7.2 abs value function
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This document provides notes for a math chapter on absolute value and reciprocal functions. It covers graphing and expressing absolute value functions as piecewise functions. Key points include: absolute value is defined as the distance from zero on the number line; absolute value functions are continuous and have the same x-intercept as the linear function inside the absolute value; taking the absolute value of a quadratic function preserves the vertex but changes the range. Care must be taken with the domains of absolute value functions depending on if the coefficient of the x-term inside is positive or negative.
7.2 abs value function
- 1. Math 20-1 Chapter 7 Absolute Value and Reciprocal Functions Teacher Notes 7.2 Absolute Value Function
- 2. 7.2 Graph an Absolute Value Function Absolute value is defined as the distance from zero in the number line. Absolute value of -6 is 6, and absolute value of 6 is 6, Both are 6 units from zero in the number line. Piecewise Definition x if x 0 if 6 then 6 x x if x 0 if –6 then – 6 7.2.1
- 3. Express the distance between the two points, -6 and 6, using absolute value in two ways. 6 6 6 6
- 4. Graph an Absolute Value Function y = | 2x – 4| continuous Method 1: Sketch Using a Table of Values x y = | 2x – 4| y = | 2x – 4| –2 |2(-2) - 4| 8 2x 4 2x 4 0 |2(0) – 4| 4 2 |2(2) - 4| 0 4 |2(4) – 4| 4 6 |2(6) – 4| 8 Domain :{x | x R} The x-intercept occurs at the point (2, 0) Range :{y | y 0, y R} The y-intercept occurs at the point (0, 4) The x-intercept of the linear function is the x-intercept of the corresponding absolute value function. This point may be called an invariant point. 7.2.2
- 5. Graph an Absolute Value Function y = | 2x – 4| Method 2: Using the Graph of the Linear Function y = 2x - 4 1. Graph y = 2x - 4 Slope = 2 Y-intercept = -4 x-intercept = 2 2. Reflect in the x-axis the part of the graph of y = 2x – 4 that is below the x-axis. 3. Final graph y = |2x – 4| 7.2.3
- 6. Express as a piecewise function. y = |2x – 4| Recall the basic definition of absolute value. The expression in the abs is a line x if x 0 with a slope of +1 and x-intercept of 0. x –x if x 0 function pieces −(x) x 0 domain x > 0 domain x < 0 has negative y-values, It must be reflected in the x-axis, multiply by -1 Note that 0 is the invariant point and can be determined by making the expression contained within the absolute value symbols equal to 0. 7.2.4
- 7. Express as a piecewise function. y = |2x – 4| Invariant point: =0 expression expression x- intercept 2x – 4 = 0 Slope positive x=2 function pieces |2x - 4| −(2x - 4) (2x - 4) 2x - 4 – 2 + expression Domain x < 2 Domain x > 2 As a piecewise function y = |2x – 4| would be 2x 4 if x 2 2x 4 (2x 4) if x 2 7.2.5
- 8. Be Careful Graph the absolute value function y = |-2x + 3| and express it as a piecewise function Slope negative x-intercept 2x 3 0 3 x 2 + (-2x+3) – (-2x+3) Domain :{x | x R} 3 + - Range :{y | y 0, y R} Domain x < 3/2 2 x > 3/2 3 2x 3 if x Piecewise function: 2 2x 3 3 ( 2 x 3) if x 2 7.2.6
- 9. True or False: The domain of the function y = x + 2 is always the same as the domain of the function y = |x + 2|. True The range of the function y = x + 2 is always the same as the range of the function y = |x + 2|. False Suggested Questions: Page 375: 1a, 2, 3, 5a,b, 6a,c,e, 9a,b, 12, 16, 7.2.7
- 10. Part B: Abs of Quadratic Functions Create a Table of Values to Compare y = f(x) to y = | f(x) | x y = x2 - 4 y = |x2 - 4| –3 5 5 -2 0 0 0 -4 4 x 2 2 x 2 x 2 2 0 0 3 5 5 Compare and contrast the domain and range of the function and the absolute value of the function. What is the effect of the vertex of the original function when the absolute value is taken on the function? 7.2.8
- 11. Graph an Absolute Value Function of the Form f(x) = |ax2 + bx + c| Sketch the graph of the function f ( x ) x 2 3x 4 Express the function as a piecewise function 1. Graph y = x2 – 3x - 4 vertex y x 4 x 1 (1.5, -6.25) x-int ( x 4)( x 1) 0 x 4 or x 1 2. Reflect in the x-axis the part of the graph of Domain :{x | x R} y = x2 – 3x - 4 that is below Range :{y | y 0, y R} the x-axis. How does this compare 3. Final graph y = |x2 – 3x - 4 | to the original function? 7.2.9
- 12. Express as a piecewise function. f ( x ) x 2 3x 4 expression = 0 x 2 3x 4 0 Critical points ( x 4)( x 1) 0 x-intercepts x 4 or x 1 a > 0, opens up x 2 3x 4 x 2 3x 4 ( x 2 3x 4) x 2 3x 4 + -1 – 4 + x2 - 3x - 4 expression x < -1 x>4 -1 < x < 4 x 2 3x 4 if -1 x 4 Piecewise function: x 2 3x 4 ( x 2 3x 4) if -1<x 4 7.2.10
- 13. Be Careful with Domain Graph the absolute value function y = |-x2 + 2x + 8| and express it as a piecewise function expression = 0 vertex -x 2 2 x 8 0 Domain (1, 9) Critical points x 2 2x 8 0 ( x 4)( x 2) 0 x-intercepts x 4 or x 2 a < 0, opens down x2 2x 8 x2 2x 8 x2 2x 8 x < -2 -2 4 -2 < x < 4 x>4 2 x2 2x 8 if -2 x 4 Piecewise function: x 2x 8 ( x2 2 x 8) if -2>x 4 7.2.11
- 14. Absolute Value as a Piecewise Function Match the piecewise definition with the graph of an absolute value function. 7.2.12
- 15. Suggested Questions: Page 375: 4, 7b, 8b,c,d, 10a,c, 11b,d, 13, 15, 20, 7.2.12
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