6-5B Graphing Absolute Value Equations Algebra 1 Glencoe McGraw-HillLinda Stamper

6-5B Graphing Absolute Value Equations

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Algebra 1 Glencoe McGraw-Hill Linda Stamper

Graphs of Absolute Value Equations

An absolute Value equation is

.cbxay

Every absolute value equation has a V-shaped graph.

x

y

The V-shape opens up if the value of a is positive.

x

y

The V-shape opens down if the value of a is negative.

Determine whether the graph opens up or down.

1x2y

up

4x23

y

down

42xy

down

What is the value of

“a”?

On a graph that opens up, the vertex is the lowest point.On a graph that opens down, the vertex is the highest point.The vertical line passing through the vertex that divides the graph into two symmetric parts is called the line of symmetry.

x

y

x

y

axis of symmetry

line of symmetry

•

•

More Absolute Value Graphs

The vertex is the lowest or highest point on the graph.

x

y

x

y

•

•

axis of symmetry

line of symmetry

Graphing an Absolute Value Equation

1. Find the x-coordinate of the vertex, by finding the value of x for which x + b = 0

2. Make a table of values. Using x-values, calculate at least two values to the left and two values to the right of the vertex.3. Plot the points given in the table and draw a V-shaped graph (opening up or down) through the points.

cbxay

Note: If all of your values are on one side of the vertex, you will graph a line.

24x21

y

Find the coordinates of the vertex of the graph.

The absolute value equations is

Set the expression inside the absolute value bars equal to zero and solve for x.

04x

4 x

The y coordinate of the vertex is the value given for c.

Answer is an ordered pair.

2

cbxay

,4

4 4

25x32

y

Find the coordinates of the vertex of the graph.

05x

5x

2,5

5 5

Example 1

Example 2 63x4y

03x

3x 6,3

3 3

Graphing an Absolute Value Equation

1. Find the x-coordinate of the vertex, by finding the value of x for which x + b = 0

2. Make a table of values. Using x-values, calculate at least two values to the left and two values to the right of the vertex.3. Plot the points given in the table and draw a V-shaped graph (opening up or down) through the points.

cbxay

Note: If all of your values are on one side of the vertex, you will graph a line.

Sketch the graph of

Make a table of values. y,x

4 2c 2,4

2

0

6

8

x

vertex

24x21

y

24x21

y

To avoid graphing fractions,

choose values that will create

absolute values divisible

by two.

Find the vertex. (-4,-2)

24621

y

2221

y

2221

y

21y

1y 1,6

0,8

1,2

0,0

Sketch the graph of


4 2c 2,4

2

0

6

8

x

vertex

24x21

y

24x21

y


1,6

0,8

1,2

0,0

The “y” values will matchy,matchy!

Sketch the graph of


4 2c 2,4

3

2

5

6

x

vertex

24x2y

24x2y

If “a” is a whole

number, choose

values in numerical

order!


Sketch the graph for each of the following.

xy Example 3 Example 4

xy

Example 5

2xy

Will the graph open

up or down?

Example 3 Sketch the graph of

.xy Find the x-coordinate of the vertex. 0x

What is the value for “c”?What is the ordered pair for the vertex?

0,0


Reminder: When evaluating the

absolute value expression, the

amount will always be positive. Why?


.xy

Make a table of values.

xy y,x

22

2

y

2,21

1

1

y

1,1

0 0c 0,0

1

1

1

y

1,12

2

2

y

2,2

x

vertex


.xy


xy y,x

22

2

y

2,21

1

1

y

1,1

0 0c 0,0

1

1

1

y

1,12

2

2

y

2,2

x

vertex

matchy,

matchy!


.xy


xy y,x

22

2

y

2,21

1

1

y

1,1

0 0c 0,0

1

1

1

y

1,12

2

2

y

2,2

x

x

y

•••••

When constructing your ray, it may be helpful to begin at the vertex.

vertex


xy Example 3 Example 4

xy

Example 5

2xy

Will the graph open

up or down?

Example 4 Sketch the graph of .xy Find the x-coordinate of the vertex. 0x Set the expression inside the absolute value bars equal to zero and solve for x.

Since the negative sign is outside the

absolute value bars, the value of y can

be negative.

Example 4 Sketch the graph of .xy

Make a table of values. xy y,x

0 0c 0,0

1

1

1

y

1,12

2

2

y

2,2

11

1

y

1,1

22

2

y 2,2

x

vertex



0 0c 0,0

1

1

1

y

1,12

2

2

y

2,2

11

1

y

1,1

22

2

y 2,2

x

vertex

matchy,

matchy!



0 0c 0,0

1

1

1

y

1,12

2

2

y

2,2

11

1

y

1,1

22

2

y 2,2

x

x

y

•••••

How does this graph compare to the first graph?

vertex

Will the graph open up or down?

Example 5 Sketch the graph of .2xy

Find the x-coordinate of the vertex.

2x

22

02x


Example 5 Sketch the graph of .2xy

Make a table of values. 2xy y,x

2 0c 0,2

3 1

23

y

1,34

2

24

y

2,4

1 1

21

y

1 ,1

0 2

20

y

2,0

x

x

y

•••••

The vertex shifted 2 spaces to the right (2,0).


vertex

matchy,

matchy!


4xy Example 6 Example 7

4xy

Example 8

x3y

Example 9

32x21

y

Example 10

21x2y

Example 6

4xy

4xy y,x

4 0c 0,4

3 1

43y

1,32

2

42y

2,2

5 1

45y

1,5

6 2

46y

2,6

x

x

y

•••••

The vertex shifted 4 spaces to the left (–4,0).


vertex

Example 7

4xy

4xy y,x

0 4c 4,0

1 5

41

y

5,1

2 6

42

y

6,2

1 5

41

y

5,1

2 6

42

y

6,2

x

x

y

•••••

The vertex shifted 4 spaces upward (0,4).


vertex

3 is a multiplier of the absolute

value expression.

Example 8

x3y

x3y y,x

0 0c 0,0

1 3

13

y

3,12

6

23

y

6,2

1 3

13

y

3,1

2 6

23

y

6,2

x

x

y

•••

••

The value of “a” made the V narrower.


vertex

Choose values for x

that will result in

whole number values for y.

Example 9

32x21

y

3221 xy y,x

2 3c 3,2

0

2

31

3221

32021

y

2,0

2

1

32

3421

32221

y

1,2

x

32421 y4 2,4

32621 y6 1,6

x

y

• • • • •

Example 10

21x2y

212 xy y,x

1 2c 2,1

2

0

22

212

2122

y

0,2

3

2

24

222

2132

y

2,3

0

0

212

2102

y

0,0

1

2

24

222

2112

y

2,1

x

x

y

•••••

6-A11 Pages 325-327 # 27-30, 49-51 and Handout A–11.

Documents

6-5B Graphing Absolute Value Equations Algebra 1 Glencoe McGraw-HillLinda Stamper