1. Digital Lesson
Shifting Graphs

2. Example: Shift, Reflection
The graphs of many functions are transformations of the graphs of very basic functions.
Example: The graph of y = x is the graph of y = x2 shifted upward three units. This is a vertical shift. x y -4 4 -8 8
y = x2 + 3
y = x2
The graph of y = -x2 is the reflection of the graph of y = x2 in the x-axis.
y = -x2
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Example: Shift, Reflection

3. Vertical Shifts
If c is a positive real number, the graph of f(x) + c is the graph of y = f(x) shifted upward c units.
If c is a positive real number, the graph of f(x) – c is the graph of y = f(x) shifted downward c units. x y
f(x) + c
f(x) +c
f(x) – c -c
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Vertical Shifts

4. Example: Vertical Shifts
Example: Use the graph of f (x) = |x| to graph the functions f (x) = |x| + 3 and f (x) = |x| – 4. x y -4 4 8
f (x) = |x| + 3
f (x) = |x|
f (x) = |x| – 4
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Example: Vertical Shifts

5. Horizontal Shifts
If c is a positive real number, then the graph of f (x – c) is the graph of y = f (x) shifted to the right c units. x y -c +c
If c is a positive real number, then the graph of f (x + c) is the graph of y = f (x) shifted to the left c units.
y = f (x + c)
y = f (x)
y = f (x – c)
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Horizontal Shifts

6. Example: Horizontal Shifts
Example: Use the graph of f (x) = x3 to graph f (x) = (x – 2)3 and f (x) = (x + 4)2 . x y -4 4
f(x) = x3
f (x) = (x + 4)2
f (x) = (x – 2)3
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Example: Horizontal Shifts

7. Example: Vertical and Horizontal Shifts
Example: Graph the function using the graph of
First make a vertical shift 4 units downward.
Then a horizontal shift 5 units left. -4 x y 4 -4 y 4 x
(4, 2)
(0, 0)
(-1, -2)
(4, -2)
(0, -4)
(-5, -4)
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Example: Vertical and Horizontal Shifts

8. Reflection in the y-Axis and x-Axis.
The graph of a function may be a reflection of the graph of a basic function.
The graph of the function y = f (-x) is the graph of y = f (x) reflected in the y-axis. x y
y = f (-x)
y = f (x)
The graph of the function y = -f (x) is the graph of y = f (x) reflected in the x-axis.
y = -f (x)
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Reflection in the y-Axis and x-Axis.

9. Example: Graph y = -(x + 3)2 using the graph of y = x2.
First reflect the graph in the x-axis.
Then shift the graph three units to the left. x y -4 4 x y 4
y = x2
(-3, 0)
y = - (x + 3)2
y = - x2
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Example: Reflections

10. Vertical Stretching and Shrinking
If c > 1 then the graph of y = c f (x) is the graph of y = f (x) stretched vertically by c.
If 0 < c < 1 then the graph of y = c f (x) is the graph of y = f (x) shrunk vertically by c.
- 4 x y 4
y = x2
y = 2x2
Example: y = 2x2 is the graph of y = x2 stretched vertically by 2.
is the graph of y = x2 shrunk vertically by .
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Vertical Stretching and Shrinking

11. Example: Vertical Stretch and Shrink
Horizontal Stretching and Shrinking
If c > 1, the graph of y = f (cx) is the graph of y = f (x) shrunk horizontally by c.
If 0 < c < 1, the graph of y = f (cx) is the graph of y = f (x) stretched horizontally by c.
y = |2x|
Example: y = |2x| is the graph of y = |x| shrunk horizontally by 2.
- 4 x y 4
y = |x|
is the graph of y = |x| stretched horizontally by .
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Example: Vertical Stretch and Shrink

12. Example: Multiple Transformations
Example: Graph given the graph y = x3.
Graph y = x3 and do one transformation at a time.
- 4 4 8 x y
- 4 4 8 x y
Step 3:
Step 1: y = x3
Step 4:
Step 2: y = (x + 1)3
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Example: Multiple Transformations