Shifts or Translations
A vertical shift raises or lowers the entire graph the same value.
• f(x) + c represents a vertical shift
– The graph of f(x) shifts up c units if c > 0.
– The graph of f(x) shifts down |c| units if c < 0.
∙ Why?
Examples
1. f ( x)
2.
x
g ( x)
x
2
h( x )
x 5
f ( x)
1
x2
g ( x)
1
x2
2
h( x )
1
x2
5
A horizontal shift moves the entire graph left or right the same value.
• f(x – c) represents a horizontal shift.
– The operation is subtraction.
– The graph of f(x) shifts right c units if c > 0.
– The graph of f(x) shifts left |c| units if c < 0.
• Why?
Examples
1. f ( x) x 3
g ( x) ( x 2)3
h( x) ( x 5)3
2.
f ( x) | x |
g ( x) | x 2 |
h( x ) | x 5 |
Reflections
A vertical reflection flips the graph over the x-axis.
∙ -f(x) represents a vertical reflection.
– The portion of the graph of f(x) that was above the x-axis will now be below the x-axis.
– The portion of the graph of f(x) that was below the x-axis will now be above the x-axis.
• Why?
Examples
1.
2.
f ( x)
1
x
g ( x)
1
x
f ( x)
x
g ( x)
x
A horizontal reflection flips the graph over the y-axis.
∙ f(-x) represents a vertical reflection.
– The portion of the graph of f(x) that was right of the y-axis will now be left of the y-axis.
– The portion of the graph of f(x) that was left of the y-axis will now be right of the y-axis.
• Why?
• It doesn’t change the graphs of even functions.
• It is only needed for the even indexed radical and greatest integer functions for now.
Examples
1. f ( x)
x
g ( x)
2.
f ( x)
g ( x)
x
x
x
Dilations
Let c > 0.
A vertical dilation moves the points of the graph away or towards the x-axis by a constant factor.
• cf(x) represents a vertical dilation.
– All points of the graph of f(x) are c times farther from the x-axis if c > 1, called a stretch.
– All points of the graph of f(x) are 1/c times closer to the x-axis if c < 1, called a
compression.
• Why?
Examples
1. f ( x)
2.
3
x
g ( x)
43 x
h( x )
13
x
2
f ( x)
x
g ( x) .5 x
h( x )
4 x
A horizontal dilation moves the points of the graph away or towards the y-axis by a constant
factor.
• f(cx) represents a vertical dilation.
– All points of the graph of f(x) are 1/c times closer to the y-axis if c > 1, called a
compression.
– All points of the graph of f(x) are c times farther from the y-axis if c < 1, called a stretch.
• Why?
• It is only needed for the greatest integer function for now.
Example
f ( x) x
g ( x)
.5 x
h( x )
4x
In general, af(b(x – c)) + d
a. Vertical dilation and possibly reflection.
b. Horizontal dilation and possibly reflection.
c. Horizontal shift.
d. Vertical shift.
To graph af(b(x – c)) + d,
1. Identify the function f(x).
2. Start with enough points to graph f(x) (at least 3).
3. Multiply all x-coordinates by 1/b.
4. Add c to the new x-coordinates.
5. Multiply all y-coordinates by a.
6. Add d to the new y-coordinates.
– If given af(bx – c) + d, you must factor out b from bx – c yielding af(b(x – c/b)) + d.
Examples
1.
f ( x) 3
2x 5 1
2.
f ( x)
3
1
x 7
2
1