2.2 F15 O’Brien
CA 6th ed HLR
2.2: Vertical and Horizontal Shifts of Graphs
I.
Review of Function Library
A.
Identity Function
f(x) = x
 ,  
 ,  
x
y
-2
-2
-1
-1
0
0
1
1
2
2
x
y
-2
4
-1
1
0
0
1
1
2
4
x
y
-2
-8
-1
-1
0
0
1
1
2
8
Domain: 0, 
x
y
Range: 0, 
0
0
1
1
4
2
9
3
16
4
Domain:
Range:
 ,  
Increasing Interval:
Decreasing Interval: None
Constant Interval: None
Continuous on  ,  
B.
f(x) = x2
Squaring Function
Domain:
Range:
 ,  
0, 
0,  
Decreasing Interval:  , 0
Increasing Interval:
Constant Interval: None
Continuous on  ,  
C.
Cubing Function
Domain:
Range:
f(x) = x3
 ,  
 ,  
Increasing Interval:
 ,  
Decreasing Interval: None
Constant Interval: None
Continuous on  ,  
D.
Square Root Function
Increasing Interval:
f(x)  x
0,  
Decreasing Interval: None
Constant Interval: None
Continuous on 0, 
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2.2 F15 O’Brien
CA 6th ed HLR
E.
f(x)  3 x
Cube Root Function
Domain:
Range:
 ,  
 ,  
Increasing Interval:
 ,  
Decreasing Interval: None
Constant Interval: None
Continuous on  ,  
F.
Absolute Value Function
Domain:
Range:
0,  
Decreasing Interval:  , 0
Constant Interval: None
Continuous on  ,  
II.
y
-8
-2
-1
-1
0
0
1
1
8
2
f x   x
 ,  
0, 
Increasing Interval:
x
x
y
-2
-2
-1
1
0
0
1
1
2
1
Vertical Shifts
If c > 0, then the graph of y = f(x) + c is obtained by shifting the graph of y = f(x) up c units.
The graph of y = f(x) – c is obtained by shifting the graph of y = f(x) down c units.
Examples 1, 2, and 3
Hints: When you graph parabolas and absolute value graphs, make the vertex the center of your t-chart.
Then select 2 x-values to the left of the vertex and 2 x-values to the right of the vertex. Plug those
x-values into the function and find their corresponding y-values.
When you graph cubics and cube root graphs, make the inflection point the center of your t-chart.
Then select 2 x-values to the left of the inflection point and 2 x-values to the right of the inflection point.
Plug those x-values into the function and find their corresponding y-values.
Use three points to graph a line. You can use the x-intercept, the y-intercept, and a third point.
Or you an use the y-intercept and the slope to find two additional points to use to graph the line.
To graph a square root function, chose five x-values which will make the radicand a perfect square.
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CA 6th ed HLR
III.
2.2 F15 O’Brien
Horizontal Shifts
If c > 0, the graph of y = f(x – c) is obtained by shifting the graph of y = f(x) to the right c units.
The graph of y = f(x + c) is obtained by shifting the graph of y = f(x) to the left c units.
Be careful when translating graphs horizontally. Horizontal shifts are counterintuitive.
When you see subtraction inside the parentheses, the shift is to the right. When you see addition inside the
parentheses, the shift is to the left. To determine the direction and magnitude of horizontal shifts, find the value of
x that would cause the expression in parentheses to equal zero.
Examples 4, 5, and 6
IV.
Combinations of Vertical and Horizontal Shifts
When both a horizontal shift and a vertical shift occur to the same function, the horizontal shift occurs
first, and then the vertical shift. This follows our standard order of operations, PEMDAS. We do what is
in parentheses first.
Examples 7, 8, and 9
V.
Effects of Shifts on Domain and Range
Horizontal shifts sometimes effect the domain of a function and vertical shifts sometimes effect the range
of a function. The magnitude and direction of the changes will depend on the value of the constants we
are adding or subtracting and what type of function we have.
Examples 10 and 11
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CA 6th ed HLR
2.2 F15 O’Brien
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CA 6th ed HLR
2.2 F15 O’Brien
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