Math 1314
2.5 Transformations of Functions
Section 2.5 Notes
Basic Functions: Must know these!!!
1. The identity function f(x) = x
2. The squaring function f(x) = x2
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3. The square root function f ( x )  x
Section 2.5 Notes
4. The absolute value function f(x) = |x|
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5. The cubic function f(x) = x3
Section 2.5 Notes
6. The cube root function f ( x )  3 x
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Section 2.5 Notes
We will now see how certain transformations (operations) of a function change its graph. This will give us a
better idea of how to quickly sketch the graph of certain functions.
The transformations are (1) translations, (2) reflections, and (3) stretching.
Vertical Translation
Observation: Let’s graph the functions f(x) = x2, g(x) = x2 + 3, h(x) = x2 – 2.
Vertical Translation
For b> 0,
The graph of y = f(x) + b is the graph of y = f(x) shifted upb units;
The graph of y = f(x) b is the graph of y = f(x) shifted downb units.
Horizontal Translation
Observation: Let’s graph the functions f(x) = x2, g(x) = (x + 2)2, h(x) = (x – 2)2.
Horizontal Translation
For d> 0,
The graph of y = f(x  d) is the graph of y = f(x) shifted right d units;
The graph of y = f(x + d) is the graph of y = f(x) shifted left d units.
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Section 2.5 Notes
Example: Give the function of each graph.
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Section 2.5 Notes
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Reflections
Section 2.5 Notes
 The graph of f(x) is the reflection of the graph of f(x) across the x-axis.
 The graph of f(x) is the reflection of the graph of f(x) across the y-axis.
 If a point (x, y) is on the graph of f(x), then (x, y) is on the graph of f(x), and
(x, y) is on the graph of f(x).
Example: Use the basic function to sketch the graph of the function f ( x )  x
Basic function: y   x
To sketch the graph of f ( x )   x , reflect the graph of the basic function y  x over the y-axis.
Vertical Stretching and Shrinking
Observation: Let’s graph the functions f(x) = x2, g(x) = 2x2 , h(x) = 1/2x2
Vertical Stretching and Shrinking
The graph of af(x) can be obtained from the graph
of f(x) by
stretching vertically for |a| > 1, or
shrinking vertically for 0 < |a| < 1.
For a< 0, the graph is also reflected across the x-axis.
(The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)
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Horizontal Stretching or Shrinking
Section 2.5 Notes
Observation: Let’s graph the functions f(x) = x2, g(x) = (2x)2, h(x) = (1/2x)2
Horizontal Stretching or Shrinking
The graph of y = f(cx) can be obtained from the graph
of y = f(x) by
shrinking horizontally for |c| > 1, or
stretching horizontally for 0 < |c| <1.
For c< 0, the graph is also reflected across the y-axis.
(The x-coordinates of the graph of y = f(cx) can be obtained by dividing the x-coordinates of the graph ofy =f(x)
by c.)
Example: The point (-12, 4) is on the graph of y = f(x). Find a point on the graph of y = g(x).
 g(x) = f(x – 2) and the point is (-10, 4)
 g(x)= 4f(x)and the point is (-12, 16)
 g(x) = f(½x)and the point is (-12, -4)
 g(x) = -f(x)and the point is (-24, 4)
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Summary
Section 2.5 Notes
1/ y = f(x) + C
 C > 0 moves it up
 C < 0 moves it down
2/y = f(x + C)

C > 0 moves it left
 C < 0 moves it right
3/y = C·f(x)
 C > 1 stretches it vertically (in the y-direction)
 0 < C < 1 compresses (shrinks vertically) it
4/y = f(Cx)
 C > 1 compresses (shrinks horizontally) it in the x-direction
 0 < C < 1 stretches it horizontally
5/y = -f(x) Reflects it about x-axis
6/y = f(-x) Reflects it about y-axis
Use the translation to sketch the graph of the function
Basic function: y  x
To sketch the graph of f ( x )  x  3 , shift the graph of the basic function y  x to the left 3 units.
Examples: Use the basic graph to sketch the graph of the following:
1. f(x) = x2 – 5
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2. f ( x )  x  3
3.
Section 2.5 Notes
f ( x)   x
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4.
Section 2.5 Notes
f ( x)  x  2  1
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Section 2.5 Notes
Examples: Use the given (non-basic) function f(x) to sketch the following:
1. y = f(x) – 4
2. y = f(x + 5)
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3.
y   f ( x  2)  5
4.
y  2 f ( x  3)  1
Section 2.5 Notes
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Examples:
Section 2.5 Notes
1. Use the transformations of functions to sketch the graph of function
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f ( x)   x  2  4
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Indicate:
a.
Basic shape:
b. Horizontal shift:
c. Stretching / Compressing:
d. Reflection:
e. Vertical shift:
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Section 2.5 Notes
2. Use the transformations of functions to sketch the graph of function
f ( x )  ( x  3) 2  4
Indicate:
a. Basic function:
b. Horizontal shift:
c. Compression/Stretching:
d. Reflection:
e. Vertical shift:
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Section 2.5 Notes
3. Use the transformations of functions to sketch the graph of function
f ( x)  2 x  3  1
Indicate:
a. Basic function:
b. Horizontal shift:
c. Compression-Stretching:
d. Reflection:
e. Vertical shift:
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Section 2.5 Notes
4. Use the transformations of functions to sketch the graph of function
f ( x)  3  x  2
Indicate:
a. Basic function:
b. Horizontal shift:
c. Compression-Stretching:
d. Reflection:
e. Vertical shift:
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