1. Warm Up
Find the solutions for each absolute value equation:

2. Graphing Absolute Value Equations
Math 8H
Graphing Absolute Value Equations
Algebra Glencoe McGraw-Hill JoAnn Evans

3. The ABSOLUTE VALUE of a real number is the distance between the origin and the point representing the real number.
The number 5 is five spaces from 0, the origin.
|x| = x when x > 0
0 is zero spaces from itself.
|0| = 0
The number -3 is three spaces from 0, the origin.
|x| = -x when x < 0
Distance is not negative; the absolute value of a number will never be negative.

4. Graph the equation: y = |x|
x y
Every absolute value equation will graph into a v-shape. The VERTEX is the point of the v-shaped graph. Some will open up, others will open down.

5. How does this graph differ from y = |x|?
Graph y = -|x|
Graph y = |x - 2|
How does this graph differ from y = |x|?
How does this graph differ from y = |x|?
x y
x y -2 -1 1 2 -2 -1 -2 2 4 6 4 2
vertex
vertex

6. How does this graph differ from y = |x|?
Graph y = |x| + 1
Graph y = |x| - 3
How does this graph differ from y = |x|?
How does this graph differ from y = |x|?
x y
x y -2 -1 1 2 3 2 1 -2 -1 1 2 -1 -2 -3
vertex
vertex

7. How does this graph differ from y = |x|?
Graph y = |x + 2|
Graph y = |x - 1|
How does this graph differ from y = |x|?
How does this graph differ from y = |x|?
x y
x y -4 -3 -2 -1 2 1 -1 1 2 3 2 1
vertex
vertex

8. Why is this useful information
It’s possible to tell what the x value of the vertex will be just by looking at the absolute value equation.
The value of x that will make the expression INSIDE the absolute value symbol equal to ZERO will be the x-value of the vertex of the graph.
Why is this useful information
Knowing the x-value of the vertex will help you to efficiently select x-values for the table of values. You need several values on either side of the vertex in order to see the v-shape appear.

9. To Sketch the Graph of an Absolute Value Equation:
Find the value of x that will make the expression inside the absolute value symbol equal to zero. Place this value of x in the middle of your table of values.
Choose two values of x less than this number and two values of x greater than this number.
Calculate the corresponding y values and sketch the resulting v-shaped graph. If the x values are evenly spaced on either side of the x value of the vertex, the y values should show a pattern.

10. -2 Sketch the graph of y = |x + 2| - 3
What value of x will make the expression inside the absolute value sign equal to 0? -2 x y -4 -3 -2 -1 -1
-2 is the x value of the vertex. Place it in the middle of the table. Choose 2 values less and 2 values more, evenly spacing them. -2 -3 -2 -1

11. What value of x will make the expression inside the absolute value sign equal to 0?
Sketch the graph of y = -2|x - 1| + 2 x y 1 -1 1 2 3 -2
Place 1 in the middle of the table. Choose 2 values less and 2 values more, evenly spaced. 2 -2
When there’s a negative coefficient before the absolute value symbol, the graph will open down.

12. What value of x will make the expression inside the absolute value sign equal to 0?
Sketch the graph of x y -2 -4 -3 -2 -1 -2
Place -2 in the middle of the table. Choose 2 values less and 2 values more, evenly spaced.
-2.5 -3
-2.5 -2
When there’s a positive coefficient before the absolute value symbol, the graph will open up.

13. Is there a way to easily tell what the y value of the vertex will be?
y = |x| + 1
What will the x value of the vertex be? 1
If x is 0, what is y?
y = |x - 2| - 5
What will the x value of the vertex be? 2 -5
If x is 2, what is y?
y = |x + 3| - 4 -3
What will the x value of the vertex be? -4
If x is -3, what is y? 1
y = 2|x - 1| + 7
What will the x value of the vertex be? 7
If x is 1, what is y?

14. What will be the coordinates of the vertex?
(0, 3)
(-8, 0)
(0, -5)
(-9, -14)
(-2, 0)
(2, 6)
(1, 5)
y = |x| + 3
y = |x + 8|
y = |x| - 5
y = |x + 9| - 14
y = -5|x + 2|
y = 2|2x – 4| + 6
y = -|x – 1| + 5