# Symmetry and Transformations Worksheet - Page 2

Symmetry and Transformations Worksheet
• Page 2
11.
Which of the following is true for the function $f$($x$) = $x$4 + 2$x$2 + 2 ?
 a. not symmetric b. symmetric about the point $y$ = 2 c. symmetric about origin d. symmetric about $y$ - axis e. an odd function

#### Solution:

f(x) = x4 + 2x2 + 2

f(- x) = (- x)4 + 2(- x)2 + 2

f(- x) = x4 + 2x2 + 2 = f(x)

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

So, f(x) is symmetric about the y - axis.
[Even function is symmetric about y - axis.]

12.
Which of the following is true for the function $f$($x$) = $e$$x$2?
I. symmetric about $y$ - axis
III. not symmetric
IV. $f$($x$) is an even function
V. $f$(- $x$) = - $f$($x$)
 a. V only b. II only c. III only d. I and IV e. III and V

#### Solution:

f(x) = ex2

f(- x) = e(- x)2

f(- x) = f(x)

f(x) is an even function.

So, f(x) is symmetric about y - axis.

13.
Which of the following is true for the function $f$($x$) = $\frac{1}{{x}^{2}+1}$ ?
I. $f$($x$) is an even function
III. not symmetric
IV. symmetric about $y$ - axis
V. $f$(- $x$) = - $f$($x$)
 a. III and V b. II only c. III only d. I and IV e. V only

#### Solution:

f(x) = 1x2+1

f(- x) = 1(-x)2+1 = 1x2+1

f(- x) = f(x)

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

So, f(x) is symmetric about y - axis.

14.
Which of the following is true for an odd function $f$($x$) ?
II. not symmetric
III. $f$(- $x$) = - $f$($x$)
IV. symmetric about $y$ - axis
V. $f$(- $x$) = $f$($x$)
 a. I and III b. IV and V c. V only d. II only e. IV only

#### Solution:

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

The graph of an odd function is symmetric about origin.

15.
Which of the following is true for the function $f$($x$) = | $x$5 + 3$x$ | ?
I. symmetric about $y$ - axis
II. not symmetric
III. $f$($x$) is an even function
IV. $f$(- $x$) = - $f$($x$)
 a. IV and V b. II only c. II and IV d. I and III e. IV only

#### Solution:

f(x) = | x5 + 3x |

f(- x) = | (- x)5 + 3(- x) |

f(- x) = | - (x5 + 3x) |

f(- x) = | x5 + 3x | = f(x)

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

So, f(x) is symmetric about y - axis.

16.
Which of the following is true for the function $f$($x$) = $\frac{x}{{x}^{2}+1}$ ?
I. $f$($x$) is an odd function
II. symmetric about $y$ - axis
III. not symmetric
V. $f$(- $x$) = $f$($x$)
 a. I and IV b. III only c. II only d. III and IV e. V only

#### Solution:

f(x) = xx2+1

f(- x) = -x(-x)2+1 = - xx2+1

f(- x) = - f(x)

f(x) is an odd function.

So, f(x) is symmetric about origin.

17.
Which of the following is true for the function $f$($x$) = $x$5 + $x$4 + $x$ + 1 ?
I. $f$($x$) is an even function
II. $f$($x$) is an odd function
III. $f$($x$) is neither even nor odd
V. symmetric about $y$ - axis
 a. IV only b. V only c. I and V d. II and IV e. III only

#### Solution:

f(x) = x5 + x4 + x + 1

f(- x) = (- x)5 + (- x)4 + (- x) + 1

f(- x) = - x5 + x4 - x + 1

f(- x) ≠ f(x), f(- x) ≠ - f(x)

The function is neither even nor odd.

18.
Which of the following is true for the function $f$($x$) = sin 2$x$ ?
I. $f$($x$) is an odd function
II. $f$($x$) is symmetric about origin
III. $f$($x$) is symmetric about $y$ - axis
IV. $f$($x$) is not symmetric
V. $f$(- $x$) = $f$($x$)
 a. V only b. I and II c. III only d. III and IV e. IV only

#### Solution:

f(x) = sin 2x

f(- x) = sin 2(- x) = - sin 2x

f(- x) = - f(x)

f(x) is an odd function.
[If f(- x) = - f(x), then f(x) is an odd function.]

So, f(x) is symmetric about origin.

19.
Identify the figure that has the dotted line dividing the letter into two symmetrical halves.

 a. Figure 3 b. Figure 2 c. Figure 4 d. Figure 1

#### Solution:

Line of symmetry is a line that divides the figure into two parts which are mirror images of each other.

Among the choices, the dotted line through the letter M divides it into two parts which are mirror images of each other.

The lines passing through the letters L, P, and Q do not divide them into two parts that would be mirror images of each other.

So, the dotted line through the letter M is a line of symmetry.

20.
Which of the figures (alphabets) has a line of symmetry?

 a. all the three figures b. Figure 2, Figure 1 c. Figure 1 d. Figure 3, Figure 1

#### Solution:

Line of symmetry is a line that divides the figure into two parts, which are mirror images of each other.

The dotted line for all the figures (alphabets) divides the figures into two parts, which are mirror images of each other.

So, the dotted lines for all the figures are the lines of symmetry.