# Symmetry and Transformations Worksheet

Symmetry and Transformations Worksheet
• Page 1
1.
Use function notation to represent the transformation of the graph of $h$($x$) shown with dotted lines.

 a. $\frac{1}{2}$$h$($x$) b. $h$($x$) + 2 c. $h$($x$) - 2 d. 2$h$($x$) e. $h$($x$)

#### Solution:

The graph with dotted lines is obtained by shifting vertically 2 units upward of the graph of h(x).

So, the function notation representing the transformation of h(x) is h(x) + 2.

2.
Use function notation to represent the transformation of the graph of $f$($x$) shown with dotted lines.

 a. $f$ ($x$ + 4) b. $\frac{1}{4}$ $f$ ($x$) c. $f$ ($x$) + 4 d. 4 $f$ ($x$) e. $f$ (4$x$)

#### Solution:

The graph with dotted lines is obtained by stretching vertically the graph of f(x) by a scale factor of 4.

So, the function notation representing the transformation of f (x) is 4 f (x).

3.
Use function notation to represent the transformation of the graph of $f$ ($x$) shown with dotted lines.

#### Solution:

The graph of f(x) is translated 5 units to the left to obtain the graph with dotted lines.

So, the function notation representing the transformation of f (x) is f (x + 5).

4.
Describe the graph of $\frac{1}{5}$ $g$($x$) that can be obtained by tranformation of the graph of $g$($x$).
 a. Horizontally stretching by a scale factor of 5 b. Vertically shifting $\frac{1}{5}$ units upward c. Horizontally compressing by a scale factor of $\frac{1}{5}$ d. Vertically compressing by a scale factor of $\frac{1}{5}$ e. Vertically stretching by a scale factor of 5

#### Solution:

The graph of 1 / 5 g(x) is obtained by compressing vertically of the graph of g(x) by a scale factor of 1 / 5.

5.
Describe the graph of $h$($\frac{1}{3}$$x$) that can be obtained by tranformation of the graph of $h$($x$).
 a. Vertically stretching by a scale factor of 3 b. Horizontally shifting 3 units upward c. Horizontally stretching by a scale factor of $\frac{1}{3}$ d. Vertically compressing by a scale factor of $\frac{1}{3}$ e. Horizontally stretching by a scale factor of3

#### Solution:

The graph of h(1 / 3x) is obtained by stretching horizontally the graph of h( x) by a scale factor of 1 / 3.

6.
Describe the graph of $f$ ($x$ - 10) that can be obtained by the tranformation of the graph of $f$ ($x$).
 a. Horizontal shift of 10 units to the right b. Vertical shift of 10 units upward c. Horizontal shift of 10 units to the left d. Vertical shift of 10 units downward e. Horizontally compressing by a scale factor of 10

#### Solution:

The graph of f (x - 10) is obtained by the horizontal shift of 10 units to the right of the graph of f (x).

7.
Given the function $h$($x$), explain how the transformation $h$(- $x$) changes the graph of $h$($x$).
 a. Translate 1 unit down b. Reflection about the origin c. Reflection over the $x$-axis d. Translate 1 unit to the left e. Reflection over the $y$-axis

#### Solution:

The graph of h(- x) is the reflection over the y-axis of the graph of h(x).

8.
Use function notation to represent the transformation of the graph of $f$($x$) shown with dotted lines.

 a. 4$f$ ($x$) b. $f$($x$ + 4) c. $\frac{1}{4}$ $f$($x$) d. $f$(4$x$) e. $f$($x$) + 4

9.
Which of the following is true for the graph of even function?
 a. symmetric with respect to the origin b. symmetric with respect to the $x$-axis c. symmetric with respect to the both origin and $y$-axis d. symmetric with respect to the $y$-axis e. symmetric with respect to both $x$ and $y$-axes

#### Solution:

If f(- x) = f(x) , then the function f(x) is an even function.

The graph of an even function is symmetric with respect to the y - axis.

10.
Which of the following is true for the function $f$($x$) = $x$3 + $x$ ?
 a. $f$($x$) is an odd function b. $f$($x$) is an even function c. $f$($x$) is symmetric about origin d. $f$($x$) is symmetric about $y$ - axis e. Both A and C

#### Solution:

f(x) = x3 + x

f(- x) = (- x)3 + (- x) = - x3 - x

f(- x) = - (x3 + x) = - f(x)

So, f(x) is an odd function and is symmetric about origin.
[If f(- x) = - f(x), then f(x) is an odd function.]