# Algebra II : Hyperbolic Functions

## Example Questions

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### Example Question #1 : Hyperbolic Functions

Which of the following equations represents a vertical hyperbola with a center of  and asymptotes at ?

Explanation:

First, we need to become familiar with the standard form of a hyperbolic equation:

The center is always at . This means that for this problem, the numerators of each term will have to contain and .

To determine if a hyperbola opens vertically or horizontally, look at the sign of each variable. A vertical parabola has a positive term; a horizontal parabola has a positive term. In this case, we need a vertical parabola, so the term will have to be positive.

(NOTE: If both terms are the same sign, you have an ellipse, not a parabola.)

The asymptotes of a parabola are always found by the equation , where  is found in the denominator of the term and is found in the denominator of the term. Since our asymptotes are , we know that must be 4 and must be 3. That means that the number underneath the term has to be 16, and the number underneath the term has to be 9.

### Example Question #2 : Hyperbolic Functions

What is the shape of the graph depicted by the equation:

Oval

Hyperbola

Parabola

Circle

Hyperbola

Explanation:

The standard equation of a hyperbola is:

### Example Question #3 : Hyperbolic Functions

Express the following hyperbolic function in standard form:

Explanation:

In order to express the given hyperbolic function in standard form, we must write it in one of the following two ways:

From our formulas for the standard form of a hyperbolic equation above, we can see that the term on the right side of the equation is always 1, so we must divide both sides of the given equation by 52, which gives us:

Simplifying, we obtain our final answer in standard form:

### Example Question #4 : Hyperbolic Functions

Which of the following answers best represent ?

Explanation:

The correct definition of hyperbolic sine is:

Therefore, by multiplying 2 by both sides we get the following answer,

### Example Question #5 : Hyperbolic Functions

Which of the following best represents , if the value of  is zero?

Explanation:

Find the values of hyperbolic sine and cosine when x is zero.  According to the properties:

Therefore:

### Example Question #6 : Hyperbolic Functions

What is the value of ?

Explanation:

The hyperbolic tangent will need to be rewritten in terms of hyperbolic sine and cosine.

According to the properties:

Therefore:

### Example Question #7 : Hyperbolic Functions

Simplify:

Explanation:

The following is a property of hyperbolics that is closely similar to the problem.

We will need to rewrite this equation by taking a negative one as the common factor, and divide the negative one on both sides.

Substitute the value into the problem.

### Example Question #8 : Hyperbolic Functions

Which of the following is the correct expression for a hyperbola that is shifted   units up and   to the right of ?

Explanation:

The parent function of a hyperbola is represented by the function  where  is the center of the hyperbola. To shift the original function up by   simply add . To shift it to the right   take away .

### Example Question #9 : Hyperbolic Functions

Find the foci of the hyperbola:

Explanation:

Write the standard forms for a hyperbola.

OR:

The standard form is given in the second case, which will have different parameters compared to the first form.

Center:

Foci:  , where

Identify the coefficients  and substitute to find the value of .

### Example Question #10 : Hyperbolic Functions

Given the hyperbola , what is the  value of the center?

Explanation:

In order to determine the center, we will first need to rewrite this equation in standard form.

Isolate 41 on the right side.  Subtract  and add  on both sides.

The equation becomes:

Group the x and y terms.  Be careful of the negative signs.

Pull out a common factor of 4 on the second parentheses.

Complete the square twice.  Divide the second term of each parentheses by two and square the quantity.  Add the terms on both sides.

This equation becomes:

Factorize the left side and simplify the right.

Divide both sides by nine.

The equation is now in the standard form of a hyperbola.

The center is at:

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