codomain, range, one-to-one, many-to-one, one-to-many and many-to-many relations.
This module introduces linear functions of the form f(x) = mx + b. Learners will develop skills to determine the slope, trend, intercepts, and points of linear functions. The module is designed to help learners:
1) Determine slope, trend, intercepts, and points of a linear function given f(x) = mx + b
2) Determine f(x) = mx + b given various conditions like slope and intercepts, two points, etc.
The document provides lessons that explain how to find these properties of linear functions through examples. Practice problems are also included for learners to test their understanding.
This document provides derivatives of various functions. There are 100 problems presented without showing the step-by-step work to arrive at the solutions. The solutions are provided directly in simplified form.
This document provides 98 examples of functions and their derivatives. The functions include polynomials, trigonometric functions like sine, cosine, tangent, inverse trigonometric functions, exponential functions, logarithmic functions, and combinations of these functions.
The document discusses function graphs and the rules for transforming graphs of basic functions like f(x)=x^2 and f(x)=x^3. It explains how to sketch the graphs of transformed functions like f(x)+2, f(x-2), -f(x), and f(-x) based on shifting, reflecting, and stretching/squashing the original graph. Examples are provided to demonstrate applying these rules to sketch related function graphs like y=f(x-2)+1.
This module introduces linear functions. It defines a linear function as one that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. It explains how to graph linear functions given two points, the x- and y-intercepts, the slope and a point, or the slope and y-intercept. The document provides examples and practice problems for students to learn how to represent linear functions in different forms, rewrite them between standard and slope-intercept form, and graph them based on given information.
The document shows the step-by-step process of graphing two equations: y=x+4 and y=x-2. Points are plotted on the xy-plane for each equation and connected to show the linear graphs. The final graph shows the two lines y=x+4 and y=x-2 plotted on the same xy-coordinate plane.
This document provides 10 practice problems for solving linear systems by graphing. The problems involve setting up systems of linear equations from given equations and points, then finding equations of lines parallel or perpendicular to given lines that pass through specified points.
1. The document discusses functions and relations through examples and questions.
2. It covers finding the value of functions, solving equations involving functions, and evaluating composite functions.
3. Key concepts covered include domain, codomain, range, one-to-one, many-to-one, one-to-many and many-to-many relations.
This module introduces linear functions of the form f(x) = mx + b. Learners will develop skills to determine the slope, trend, intercepts, and points of linear functions. The module is designed to help learners:
1) Determine slope, trend, intercepts, and points of a linear function given f(x) = mx + b
2) Determine f(x) = mx + b given various conditions like slope and intercepts, two points, etc.
The document provides lessons that explain how to find these properties of linear functions through examples. Practice problems are also included for learners to test their understanding.
This document provides derivatives of various functions. There are 100 problems presented without showing the step-by-step work to arrive at the solutions. The solutions are provided directly in simplified form.
This document provides 98 examples of functions and their derivatives. The functions include polynomials, trigonometric functions like sine, cosine, tangent, inverse trigonometric functions, exponential functions, logarithmic functions, and combinations of these functions.
The document discusses function graphs and the rules for transforming graphs of basic functions like f(x)=x^2 and f(x)=x^3. It explains how to sketch the graphs of transformed functions like f(x)+2, f(x-2), -f(x), and f(-x) based on shifting, reflecting, and stretching/squashing the original graph. Examples are provided to demonstrate applying these rules to sketch related function graphs like y=f(x-2)+1.
This module introduces linear functions. It defines a linear function as one that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. It explains how to graph linear functions given two points, the x- and y-intercepts, the slope and a point, or the slope and y-intercept. The document provides examples and practice problems for students to learn how to represent linear functions in different forms, rewrite them between standard and slope-intercept form, and graph them based on given information.
The document shows the step-by-step process of graphing two equations: y=x+4 and y=x-2. Points are plotted on the xy-plane for each equation and connected to show the linear graphs. The final graph shows the two lines y=x+4 and y=x-2 plotted on the same xy-coordinate plane.
9-7 Graphing Points in Coordinate PlaneRudy Alfonso
The document explains how to graph points on a coordinate grid using ordered pairs. It defines the x-axis as the horizontal axis and y-axis as the vertical axis. The first number in an ordered pair represents the distance from the origin on the x-axis, while the second number represents the distance from the origin on the y-axis. Several examples are given of locating points from their ordered pair coordinates.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
The question defines two quartic functions, f(x) and g(x), and states they both cross the x-axis at -2. It is asked to determine the values of a and b in the functions. By setting each function equal to 0 and solving the simultaneous equations, the values are found to be a = -3 and b = -28.
The document discusses finding patterns in tables of values to write equations. It shows a table where the y-values are found by adding 5 to the x-values. By rewriting the table vertically and using number lines, it is evident that the relationship is y = x + 5, as the y-values are always 5 greater than the corresponding x-values. The document demonstrates finding the rule by recognizing that each value moves left by 3 when switching between the x and y columns.
The document provides instructions for a mathematics test that is 1 1/2 hours long and consists of 75 questions worth a total of 225 marks. For each correct answer, 3 marks are awarded, and for each wrong answer, 1 mark is deducted. It then lists 34 math problems as sample questions on topics including relations, functions, complex numbers, matrices, series, and calculus.
This document discusses polynomial functions and the remainder theorem. It provides examples of using long division and the remainder theorem to find the remainder when dividing one polynomial by another. Specifically, it shows:
1) How to perform long division of polynomials by working from highest to lowest degree terms and subtracting appropriately. There may be a remainder left over.
2) The remainder theorem states that if a polynomial P(x) is divided by (x-c), the remainder is equal to P(c).
3) Examples of using long division and directly applying the remainder theorem to find the remainder of dividing one polynomial by another.
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
1. The document discusses how to sketch logarithmic graphs of the form y = a logb(x + c) by using mini log rules to find two key points and then sketching the curve between them.
2. It provides examples of sketching graphs like y = 4 log5x, y = 6 log7x, and y = 2 log3(x - 1), explaining how to find the points where the logarithm equals 1 and is 0.
3. The document emphasizes that mini log rules are very helpful for finding the two points needed to sketch nasty log graphs.
This document contains two mathematics quizzes covering sequences and series. Quiz 1 has three problems: (1) expressing a fraction in partial fractions and finding the expansion and convergence of a series, (2) using the method of differences to find sums of series, (3) expressing a recurring decimal as a rational number. Quiz 2 has three problems: (1) finding terms in a binomial expansion, (2) expanding a binomial expression and stating the valid range, (3) proving an equality for small x and using it to evaluate an expression.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key points are eliminating variables to reduce the equations, then solving the reduced equations to find the values of the variables.
This document defines polynomials and discusses their key properties. A polynomial is an expression of the form anxn + an-1xn-1 + ... + a1x + a0, where the coefficients an, an-1, ..., a1, a0 are real numbers. The degree of a polynomial is the highest exponent in the expression, and the leading coefficient and term refer to the coefficient and term with the highest degree. Addition and multiplication of polynomials follow the distributive property, and the degree and leading term of a product are determined by the individual polynomials' degrees and leading terms.
This module discusses quadratic functions and their application to solving word problems. Students will learn to 1) recall steps to solve word problems, 2) translate problems to symbolic expressions, and 3) apply quadratic equations. Examples provided cover number, geometry, and motion problems. Students must first translate word problems into equations before solving. Key steps include identifying unknowns, writing the equation, solving, and checking solutions. Practice problems are provided to help students apply the concepts.
This PowerPoint covers about 90% of the material that will be on the Chapter 8 test. It includes steps to solve linear equations for y in terms of x, naming the slope (m) and y-intercept (b), identifying ordered pairs, determining if a relation is a function, writing rules for linear functions from tables of values, and finding the slope.
1. Indices involve rules for exponents like xa+b = xaxb and (xa)b = xab. Solving exponential equations uses these rules.
2. Graph transformations include translations, stretches, reflections, and asymptotes. Translations replace x with (x-a) and y with (y-b).
3. Sequences are functions with successive terms defined by a rule. Geometric sequences multiply successive terms by a constant ratio while arithmetic sequences add a constant.
This module introduces polynomial functions of degree greater than 2. It covers identifying polynomial functions from relations, determining the degree of a polynomial, finding quotients of polynomials using division algorithm and synthetic division, and applying the remainder and factor theorems. The document provides examples and practice problems for each topic. It aims to teach students how to work with higher degree polynomial functions.
1) The document discusses completing the square, which involves rewriting quadratic expressions in the form (x + a)2 + b to find maximum and minimum values.
2) Examples are provided of completing the square for expressions like x2 + 8x + 3 and 2x2 + 4x + 11.
3) The technique of setting the expression equal to 0 and solving for x is described as a way to find the minimum value and the corresponding x-value that produces it.
11 x1 t13 07 products of intercepts (2012)Nigel Simmons
This document discusses the product of intercepts theorem for intersecting chords and secants of a circle. It states that the product of the intercepts formed by two chords or secants on one side of their point of intersection is equal to the product of the intercepts on the other side (AX * BX = CX * DX). It also notes that for secants, the intercepts are formed by lines extending the secants outside the circle. Additionally, it presents the related theorem that the square of a tangent segment is equal to the product of its intercepts (AX^2 = CX * DX).
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
11 x1 t05 06 line through pt of intersection (2013)Nigel Simmons
The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
9-7 Graphing Points in Coordinate PlaneRudy Alfonso
The document explains how to graph points on a coordinate grid using ordered pairs. It defines the x-axis as the horizontal axis and y-axis as the vertical axis. The first number in an ordered pair represents the distance from the origin on the x-axis, while the second number represents the distance from the origin on the y-axis. Several examples are given of locating points from their ordered pair coordinates.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
The question defines two quartic functions, f(x) and g(x), and states they both cross the x-axis at -2. It is asked to determine the values of a and b in the functions. By setting each function equal to 0 and solving the simultaneous equations, the values are found to be a = -3 and b = -28.
The document discusses finding patterns in tables of values to write equations. It shows a table where the y-values are found by adding 5 to the x-values. By rewriting the table vertically and using number lines, it is evident that the relationship is y = x + 5, as the y-values are always 5 greater than the corresponding x-values. The document demonstrates finding the rule by recognizing that each value moves left by 3 when switching between the x and y columns.
The document provides instructions for a mathematics test that is 1 1/2 hours long and consists of 75 questions worth a total of 225 marks. For each correct answer, 3 marks are awarded, and for each wrong answer, 1 mark is deducted. It then lists 34 math problems as sample questions on topics including relations, functions, complex numbers, matrices, series, and calculus.
This document discusses polynomial functions and the remainder theorem. It provides examples of using long division and the remainder theorem to find the remainder when dividing one polynomial by another. Specifically, it shows:
1) How to perform long division of polynomials by working from highest to lowest degree terms and subtracting appropriately. There may be a remainder left over.
2) The remainder theorem states that if a polynomial P(x) is divided by (x-c), the remainder is equal to P(c).
3) Examples of using long division and directly applying the remainder theorem to find the remainder of dividing one polynomial by another.
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
1. The document discusses how to sketch logarithmic graphs of the form y = a logb(x + c) by using mini log rules to find two key points and then sketching the curve between them.
2. It provides examples of sketching graphs like y = 4 log5x, y = 6 log7x, and y = 2 log3(x - 1), explaining how to find the points where the logarithm equals 1 and is 0.
3. The document emphasizes that mini log rules are very helpful for finding the two points needed to sketch nasty log graphs.
This document contains two mathematics quizzes covering sequences and series. Quiz 1 has three problems: (1) expressing a fraction in partial fractions and finding the expansion and convergence of a series, (2) using the method of differences to find sums of series, (3) expressing a recurring decimal as a rational number. Quiz 2 has three problems: (1) finding terms in a binomial expansion, (2) expanding a binomial expression and stating the valid range, (3) proving an equality for small x and using it to evaluate an expression.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key points are eliminating variables to reduce the equations, then solving the reduced equations to find the values of the variables.
This document defines polynomials and discusses their key properties. A polynomial is an expression of the form anxn + an-1xn-1 + ... + a1x + a0, where the coefficients an, an-1, ..., a1, a0 are real numbers. The degree of a polynomial is the highest exponent in the expression, and the leading coefficient and term refer to the coefficient and term with the highest degree. Addition and multiplication of polynomials follow the distributive property, and the degree and leading term of a product are determined by the individual polynomials' degrees and leading terms.
This module discusses quadratic functions and their application to solving word problems. Students will learn to 1) recall steps to solve word problems, 2) translate problems to symbolic expressions, and 3) apply quadratic equations. Examples provided cover number, geometry, and motion problems. Students must first translate word problems into equations before solving. Key steps include identifying unknowns, writing the equation, solving, and checking solutions. Practice problems are provided to help students apply the concepts.
This PowerPoint covers about 90% of the material that will be on the Chapter 8 test. It includes steps to solve linear equations for y in terms of x, naming the slope (m) and y-intercept (b), identifying ordered pairs, determining if a relation is a function, writing rules for linear functions from tables of values, and finding the slope.
1. Indices involve rules for exponents like xa+b = xaxb and (xa)b = xab. Solving exponential equations uses these rules.
2. Graph transformations include translations, stretches, reflections, and asymptotes. Translations replace x with (x-a) and y with (y-b).
3. Sequences are functions with successive terms defined by a rule. Geometric sequences multiply successive terms by a constant ratio while arithmetic sequences add a constant.
This module introduces polynomial functions of degree greater than 2. It covers identifying polynomial functions from relations, determining the degree of a polynomial, finding quotients of polynomials using division algorithm and synthetic division, and applying the remainder and factor theorems. The document provides examples and practice problems for each topic. It aims to teach students how to work with higher degree polynomial functions.
1) The document discusses completing the square, which involves rewriting quadratic expressions in the form (x + a)2 + b to find maximum and minimum values.
2) Examples are provided of completing the square for expressions like x2 + 8x + 3 and 2x2 + 4x + 11.
3) The technique of setting the expression equal to 0 and solving for x is described as a way to find the minimum value and the corresponding x-value that produces it.
11 x1 t13 07 products of intercepts (2012)Nigel Simmons
This document discusses the product of intercepts theorem for intersecting chords and secants of a circle. It states that the product of the intercepts formed by two chords or secants on one side of their point of intersection is equal to the product of the intercepts on the other side (AX * BX = CX * DX). It also notes that for secants, the intercepts are formed by lines extending the secants outside the circle. Additionally, it presents the related theorem that the square of a tangent segment is equal to the product of its intercepts (AX^2 = CX * DX).
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
11 x1 t05 06 line through pt of intersection (2013)Nigel Simmons
The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
The document discusses the slope (gradient) of a line and how to calculate it. It provides four methods to calculate slope:
(1) The rise over the run between two points (vertical change over horizontal change)
(2) The change in y-values over the change in x-values between two points using a formula
(3) The slope of a line is equal to the tangent of the angle of inclination
(4) The relationship between slopes of parallel and perpendicular lines. Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is -1. An example problem demonstrates finding the value of a that results in two lines being parallel or perpendicular.
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses the oblique asymptote of a hyperbola. Additionally, it shows that the graph y=x is increasing for all x≥0. It proves that the sum of the first n positive integers is greater than or equal to the integral of x from 0 to n. Finally, it uses mathematical induction to show an inequality relating the sum of the first n positive integers to 4n+3/6.
The document discusses the formula for calculating the perpendicular distance from a point to a line. It states that the perpendicular distance is the shortest distance. The formula is given as d = (Ax1 + By1 + C)/√(A2 + B2), where (x1, y1) are the coordinates of the point and Ax + By + C = 0 is the equation of the line. An example is worked through to find the equation of a circle given its tangent line and center point. It also discusses how the sign of (Ax1 + By1 + C) indicates which side of the line a point lies.
All straight lines can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. Alternatively, lines can be written in the general form of Ax + By + C = 0, where A, B, and C are integers or surds. Lines parallel to the x-axis have the form y = c, where c is a constant. Lines parallel to the y-axis have the form x = k, where k is a constant. The example shows finding the equation of a line perpendicular to another line in general form.
The document proves that the alternate segment theorem, which states that any angle in an alternate segment of a circle is equal to an angle formed by the tangent and chord at the point of contact. It does this by joining a radius to the point of contact P and extending it to meet the circle at Z. It then uses properties of angles on tangents, in semicircles, and angle sums in triangles to show that ∠APY = ∠ABP.
The document discusses several theorems related to tangents of circles:
1) The angle between a tangent line and radius drawn to the point of contact is 90 degrees.
2) Equal tangents can be drawn from any external point to a circle, and the line joining the point to the center is an axis of symmetry.
3) If two circles share a common tangent, the centers and point of contact must be collinear.
The document defines various terms related to circle geometry, including radius, diameter, chord, secant, tangent, arc, sector, segment, and more. It also presents several theorems regarding chords and arcs, such as a perpendicular from the circle's center bisects a chord, equal chords subtend equal angles at the center, and more. Diagrams with proofs are provided.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
Slope is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. It is calculated by taking the change in the y-values (rise) and dividing by the change in the x-values (run). Examples are provided to demonstrate calculating the slope between two points and interpreting what different values mean in terms of the line's orientation.
The document discusses using the point-slope formula to find the equation of lines given certain conditions. It provides examples of finding the equation of a line passing through two points, the equation of a line parallel to another line passing through a point, and the equation of a line perpendicular to another line passing through a point. It also briefly mentions using lines to prove concurrency.
This document provides notes on functions and quadratic equations from Additional Mathematics Form 4. It includes:
1) Definitions of functions, including function notation f(x) and the relationship between objects and images.
2) Methods for solving quadratic equations, including factorisation, completing the square, and the quadratic formula.
3) Properties of quadratic functions like finding the maximum/minimum value and sketching the graph.
4) Solving simultaneous equations involving one linear and one non-linear equation through substitution.
5) Conversions between index and logarithmic forms and basic logarithm laws.
This document provides notes on key concepts in additional mathematics including:
1) Functions such as f(x) = x + 3 and finding the object and image of a function.
2) Solving quadratic equations using factorisation and the quadratic formula. Types of roots are discussed.
3) Sketching quadratic functions by finding the y-intercept, maximum/minimum values, and a third point. Quadratic inequalities are also covered.
4) Methods for solving simultaneous equations including substitution when one equation is nonlinear.
5) Properties of exponents and logarithms, and how to solve exponential and logarithmic equations.
This document contains a 5 page exam for the course CS-60: Foundation Course in Mathematics in Computing. The exam contains 17 multiple choice and numerical problems covering topics like algebra, calculus, matrices, and complex numbers. Students have 3 hours to complete the exam which is worth a total of 75 marks. Question 1 is compulsory, and students must attempt any 3 questions from questions 2 through 6. The use of a calculator is permitted.
The document provides information about linear equations and their graphs. It defines linear equations and discusses how to write equations in slope-intercept form, point-slope form, and standard form. It also describes how to graph linear equations by plotting intercepts and using slope. Key topics covered include finding the slope between two points, determining if lines are parallel or perpendicular based on their slopes, and recognizing the intercepts on a graph of a linear equation in two variables.
Algebra i ccp quarter 3 benchmark review 2013 (2)MsKendall
This document provides a review packet for Algebra 1 with questions on finding equations of lines in slope-intercept form, graphing linear equations, finding x- and y-intercepts, writing equations of lines given slope and a point, solving systems of equations by graphing and substitution, writing equations of parallel and perpendicular lines, and rewriting equations between different forms. There are over 50 short problems covering various topics involving linear equations and systems of linear equations.
The document contains solutions to problems involving finding equations of lines. It provides step-by-step workings showing how to:
1) Find the equation of a line parallel to another line and passing through a given point.
2) Find the equation of a line passing through a point and parallel to the line joining two other points.
3) Find the equation of a line passing through a point and perpendicular to another line.
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key point is that simultaneous equations can be solved when there are as many equations as unknown variables.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example problem demonstrating these steps, showing the elimination of variables through multiplication and addition of the equations until a single variable remains that can be solved for. The document notes that simultaneous equations with the same number of equations as variables can always be solved to find the values of the variables.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example problem demonstrating these steps, eliminating variables through multiplication and addition of the equations until a single variable remains that can be solved for. The document notes that simultaneous equations with the same number of variables and equations can always be solved using this method.
The document discusses different methods to find the equation of a straight line. It provides examples of finding the equation when given:
1) Two points that the line passes through.
2) A point and the slope of the line.
3) The x-intercept and y-intercept.
4) When the x-intercept and y-intercept are equal.
5) When given the y-intercept and slope.
Worked examples are provided for each method.
The document discusses how to identify the slope and y-intercept of a line given in standard form. It shows working through examples of changing lines from standard form (Ax + By = C) to slope-intercept form (y = mx + b). Through solving the equations for y, the slope (m) and y-intercept (b) can be determined. Graphing lines on a coordinate plane is also demonstrated.
The document discusses parallel and perpendicular lines, explaining that parallel lines have the same slope while perpendicular lines have slopes that are negative reciprocals. It provides examples of finding the slope of parallel and perpendicular lines and writing equations of lines given properties like a point and slope. Formulas for slope-intercept and point-slope forms of linear equations are also presented.
The document discusses key concepts about parallel and perpendicular lines including how to determine if two lines are parallel or perpendicular based on their slopes, and how to write equations of parallel and perpendicular lines. It provides examples of finding slopes of parallel and perpendicular lines and writing equations to represent lines given certain conditions. Formulas for slope-intercept and point-slope forms are also presented.
The document provides steps to determine the equation of a line given certain information:
1) If the slope is -5 and y-intercept is 8, the equation is y = -5x + 8.
2) If the slope is 4 and the line passes through point (-2, -3), the equation is y = 4x + 5.
3) If a line passes through (3, -5) and is parallel to y = -2x + 9, the equation is y = -2x + 1.
Soalan kuiz matematik tambahan ting empat 2006zabidah awang
This document contains 30 multiple choice mathematics questions related to quadratic equations, functions, and their inverses. The questions cover topics such as finding the inverse of a function, determining the roots of a quadratic equation, finding the range of values for variables in equations, and relating the roots and coefficients of related quadratic equations.
This document provides derivatives of various functions. There are 100 problems given without showing the step-by-step work. The solutions are provided directly without simplification.
The document provides steps for finding the gradient of a curve at a given point by differentiating and substituting the x-value. It also outlines the process of finding the equation of a tangent line to a curve at a point, which involves determining the gradient, substituting the x and y coordinates to find the value of c, and writing the equation in y=mx+c form. Examples are worked through applying these steps to find gradients, tangent lines, and solving related problems.
Similar to 11 x1 t05 04 point slope formula (2013) (20)
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
The document discusses nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. These solutions form a regular n-sided polygon with vertices on the unit circle when placed on an Argand diagram. As an example, it shows that the solutions to z5 = 1 are the fifth roots of unity located at angles that are integer multiples of 2π/5 around the unit circle. It then proves that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4 and ω5 are also roots. Finally, it proves that 1 + ω + ω2 + ω3 + ω4 = 0.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
🔥🔥🔥🔥🔥🔥🔥🔥🔥
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 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
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.
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
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.
3. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
4. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m
3 2
10
5
2
5. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
10
5
2
6. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
7. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
(ii) Find the equation of the line passing through (2,–3) and is parallel to
3x + 4y – 5 =0
8. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
(ii) Find the equation of the line passing through (2,–3) and is parallel to
3x + 4y – 5 =0
3 5
y x
4 4
9. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
(ii) Find the equation of the line passing through (2,–3) and is parallel to
3x + 4y – 5 =0
3 5
y x
4 4
3
required m
4
10. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
(ii) Find the equation of the line passing through (2,–3) and is parallel to
3x + 4y – 5 =0 3
y 3 x 2
3 5 4
y x
4 4
3
required m
4
11. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
(ii) Find the equation of the line passing through (2,–3) and is parallel to
3x + 4y – 5 =0 3
y 3 x 2
3 5 4
y x 4 y 12 3 x 6
4 4
3 3x 4 y 6 0
required m
4
12. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
(ii) Find the equation of the line passing through (2,–3) and is parallel to
3x + 4y – 5 =0 3 OR
y 3 x 2
3 5 4
y x 4 y 12 3 x 6
4 4
3 3x 4 y 6 0
required m
4
13. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
(ii) Find the equation of the line passing through (2,–3) and is parallel to
3x + 4y – 5 =0 3 OR 3x 4 y k 0
y 3 x 2
3 5 4
y x 4 y 12 3 x 6
4 4
3 3x 4 y 6 0
required m
4
14. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
(ii) Find the equation of the line passing through (2,–3) and is parallel to
3x + 4y – 5 =0 3 OR 3x 4 y k 0
y 3 x 2
3
y x
5 4 2, 3 : 3 2 4 3 k 0
4 4 4 y 12 3 x 6
3 3x 4 y 6 0
required m
4
15. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
(ii) Find the equation of the line passing through (2,–3) and is parallel to
3x + 4y – 5 =0 3 OR 3x 4 y k 0
y 3 x 2
3
y x
5 4 2, 3 : 3 2 4 3 k 0
4 y 12 3 x 6
4 4 6 k 0
3x 4 y 6 0
required m
3 k 6
4
16. Point Slope Formula
y y1 m x x1
e.g. (i) Find the equation of the line passing through (–3,4) and (2,–6)
46
m y 4 2 x 3
3 2
y 4 2 x 6
10
2x y 2 0
5
2
(ii) Find the equation of the line passing through (2,–3) and is parallel to
3x + 4y – 5 =0 3 OR 3x 4 y k 0
y 3 x 2
3
y x
5 4 2, 3 : 3 2 4 3 k 0
4 y 12 3 x 6
4 4 6 k 0
3x 4 y 6 0
required m
3 k 6
4 3x 4 y 6 0
17. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
18. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
9 6
y x
4 4
19. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
9 6
y x
4 4
4
required m
9
20. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
4
9 6 y 4 x 6
y x 9
4 4
4
required m
9
21. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
4
9 6 y 4 x 6
y x 9
4 4 9 y 36 4 x 24
4
required m 4 x 9 y 60 0
9
22. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
OR
4
9 6 y 4 x 6
y x 9
4 4 9 y 36 4 x 24
4
required m 4 x 9 y 60 0
9
23. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
OR 4x 9 y k 0
4
9 6 y 4 x 6
y x 9
4 4 9 y 36 4 x 24
4
required m 4 x 9 y 60 0
9
24. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
OR 4x 9 y k 0
4
9
y x
6 y 4 x 6 6, 4 : 4 6 9 4 k 0
9
4 4 9 y 36 4 x 24
4
required m 4 x 9 y 60 0
9
25. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
OR 4x 9 y k 0
4
9
y x
6 y 4 x 6 6, 4 : 4 6 9 4 k 0
9
4 4 9 y 36 4 x 24 60 k 0
4 k 60
required m 4 x 9 y 60 0
9
26. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
OR 4x 9 y k 0
4
9
y x
6 y 4 x 6 6, 4 : 4 6 9 4 k 0
9
4 4 9 y 36 4 x 24 60 k 0
4 k 60
required m 4 x 9 y 60 0
9 4 x 9 y 60 0
27. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
OR 4x 9 y k 0
4
9
y x
6 y 4 x 6 6, 4 : 4 6 9 4 k 0
9
4 4 9 y 36 4 x 24 60 k 0
4 k 60
required m 4 x 9 y 60 0
9 4 x 9 y 60 0
To prove three lines (l, m, n) are concurrent;
28. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
OR 4x 9 y k 0
4
9
y x
6 y 4 x 6 6, 4 : 4 6 9 4 k 0
9
4 4 9 y 36 4 x 24 60 k 0
4 k 60
required m 4 x 9 y 60 0
9 4 x 9 y 60 0
To prove three lines (l, m, n) are concurrent;
(i ) solve l and m simultaneously
29. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
OR 4x 9 y k 0
4
9
y x
6 y 4 x 6 6, 4 : 4 6 9 4 k 0
9
4 4 9 y 36 4 x 24 60 k 0
4 k 60
required m 4 x 9 y 60 0
9 4 x 9 y 60 0
To prove three lines (l, m, n) are concurrent;
(i ) solve l and m simultaneously
(ii ) substitute point of intersection into n
30. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
OR 4x 9 y k 0
4
9
y x
6 y 4 x 6 6, 4 : 4 6 9 4 k 0
9
4 4 9 y 36 4 x 24 60 k 0
4 k 60
required m 4 x 9 y 60 0
9 4 x 9 y 60 0
To prove three lines (l, m, n) are concurrent;
(i ) solve l and m simultaneously
(ii ) substitute point of intersection into n
(iii ) if it satisfies the equation, then the lines are concurrent
31. (ii) Find the equation of the line passing through (6,4) and is
perpendicular to 9x – 4y + 6 =0
OR 4x 9 y k 0
4
9
y x
6 y 4 x 6 6, 4 : 4 6 9 4 k 0
9
4 4 9 y 36 4 x 24 60 k 0
4 k 60
required m 4 x 9 y 60 0
9 4 x 9 y 60 0
To prove three lines (l, m, n) are concurrent;
(i ) solve l and m simultaneously
(ii ) substitute point of intersection into n
(iii ) if it satisfies the equation, then the lines are concurrent
Exercise 5D; 1e, 2c, 4abc (i), 5b, 7d, 9, 11, 13, 15, 17ab (i),
18ab (ii), 19, 22, 23c, 26*