### All Precalculus Resources

## Example Questions

### Example Question #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions

Given the function , determine the equation of all the vertical asymptotes across the domain. Let be an integer.

**Possible Answers:**

**Correct answer:**

For the function , it is not necessary to graph the function. The y-intercept does not affect the location of the asymptotes.

Recall that the parent function has an asymptote at for every period.

Set the inner quantity of equal to zero to determine the shift of the asymptote.

This indicates that there is a zero at , and the tangent graph has shifted units to the right. As a result, the asymptotes must all shift units to the right as well. The period of the tangent graph is .

### Example Question #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions

Which of the choices represents asymptote(s), if any?

**Possible Answers:**

**Correct answer:**

Factor the numerator and denominator.

Notice that the terms will cancel. The hole will be located at because this is a removable discontinuity.

The denominator cannot be equal to zero. Set the denominator to find the location where the x-variable cannot exist.

The asymptote is located at .

### Example Question #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions

Find the vertical asymptote of the equation.

**Possible Answers:**

There are no vertical asymptotes.

**Correct answer:**

To find the vertical asymptotes, we set the denominator of the function equal to zero and solve.

Given the function , determine the equation of all vertical asymptotes across the domain. Let be any integer.

Given the function , determine the equation of all vertical asymptotes across the domain. Let be any integer.

Which of the following represents the asymptotes for the general parent function ?

**Correct answer:**

If you do not have these asymptotes memorized, they can be easily derived. Write in terms of .

Now we need to solve for since it is the denominator of the function. When the denominator of a function is equal to zero, there is a vertical asymptote because that function is then undefined.

when . So for any integer , we say that there is a vertical asymptote for when .

Assume that there is a vertical asymptote for the function at , solve for from the equation of all vertical asymptotes at .

**Possible Answers:**

**Correct answer:**

We know that the parent function has vertical asymptotes at . So now we will set the inner quantity of the function equal to zero to find the shift of the asymptote.

Now we will add this to the parent function equation for vertical asymptotes

Now we will set this equation for the given vertical asymptote at

True or false: There is a vertical asymptote for at

**Possible Answers:**

There is insufficient information to answer the question.

The statement is false.

The statement is true.

**Correct answer:**

The statement is true.

We know that the parent function of has asymptotes at where is any integer. Considering we can set the asymptotic equation equal to this one and solve for to see if is an integer. If is an integer, then there is a vertical asymptote here.

So when , then

, so the given statement is true.

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