absolute value functions and graphs
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Absolute Value Functions and Graphs. Lesson 2-5. Important Terms. - PowerPoint PPT PresentationTRANSCRIPT
Absolute Value Functions and
Graphs
Lesson 2-5
Important Terms• Parent function: the simplest function with these
characteristics. The equations of the function in a family resemble each other, and so do the graphs. Offspring of parent functions include translations, stretches, and shrinks.
• Translation: it shifts a graph horizontally, vertically, or both. It results in a graph of the same shape and size but possibly in a different position
• Stretch: a vertical stretch multiplies all y-values by the same factor greater than 1, thereby stretching the graph vertically
• Shrink: a vertical shrink reduces y-values by a factor between 0 and 1, thereby compressing the graph vertically
• Reflection: in the x-axis changes y-values to their opposites. When you change the y-value of a graph to their opposites, the graph reflects across the x-axis (creates a mirror image)
The Family of Absolute Value FunctionsVertical Translation
Parent function Y=|x| Y=f(x)
Translation up k units, k>0 Y=|x|+k Y=f(x)+k
Translation down k units, k<0 Y=|x|-k Y=f(x)-k
Horizontal Translation
Parent Function Y=|x| Y=f(x)
Translation right h units, h>0 Y=|x-h| Y=f(x-h)
Translation left h units, h<0 Y=|x+h| Y=f(x+k)
Combined Translation
(right h units, up k units) Y=|x-h|+k Y=f(x-h)+k
Families of Functions: Absolute Value Functions
Vertical Stretch or Shrink, and Reflection in x-axisParent function Y=|x| Y=f(x)
Reflection in x-axis Y=-|x| Y= -f(x)
Stretch (a>1) Y=a|x| Y=af(x)
Shrink (0<a<1)
Reflection in x-axis Y=-a|x| Y=-af(x)
Combined Translation
Y=a|x-h|+k Y=af(x-h)+k
Absolute Value
An Absolute Value graph is always in a “V” shape.
xy
Given the following function,
If: a > 0, then shift the graph “a” units up
If: a < 0, then shift the graph “a” units down
xy a
Given the following function,
Since a > 0, then shift the
graph “3” units up
3xy
Let’s Graph
3xy
5xy
How will the graph look?
Let’s Graph
5xy
2xy
How will the graph look?
Let’s Graph
2xy
4xy
How will the graph look?
Let’s Graph
4xy
Given the following function,
We get the expression (x - b) and equal it to zero
x - b = 0x = b
If: b > 0, then shift the graph “b” units to the right
If: b < 0, then shift the graph “b” units to the left
x by
Given the following function,
x – 1 = 0
x = 1
Since 1 > 0, then shift
the graph “1” unit right
1xy
Let’s Graph
1xy
6xy
How will the graph look?
Let’s Graph
6xy
3xy
How will the graph look?
Let’s Graph
3xy
7xy
How will the graph look?
Let’s Graph
7xy
Graphing
1 3xy
Recall: Shift “3” units up since 3 > 0then we use the expression x + 1,
and equal it to zerox + 1 = 0
x = -1Since –1 < 0, then we shift
“1” unit to the left
Let’s Graph
1 3xy
3 2xy
How will the graph look?
Let’s Graph
3 2xy
2 4xy
How will the graph look?
Let’s Graph
2 4xy
5 1xy
How will the graph look?
Let’s Graph
5 1xy
Given the following function,
For this equation, c determines
how wide or thin it will be.if: |c|>1, then the graph is closer to the y-axis
if: |c|=1, then the graph remains the same
if: 0<|c|<1, then the graph is further
from the y-axis
if c is a negative number, then the graph
will reflect on the x-axis
xy c
Given the following function,
Since |5| > 0, then the
graph is closer to the y-axis
5 xy
Let’s Graph
5 x
xy
y
4 xy
How will the graph look?
Let’s Graph
4 x
xy
y
1
2xy
How will the graph look?
Let’s Graph
1
2x
xy
y
5
4xy
How will the graph look?
Let’s Graph
5
4x
xy
y
2
3xy
How will the graph look?
Let’s Graph
2
3
x
x
x
y
y
y
Given the following function,
Since 4 > 0, shift the graph “4” units up
x – 1 = 0
x = 1
Since 1 > 0, then shift the graph
“1” unit to the right
Since |5| > 0 shift the graph
closer to the y-axis.
1 45 xy
Let’s Graph
15 4xy
53 2xy
How will the graph look?
Let’s Graph
53 2xy
42 3xy
How will the graph look?
Let’s Graph
42 3xy
31
62xy
How will the graph look?
Let’s Graph
31
62xy
45
24xy
How will the graph look?
Let’s Graph
45
24xy
29
44xy
How will the graph look?
Let’s Graph
29
44xy
52
33xy
How will the graph look?
Let’s Graph
52
33xy
14
53xy
How will the graph look?
Let’s Graph
14
53xy
Congratulations!!You just completed the
transformation of
y x