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Symmetry and Transformations Worksheet - Page 2

Symmetry and Transformations Worksheet
  • Page 2
 11.  
Which of the following is true for the function f(x) = x4 + 2x2 + 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.]


Correct answer : (4)
 12.  
Which of the following is true for the function f(x) = ex2?
I. symmetric about y - axis
II. symmetric about origin
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.


Correct answer : (4)
 13.  
Which of the following is true for the function f(x) = 1x2+1 ?
I. f(x) is an even function
II. symmetric about origin
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.


Correct answer : (4)
 14.  
Which of the following is true for an odd function f(x) ?
I. symmetric about origin
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.


Correct answer : (1)
 15.  
Which of the following is true for the function f(x) = | x5 + 3x | ?
I. symmetric about y - axis
II. not symmetric
III. f(x) is an even function
IV. f(- x) = - f(x)
V. symmetric about origin
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.


Correct answer : (4)
 16.  
Which of the following is true for the function f(x) = xx2+1 ?
I. f(x) is an odd function
II. symmetric about y - axis
III. not symmetric
IV. symmetric about origin
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.


Correct answer : (1)
 17.  
Which of the following is true for the function f(x) = x5 + x4 + x + 1 ?
I. f(x) is an even function
II. f(x) is an odd function
III. f(x) is neither even nor odd
IV. symmetric about origin
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.


Correct answer : (5)
 18.  
Which of the following is true for the function f(x) = sin 2x ?
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.


Correct answer : (2)
 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.


Correct answer : (2)
 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.


Correct answer : (1)

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