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7 , 3 7 x 4 10 x 3 7 . 1 7 , 2 x 8 9 x 2 3 . 2 2 , 8 x 9 5 x 6 . 3 Find the solutions for each absolute value equation:

Find the solutions for each absolute value equation:

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Page 1: Find the solutions for each absolute value equation:

7,37

x 410x37.1

7,2x 89x23.2

2,8x 95x6.3

Find the solutions for each absolute value equation:

Page 2: Find the solutions for each absolute value equation:

Math 8H

Graphing Absolute Value Equations

Algebra 1 Glencoe McGraw-Hill JoAnn Evans

Page 3: Find the solutions for each absolute value equation:

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

Page 4: Find the solutions for each absolute value equation:

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.

Page 5: Find the solutions for each absolute value equation:

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|?

Page 6: Find the solutions for each absolute value equation:

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|?

Page 7: Find the solutions for each absolute value equation:

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|?

Page 8: Find the solutions for each absolute value equation:

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.

Page 9: Find the solutions for each absolute value equation:

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.

Page 10: Find the solutions for each absolute value equation:

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?

Page 11: Find the solutions for each absolute value equation:

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.

Page 12: Find the solutions for each absolute value equation:

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.

Page 13: Find the solutions for each absolute value equation:

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

Page 14: Find the solutions for each absolute value equation:

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)