math 1314 section 2.5 notes 2.5 transformations of functions

Math 1314 Section 2.5 Notes

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2.5 Transformations of Functions

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 xxf )(

4. The absolute value function f(x) = |x|


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5. The cubic function f(x) = x3

6. The cube root function 3)( xxf


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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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Example: Give the function of each graph.


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Reflections

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 xxf )(

Basic function: xy

To sketch the graph of xxf )( , reflect the graph of the basic function xy 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

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

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: xy

To sketch the graph of 3)( xxf , shift the graph of the basic function xy 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. 3)( xxf

3. xxf )(


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4. 12)( xxf


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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. 5)2( xfy

4. 1)3(2 xfy


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Examples:

1. Use the transformations of functions to sketch the graph of function

422

1)( xxf

Indicate:

a. Basic shape:

b. Horizontal shift:

c. Stretching / Compressing:

d. Reflection:

e. Vertical shift:


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4)3()( 2 xxf

Indicate:

a. Basic function:


c. Compression/Stretching:

d. Reflection:

e. Vertical shift:


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132)( xxf

Indicate:

a. Basic function:


c. Compression-Stretching:

d. Reflection:

e. Vertical shift:


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4. Use the transformations of functions to sketch the graph of function 3 2)( xxf

Indicate:

a. Basic function:


c. Compression-Stretching:

d. Reflection:

e. Vertical shift:

math 1314 section 2.5 notes 2.5 transformations of functions

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