College Algebra : Graphs

Study concepts, example questions & explanations for College Algebra

varsity tutors app store varsity tutors android store

Example Questions

Example Question #141 : College Algebra

Give the equation of the horizontal asymptote of the graph of the equation 

\displaystyle f(x) =4 \log (x- 3)- 2.

Possible Answers:

\displaystyle y = 4

\displaystyle y = 2

The graph of \displaystyle f(x) does not have a horizontal asymptote.

\displaystyle y = 3

\displaystyle y = 10

Correct answer:

The graph of \displaystyle f(x) does not have a horizontal asymptote.

Explanation:

Let \displaystyle g(x) = \log x. In terms of \displaystyle g(x),

\displaystyle f(x) =4 g (x- 3)- 2

\displaystyle g(x), being a logarithmic function, has a graph without a horizontal asymptote. As \displaystyle f(x) represents the result of transformations of \displaystyle g(x), it follows that its graph does not have a horizontal asymptote, either.

Example Question #4 : Transformations

Give the equation of the vertical asymptote of the graph of the equation 

\displaystyle f(x) =4 \log (x- 3)- 2.

Possible Answers:

\displaystyle x = 3

\displaystyle x = 4

\displaystyle x = -2

\displaystyle x = 2

\displaystyle x = -3

Correct answer:

\displaystyle x = 3

Explanation:

Let \displaystyle g(x) = \log x. In terms of \displaystyle g(x),

\displaystyle f(x) =4 g (x- 3)- 2

The graph of \displaystyle g(x) has as its vertical asymptote the line of the equation \displaystyle x= 0. The graph of \displaystyle f(x) is the result of three transformations on the graph of \displaystyle g(x)- a right shift of 3 units ( \displaystyle -3 ), a vertical stretch ( \displaystyle 4g ), and a downward shift of 2 units ( \displaystyle -2 ). Of the three transformations, only the right shift affects the position of the vertical asymptote; the asymptote of \displaystyle f(x) also shifts right 3 units, to \displaystyle x = 3.

Example Question #3 : Transformations

Define 

\displaystyle f(x) = e^{x}

and

\displaystyle g(x)= -e^{x-2}.

Which two transformations must be performed in the graph of \displaystyle f(x) in order to obtain the graph of \displaystyle g(x)?

Possible Answers:

The graph of \displaystyle f(x) must be translated two units right, then reflected about the \displaystyle y-axis.

The graph of \displaystyle f(x) must be translated two units left, then reflected about the \displaystyle y-axis.

The graph of \displaystyle f(x) must be translated two units left, then reflected about the \displaystyle x-axis.

The graph of \displaystyle f(x) must be translated two units right, then reflected about the \displaystyle x-axis.

None of the other choices gives the correct response.

Correct answer:

The graph of \displaystyle f(x) must be translated two units right, then reflected about the \displaystyle x-axis.

Explanation:

\displaystyle g(x) = -f(x-2), so the graph of \displaystyle g(x) is the result of performing the following transformations:

1) \displaystyle f_{1}(x) = f(x-2) is the result of translating this graph two units right.

2) \displaystyle g(x)= -f_{1}(x) = -f(x-2) is the result of reflecting the new graph about the \displaystyle x-axis.

Example Question #61 : Graphs

The graph of a function \displaystyle f(x) is reflected about the \displaystyle x-axis, then translated upward \displaystyle \frac{8}{3} units. Which of the following is represented by the resulting graph?

Possible Answers:

\displaystyle g(x)= -f\left (-x + \frac{8}{3} \right )

\displaystyle g(x)= f(-x) + \frac{8}{3}

\displaystyle g(x)= -f\left (x- \frac{8}{3} \right )

\displaystyle g(x)= -f(x) + \frac{8}{3}

\displaystyle g(x)= -f\left (x + \frac{8}{3} \right )

Correct answer:

\displaystyle g(x)= -f(x) + \frac{8}{3}

Explanation:

Reflecting the graph of a function \displaystyle f(x) about the \displaystyle x-axis results in the graph of the function

\displaystyle f_{1}(x)= -f(x).

Translating this graph upward \displaystyle \frac{8}{3} results in the graph of the function

\displaystyle g(x)= f_{1}(x) + \frac{8}{3 }= -f(x)+ \frac{8}{3 }.

Example Question #62 : Graphs

Translate the graph of \displaystyle f(x) = \ln x upward three units to yield the graph of a function \displaystyle g(x). Which of the following is a valid way of stating the definition of \displaystyle g(x)?

Possible Answers:

\displaystyle g(x)= \ln \left ( \frac{x}{e^{3}} \right )

\displaystyle g(x)= \ln (e^{3} x)

\displaystyle g(x)= \ln x^{3}

\displaystyle g(x)= \ln (x+3)

\displaystyle g(x)= \ln (x-3)

Correct answer:

\displaystyle g(x)= \ln (e^{3} x)

Explanation:

A vertical translation of the graph of a function \displaystyle f(x) by \displaystyle k units yields the graph of the function \displaystyle g(x)+k. A translation in an upward direction is a positive translation, so setting \displaystyle f(x)= \ln x and \displaystyle k = 3, the resulting graph becomes

\displaystyle g(x)=( \ln x) +3

or

\displaystyle g(x)=3+ \ln x

Apply properties of logarithms to rewrite this as

\displaystyle g(x)= \ln e^{3} + \ln x

\displaystyle g(x)= \ln (e^{3} x).

Example Question #141 : College Algebra

Reflect the graph of \displaystyle f(x) = \ln x about the \displaystyle y-axis to yield the graph of a function \displaystyle g(x). Which of the following is a valid way of stating the definition of \displaystyle g(x)?

Possible Answers:

\displaystyle g(x)= \ln (-x)

\displaystyle g(x)= \ln \frac{1}{x}

\displaystyle g(x) =e^{x}

None of the other choices gives the correct response.

\displaystyle g(x) = \frac{1}{\ln x}

Correct answer:

\displaystyle g(x)= \ln (-x)

Explanation:

The reflection of the graph of a function \displaystyle f(x) about the \displaystyle y-axis yields the graph of the function \displaystyle g(x) = f(-x). Therefore, set \displaystyle f(x) = \ln x and substitute \displaystyle -x for \displaystyle x to yield the function

\displaystyle g(x) = \ln (-x).

Example Question #61 : Graphs

Determine the symmetry of the following equation.

 

\displaystyle y=x^2-6x+10

Possible Answers:

Symmetry along all axes. 

Symmetry along the origin.

Symmetry along the y-axis.

Symmetry along the x-axis.

Does not have symmetry.

Correct answer:

Does not have symmetry.

Explanation:

To check for symmetry, we are going to do three tests, which involve substitution. First one will be to check symmetry along the x-axis, replace \displaystyle -y=y.

\displaystyle -y=x^2-6x+10

This isn't equivilant to the first equation, so it's not symmetric along the x-axis.

Next is to substitute \displaystyle -x=x.

\displaystyle y=(-x)^2-6(-x)+10

\displaystyle y=x^2+6x+10

This is not the same, so it is not symmetric along the y-axis.

For the last test we will substitute \displaystyle -y=y, and \displaystyle -x=x

\displaystyle -y=(-x)^2-6(-x)+10

\displaystyle -y=x^2+6x+10

\displaystyle y=-x^2-6x-10

This isn't the same as the orginal equation, so it is not symmetric along the origin.

The answer is it is not symmetric along any axis.

Example Question #2 : Symmetry

Untitled

Which of the following is true of the relation graphed above?

Possible Answers:

It is an even function

It is not a function

It is an odd function

It is a function, but it is neither even nor odd.

Correct answer:

It is an odd function

Explanation:

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

Untitled

Also, it can be seen to be symmetrical about the origin. Consequently, for each \displaystyle c in the domain, \displaystyle f(-c) = -f(c) - the function is odd.

Example Question #3 : Symmetry

Relation

Which of the following is true of the relation graphed above?

Possible Answers:

It is a function, but it is neither even nor odd.

It is an even function

It is not a function

It is an odd function

Correct answer:

It is an odd function

Explanation:

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

 Relation

Also, it is seen to be symmetric about the origin. Consequently, for each \displaystyle c in the domain, \displaystyle f(-c) = -f(c) - the function is odd.

Example Question #4 : Symmetry

\displaystyle f(x) is an even function; \displaystyle f(9) = 10.

True or false: It follows that \displaystyle f(10) = 9.

Possible Answers:

True

False

Correct answer:

False

Explanation:

A function \displaystyle f(x) is even if and only if, for all \displaystyle x in its domain, \displaystyle f(-x) = f(x). It follows that if \displaystyle f(9) = 10, then 

\displaystyle f(-9)= f(9) = 10.

No restriction is placed on any other value as a result of this information, so the answer is false.

Learning Tools by Varsity Tutors