6-5b graphing absolute value equations algebra 1 glencoe mcgraw-hilllinda stamper
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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
Make a table of values. y,x
4 2c 2,4
2
0
6
8
x
vertex
24x21
y
24x21
y
Find the vertex. (-4,-2)
1,6
0,8
1,2
0,0
The “y” values will matchy,matchy!
Sketch the graph of
Make a table of values. y,x
4 2c 2,4
3
2
5
6
x
vertex
24x2y
24x2y
If “a” is a whole
number, choose
values in numerical
order!
Find the vertex. (-4,-2)
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
Set the expression inside the absolute value bars equal to zero and solve for x.
Reminder: When evaluating the
absolute value expression, the
amount will always be positive. Why?
Example 3 Sketch the graph of
.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
Example 3 Sketch the graph of
.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
matchy,
matchy!
Example 3 Sketch the graph of
.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
x
y
•••••
When constructing your ray, it may be helpful to begin at the vertex.
vertex
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 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
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
matchy,
matchy!
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
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
Set the expression inside the absolute value bars equal to zero and solve for x.
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).
How does this graph compare to the first graph?
vertex
matchy,
matchy!
Sketch the graph for each of the following.
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).
How does this graph compare to the first graph?
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).
How does this graph compare to the first graph?
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.
How does this graph compare to the first graph?
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.