### All Precalculus Resources

## Example Questions

### Example Question #21 : Rational Functions

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:**

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:**

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:**

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:**

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:**

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:**

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:**

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:**

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:

Zero: ; Asymptote:

Zero: ; Asymptotes:

Zeros: ; Asymptotes:

Zeros: ; Asymptote:

**Correct answer:**

Zero: ; Asymptote:

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:

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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