Cellini, EDLM, Math, Absolute Value

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