Algebra II : Solving and Graphing Logarithms

Study concepts, example questions & explanations for Algebra II

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Example Questions

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Example Question #3 : Graphing Logarithmic Functions

What is/are the asymptote(s) of the graph of the function  ?

Possible Answers:

 and 

 and 

Correct answer:

Explanation:

The graph of the logarithmic function

has as its only asymptote the vertical line 

Here, since , the only asymptote is the line

.

Example Question #1 : Graphing Logarithmic Functions

Which is true about the graph of 

 ?

Possible Answers:

When  ,  is twice the size as in the equation 

The range of the function is infinite in both directions positive and negative.

The domain of the function is greater than zero

All of the answers are correct

None of the answers are correct

Correct answer:

All of the answers are correct

Explanation:

There is no real number  for which 

Therefore in the equation  ,  cannot be 

However,  can be infinitely large or negative.

Finally, when   or twice as large.

Example Question #5 : Graphing Logarithmic Functions

Which of the following is true about the graph of 

Possible Answers:

It is an odd function.

It is an even function.

The range must be greater than zero.

The domain is infinite in both directions.

The graph is the mirror image of  flipped over the line 

Correct answer:

The graph is the mirror image of  flipped over the line 

Explanation:

 is the inverse of  and therefore the graph is simply the mirror image flipped over the line 

Example Question #6 : Graphing Logarithmic Functions

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

.

Possible Answers:

The graph of  does not have a horizontal asymptote.

Correct answer:

The graph of  does not have a horizontal asymptote.

Explanation:

Let 

 In terms of ,

This is the graph of  shifted left 4 units, stretched vertically by a factor of 3, then shifted up 2 units. 

The graph of  does not have a horizontal asymptote; therefore, a transformation of this graph, such as that of , does not have a horizontal asymptote either.

Example Question #7 : Graphing Logarithmic Functions

Find the equation of the vertical asymptote of the graph of the equation 

.

Possible Answers:

Correct answer:

Explanation:

Let . In terms of ,

.

The graph of  has as its vertical asymptote the line of the equation . The graph of  is the result of three transformations on the graph of - a left shift of 4 units , a vertical stretch (  ), and an upward shift of 2 units (  ). Of the three transformations, only the left shift affects the position of the vertical asymptote - the asymptote of  also shifts left 4 units, to .

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