find the solutions for each absolute value equation:
TRANSCRIPT
7,37
x 410x37.1
7,2x 89x23.2
2,8x 95x6.3
Find the solutions for each absolute value equation:
Math 8H
Graphing Absolute Value Equations
Algebra 1 Glencoe McGraw-Hill JoAnn Evans
The ABSOLUTE VALUE of a real number is the distance between the origin and
the point representing the real number.
33
00
55
The number 5 is five spaces from 0, the origin.
0 is zero spaces from itself.
The number -3 is three spaces from 0, the origin.
Distance is not negative; the absolute value of a number will never be
negative.
|x| = x when x > 0
|x| = -x when x < 0
|0| = 0
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.
Graph y = -|x| Graph y = |x - 2|
x y
vertexvertex
-2-1 0-1-2
-2-1 0 1 2
x y 42024
-2 0 2 4 6
How does this graph differ from
y = |x|?
How does this graph differ from
y = |x|?
Graph y = |x| + 1 Graph y = |x| - 3
x y
vertexvertex
32123
-2-1 0 1 2
x y -1-2-3-2-1
-2-1 0 1 2
How does this graph differ from
y = |x|?
How does this graph differ from
y = |x|?
Graph y = |x + 2| Graph y = |x - 1|
x y
vertexvertex
21012
-4-3-2-1 0
x y 21012
-1 0 1 2 3
How does this graph differ from
y = |x|?
How does this graph differ from
y = |x|?
It’s possible to tell what the x value of the vertex will be just by looking at the
absolute value equation.
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.
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.
To Sketch the Graph of an Absolute Value Equation:
1. 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.
2. Choose two values of x less than this number and two values of x greater than this number.
3. 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.
Sketch the graph of y = |x + 2| - 3
-2
-4
-3
-2
-1
0
yx-1
-2
-2
-3
-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.
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
1
When there’s a negative coefficient before the absolute value symbol, the graph will open down.
What value of x will make the
expression inside the absolute
value sign equal to 0?
-1
0
1
2
3
yx-2
0
0
2
-2
Place 1 in the middle of the
table. Choose 2 values less and 2
values more, evenly spaced.
Sketch the graph of 32x21
y
When there’s a positive coefficient before the absolute value symbol, the graph will
open up.
-2
What value of x will make the
expression inside the absolute
value sign equal to 0?
-4
-3
-2
-1
0
yx-2
-2.5
-2.5
-3
-2
Place -2 in the middle of the
table. Choose 2 values less and 2
values more, evenly spaced.
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?If x is 0, what is y?
0 1
y = |x - 2| - 5 What will the x value of the vertex be?If x is 2, what is y?
2-5
y = |x + 3| - 4 What will the x value of the vertex be?If x is -3, what is y? -4
-3
y = 2|x - 1| + 7 What will the x value of the vertex be?If x is 1, what is y? 7
1
What will be the coordinates of the vertex?
y = |x| + 3y = |x + 8|y = |x| - 5
y = |x + 9| - 14
y = -5|x + 2|
y = 2|2x – 4| + 6
y = -|x – 1| + 5
(0, 3)
(-8, 0)
(0, -5)
(-9, -14)
(-2, 0)
(2, 6)
(1, 5)