Precalculus : Find Intercepts and Asymptotes

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #1 : Find Intercepts And Asymptotes

Suppose the function below has an oblique (i.e. slant asymptote) at .

If we are given , what can we say about the relation between  and  and between  and ?

Possible Answers:

Correct answer:

Explanation:

We can only have an oblique asymptote if the degree of the numerator is one more than the degree of the denominator.  This stipulates that  must equal .  

The slope of the asymptote is determined by the ratio of the leading terms, which means the ratio of  to  must be 3 to 1.  The actual numbers are not important.

Finally, since the value of  is at least three, we know there is no intercept to our oblique asymptote.

Example Question #2 : Find Intercepts And Asymptotes

Find the -intercept and asymptote, if possible.  

Possible Answers:

Correct answer:

Explanation:

To find the y-intercept of , simply substitute  and solve for .

The y-intercept is 1.

The numerator, , can be simplified by factoring it into two binomials.

There is a removable discontinuity at , but there are no asymptotes at  since the  terms can be canceled.

The correct answer is:  

Example Question #3 : Find Intercepts And Asymptotes

Find the -intercepts of the rational function

.

 

Possible Answers:

Correct answer:

Explanation:

The -intercept(s) is/are the root(s) of the numerator of the rational functions.

In this case, the numerator is .

Using the quadratic formula,

the roots are .

Thus,  are the -intercepts.

Example Question #4 : Find Intercepts And Asymptotes

Find the vertical asymptotes of the following rational function.

Possible Answers:

No vertical asymptotes.

Correct answer:

Explanation:

Finding the vertical asymptotes of the rational function  amounts to finding the roots of the denominator, .

It is easy to check, using the quadratic formula,

that the roots, and thus the asymptotes, are .

Example Question #5 : Find Intercepts And Asymptotes

Find the y-intercept and asymptote, respectively, of the following function, if possible.  

Possible Answers:

Correct answer:

Explanation:

Before we start to simplify the problem, it is crucial to immediately identify the domain of this function .

The denominator cannot be zero, since it is undefined to divide numbers by this value.  After simplification, the equation is:

The domain is  and there is a hole at  since there is a removable discontinuity.  There are no asymptotes.  

Since it's not possible to substitute  into the original equation, the y-intercept also does not exist.

Therefore, the correct answer is:

Example Question #6 : Find Intercepts And Asymptotes

What is a vertical asymptote of the following function? 

Possible Answers:

Correct answer:

Explanation:

To find the vertical asymptote of a function, we set the denominator equal to 

With our function, we complete this process. 

The denominator is , so we begin: 

Example Question #7 : Find Intercepts And Asymptotes

What is the -intercept of the following function? 

Possible Answers:

There is no -intercept. 

Correct answer:

Explanation:

The y-intercept of a function is always found by substituting in .

We can go through this process for our function. 

Example Question #8 : Find Intercepts And Asymptotes

Which of these functions has a vertical asymptote of and a slant asymptote of ?

Possible Answers:

Correct answer:

Explanation:

In order for the vertical asymptote to be , we need the denominator to be . This gives us three choices of numerators:

If the slant asymptote is , we will be able to divide our numerator by and get with a remainder.

Dividing the first one gives us with no remainder.

Dividing the last one gives us with a remainder.

The middle numerator would give us what we were after, with a remainder of -17.

The answer is

Example Question #9 : Find Intercepts And Asymptotes

Find the zeros and asymptotes for

.

Possible Answers:

Zero: ; Asymptotes:

Zeros: ; Asymptotes:

Zero: ; Asymptote:

Zeros: ; Asymptote:

Zero: ; Asymptotes:

Correct answer:

Zero: ; Asymptote:

Explanation:

To find the information we're looking for, we should factor this equation:

This means that it simplifies to .

When the equation is in the form of a fraction, to find the zero of the function we need to set the numerator equal to zero and solve for the variable.

To find the asymptote of an equation with a fraction we need to set the denominator of the fraction equal to zero and solve for the variable.

Therefore our equation has a zero at -3 and an asymptote at -2.

Example Question #10 : Find Intercepts And Asymptotes

Find the slant and vertical asymptotes for the equation

.

Possible Answers:

Vertical asymptote: ; Slant asymptote: 

Vertical asymptote: ; Slant asymptote: 

Vertical asymptote: ; Slant asymptote:

Vertical asymptote: ; Slant asymptote:

Vertical asymptote: ; Slant asymptote: 

Correct answer:

Vertical asymptote: ; Slant asymptote: 

Explanation:

To find the vertical asymptote, just set the denominator equal to 0:

To find the slant asymptote, divide the numerator by the denominator, but ignore any remainder. You can use long division or synthetic division.

               

          

                         

                   

                                       

The slant asymptote is

.

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