2.8 absolute value functions
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1. The document discusses absolute value functions of the form y = a|x - h| + k, where a ≠ 0. 2. It explains how to find the vertex (h, k) and line of symmetry x = h of an absolute value function graph. 3. It also describes how the graph opens up or down depending on whether a is positive or negative, and how the width compares to y = |x| depending on the value of |a|.
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This document discusses quadratic functions in vertex form. It defines quadratic functions as having an x^2 term as the highest power of x. Vertex form is defined as y = a(x - h)^2 + k, where (h, k) is the vertex. The document shows how to graph functions in vertex form by sketching the parent graph y = x^2 and then performing transformations based on the a, h, and k values in the function. Several examples are worked through to demonstrate finding the vertex and axis of symmetry.
Graphing quadratic equations
1) This document discusses how to solve quadratic equations by graphing, including identifying the terms of a quadratic equation, finding the solutions by graphing, and graphing quadratic functions.
2) The key steps for graphing a quadratic function are to find the axis of symmetry using the standard form equation, find the vertex point, and find two other points to reflect across the axis of symmetry to complete the parabolic graph.
3) An example problem walks through graphing the quadratic equation y = x^2 - 4x by first finding the roots, vertex, and axis of symmetry, and then constructing a table to plot points and graph the parabola.
AA Section 6-3
The document provides instructions on graphing parabolas using vertex form and translations. It defines the vertex form of a parabola as y = a(x - h)2 + k, where (h, k) are the coordinates of the vertex. Examples show how to find the image of a parabola under a translation Th,k and graph parabolas by hand by determining the vertex and symmetrical y-values. Steps are given to graph a parabola as finding the vertex, symmetrical values, and filling in the graph.
G9_Q2_W2_L2_GraphsofQUadraticFunction.pdf
The document discusses key concepts about quadratic functions including:
1) Quadratic functions can be represented by the equation f(x) = ax^2 + bx + c and form a parabolic shape that can open up or down.
2) The axis of symmetry of a parabola passes through the vertex (highest/lowest point) and can be found using the equation x = -b/2a.
3) The vertex coordinates can be determined by substituting the axis of symmetry value into the original equation.
4) Quadratic functions can have 0, 1, or 2 x-intercepts depending on the number of real solutions to the related quadratic equation.
Quadraticfuntions
A quadratic function has the form f(x) = ax^2 + bx + c, where a != 0. The graph of a quadratic function is a parabola, which is symmetric around its axis of symmetry. Any quadratic function can be written in standard form by completing the square, where the standard form is f(x) = a(x-h)^2 + k. The vertex of the parabola is the point (h, k) and the line of symmetry is x = h.
Quadraticfuntions
A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a is not equal to zero. The graph of a quadratic function is a parabola, which is symmetric around its axis of symmetry. Any quadratic function can be written in standard form by completing the square, where the standard form is f(x) = a(x-h)^2 + k. The vertex of the parabola is the point (h, k) and finding the standard form makes it easy to identify the vertex.
6. 1 graphing quadratics
The document discusses graphing quadratic functions. It defines a quadratic function as an equation of the form y=ax^2 +bx+c where a ≠ 0, forming a U-shaped parabola. It explains that the vertex is the highest/lowest point and the axis of symmetry is the vertical line through the vertex. It provides steps for graphing a quadratic function by finding the vertex coordinates using the standard form equation, choosing x-values on each side of the vertex, calculating corresponding y-values, and plotting points connected with a smooth curve. An example problem demonstrates these steps.
Class 3.pdf
This document discusses solving simultaneous linear equations through algebraic and graphical methods. It covers:
- Algebraic methods like substitution and elimination to solve 2 equations with 2 unknowns.
- Graphing methods to find the point where two lines intersect, representing the solution.
- Conditions where equations have a unique solution, no solution, or infinitely many solutions.
- Extended examples are provided to demonstrate solving 2 and 3 equations with algebraic elimination.
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3.4 linear programming
The document discusses linear programming problems. It provides two examples of finding the minimum and maximum values of objective functions subject to certain constraints. In the first example, the objective is to minimize or maximize the function f(x,y)=3x-2y within the constraints 1≤x≤5, y≥2, and y≤x+3. In the second example, the objective is to minimize or maximize the function f(x,y)=4x+3y within the constraints y≥-x+2, y≤(1/4)x+2, and y≥2x-5. Both examples solve the problems by finding the vertices of the constrained regions and calculating the objective
3.3 a writing systems of linear inequalities
The document discusses how to graph and write systems of linear inequalities. It explains how to graph systems by determining whether lines should be dotted or solid based on the inequality symbols, and whether shading should be above or below lines based on 'y >' or 'y <'. It also explains how to write the system of inequalities given a graph by writing equations for each line and changing the equations to inequalities based on whether lines are dotted or solid and whether shading is above or below lines. Examples of these processes are provided.
3.3 a writing a systems of inequalities
This document discusses how to graph systems of linear inequalities by:
1. Writing each inequality in the system in slope-intercept form (y=mx+b)
2. Graphing the lines for each inequality, making dashed lines for > or < and solid for ≥ or ≤
3. Shading the correct region for each line (above for > and ≥, below for < and ≤)
4. The solution to the system is the overlapping region shaded by both inequalities.
3.3 a solving systems of inequalities
1. Systems of linear inequalities can be solved by graphing the inequalities on a coordinate plane.
2. Each inequality is written in slope-intercept form (y=mx+b) and graphed as either a solid or dotted line.
3. The region representing the solution of the system is found by shading the area of overlap between the graphs based on whether each inequality represents "above or below" the line.
3.3 solving systems of inequalities
The document discusses solving systems of linear inequalities by graphing the inequalities on a coordinate plane. It explains that linear inequalities should be written in slope-intercept form and the corresponding lines should be drawn as dotted or solid based on the inequality symbols. It also notes that the solution region includes points not on the lines and instructs the reader to shade above or below lines depending on the inequality. An example problem demonstrates these steps to find the region satisfying two inequalities.
3.2 a solving systems algebraically
Here are the steps to solve this system of linear equations using the elimination method:
1) Line up the equations:
-4x - 8y = 16
6x + 12y = -24
2) Add a multiple of one equation to the other to eliminate x:
-4x - 8y = 16
6x + 12y = -24
Add -1.5 times the top equation to the bottom:
-4x - 8y = 16
0 - 3y = -40
3) Solve for y:
-3y = -40
y = 13
4) Substitute y = 13 back into either original equation to find x:
-4x
3.2 solving systems algebraically
Here are the steps to solve this system of equations using elimination:
1) Line up the equations:
-4x - 8y = 16
6x + 12y = -24
2) Multiply the top equation by -3:
12x + 24y = -48
6x + 12y = -24
3) Add the equations to eliminate y:
18x = -72
4) Divide both sides by 18:
x = -4
5) Substitute x=-4 into either original equation to find y:
-4(-4) - 8y = 16
16 - 8y = 16
-8y = 0
y = 0
So
3.1 b solving systems graphically
This document provides instructions for students to solve systems of linear equations by graphing. Students are asked to write equations in slope-intercept form when possible, use slope and y-intercept to determine the number of solutions (one, many, or no solutions), and name each system based on its number of solutions. The document assigns textbook problems #35-52 for practice and is dated for Friday, October 7th.
3.1 a solving systems graphically
This document discusses how to graph and solve systems of linear equations in two variables by determining whether the system has one solution, many solutions, or no solution. It explains that a consistent system has a solution, an inconsistent system has no solution, an independent system has one solution, and a dependent system has infinitely many solutions. It also notes that a system will have one solution if the lines cross once, no solution if the lines are parallel, and infinitely many solutions if the lines are the same.
3.1 solving systems graphically
The document discusses solving systems of linear equations by graphing. It explains that a system of linear equations contains two or more linear equations using the same variables. To solve a linear system graphically, the two equations must be put into slope-intercept form and graphed on the same coordinate plane. The point where the lines intersect satisfies both equations and is the solution to the system. The solution must be verified by substituting the coordinates into the original equations.
3.4 a linear programming
The document discusses using linear programming to find the maximum and minimum values of an objective function given certain constraints. It provides an example of finding the minimum and maximum values of the function f(x,y)=3x-2y when constrained by y≥2, 1≤x≤5, and y≤x+3. The values are found by graphing the inequalities to determine the vertices and substituting the vertices into the objective function.
2.6 graphing linear inequalities
This document discusses graphing linear inequalities. It has two objectives: 1) determining if an ordered pair is a solution to an inequality, and 2) graphing linear inequalities in one and two variables.
2.5 a correlation & best fitting lines
The document discusses finding the line of best fit for a data set using linear regression. It explains that the line of best fit should have approximately the same number of points above and below the line. It also describes how the slope of the line indicates whether the correlation is positive, negative, or relatively no correlation. Finally, it introduces the correlation coefficient r as a measure of the strength of correlation, with values closer to 1 or -1 indicating a stronger correlation.
2.5 correlation & best fitting lines
This document discusses identifying correlation in data by plotting it on a scatter plot and fitting a best-fit line. It explains that a positive slope indicates positive correlation, a negative slope indicates negative correlation, and no consistent slope indicates little correlation. It also introduces the correlation coefficient r as a measure of correlation strength, with values closer to 1 or -1 indicating stronger correlation.
2.4 writing linear equations
This document discusses writing and working with linear equations in slope-intercept form (y=mx+b). It provides examples of:
1) Writing linear equations from the slope (m) and y-intercept (b)
2) Finding the slope and y-intercept of given linear equations
3) Writing linear equations given two points or given a slope and point.
2.4 writing equations of lines
1) This document discusses writing linear equations in slope-intercept form (y=mx+b) given different information: the slope (m) and y-intercept (b), a graph, the slope and a point, or two points.
2) It provides examples of writing equations of lines given: the slope and y-intercept, a graph, the slope and a point, two points, or the x- and y-intercepts.
3) The key steps are to identify the slope (m) and y-intercept (b) from the information provided, then substitute those values into the slope-intercept form equation to write the linear equation.
2.3 linear equations
The document discusses linear equations and how to graph them. It defines linear equations as having variables with exponents of 1 that are added or subtracted. It explains how to identify the slope and y-intercept of a linear equation in slope-intercept form (y=mx+b) in order to graph it as a line on the coordinate plane. Key steps include solving for y and identifying the slope (m) and y-intercept (b). Examples are provided to demonstrate finding intercepts and graphing various linear equations.
2.2 slope
1. The document discusses slope and rate of change, including how to calculate slope from two points on a line and how to interpret positive, negative, zero, and undefined slopes.
2. It provides examples of finding the slope of lines from graphs and points, and discusses how to identify parallel and perpendicular lines based on their slopes.
3. The key concepts are how to determine the steepness of a line from its slope, and that parallel lines have the same slope while perpendicular lines have slopes that are opposite reciprocals.
2.1 a relations and functions
1. The document discusses functions and their graphs.
2. It defines a vertical line test to determine if a graph represents a function and provides examples of graphs that both do and do not represent functions.
3. It introduces function notation using f(x) and provides examples of evaluating functions for given values of x.
2.1 relations and functions
This document defines key concepts related to functions and relations including:
- Relations are sets of ordered pairs with a domain and range
- A function is a special type of relation where each element in the domain is mapped to only one element in the range (no repeated x-values)
- Relations can be represented as ordered pairs, tables, graphs, or mappings
- The document provides examples of determining if a relation represents a function based on whether the domain elements are repeated
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2.8 absolute value functions
- 1. 2.8 Absolute Value Functions Today’s objective: 1. I will learn characteristics of absolute value functions. 2. I will graph absolute value functions. 3. I will write the equation for an absolute value function.
- 2. 2.8 Absolute Value Functions y = a│x – h │ + k, a≠0 The graph is shaped like a v.
- 3. Find the vertex Vertex: (h, k) h is always the opposite of the # in the absolute value bars k is always the same as in the equation
- 4. Line of symmetry x=h Shown with a dashed vertical line.
- 5. Graph opens up or down If a > 0: the graph opens up. the vertex (h, k) is the minimum. If a < 0: the graph opens down. the vertex (h, k) is the maximum.
- 6. Is the graph wider, narrower, or the same width as y = │x│. Graph is narrower if │a │> 1. Graph is wider if 0 < │a │< 1. Graph is the same width if │a │ = 1.
- 7. Example: y = 3│ x + 2│ – 5 The vertex is ( -2, -5), because the opposite of 2 is -2, and k is – 5. The line of symmetry is x = -2 The graph opens up because a > 0. The graph is narrower because│a│= 3 The slope is 3, so start at the vertex and go up 3 and to the right 1. Go back to the vertex. This time go up 3 and to the left 1.
- 8. Writing the equation for an Absolute Value Function 1. Find the vertex (h,k) 2. Substitute this into the general form: y = a│x – h │ + k 3. Find another point on the graph (x,y) and substitute these values into the general form. 4. Solve for a. 5. Write your equation. This time only substitute the values of a, h, and k.
- 9. Write the equation for this graph. 1. Vertex: (-2,0) 2. Find another point (0,2) 3. Substitute these into the equation to find a. 2 = a│0 – (-2)│+ 0 2 = a │2│ 2 = 2a a=1 4. So the equation is: y = 1│x + 2│ y =│x + 2│