### All Algebra II Resources

## Example Questions

### Example Question #1 : Asymptotes

What is the horizontal asymptote of the graph of the equation ?

**Possible Answers:**

**Correct answer:**

The asymptote of this equation can be found by observing that regardless of . We are thus solving for the value of as approaches zero.

So the value that cannot exceed is , and the line is the asymptote.

### Example Question #2 : Asymptotes

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

?

**Possible Answers:**

**Correct answer:**

An exponential equation of the form has only one asymptote - a horizontal one at . In the given function, , so its one and only asymptote is .

### 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.

### Example Question #3 : Asymptotes

Determine the asymptotes, if any:

**Possible Answers:**

**Correct answer:**

Factorize both the numerator and denominator.

Notice that one of the binomials will cancel.

The domain of this equation cannot include .

The simplified equation is:

Since the term canceled, the term will have a hole instead of an asymptote.

Set the denominator equal to zero.

Subtract one from both sides.

There will be an asymptote at only:

The answer 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 #4 : Asymptotes

Where is an asymptote located, if any?

**Possible Answers:**

**Correct answer:**

Factor the numerator and denominator.

Rewrite the equation.

Notice that the will cancel. This means that the root of will be a hole instead of an asymptote.

Set the denominator equal to zero and solve for x.

An asymptote is located at:

The answer is:

### Example Question #1 : Solving Exponential Functions

Consider the exponential function . Determine if there are any asymptotes and where they lie on the graph.

**Possible Answers:**

There is one horizontal asymptote at .

There is one vertical asymptote at .

There are no asymptotes. goes to positive infinity in both the and directions.

There is one vertical asymptote at .

**Correct answer:**

There is one horizontal asymptote at .

For positive values, increases exponentially in the direction and goes to positive infinity, so there is no asymptote on the positive -axis. For negative values, as decreases, the term becomes closer and closer to zero so approaches as we move along the negative axis. As the graph below shows, this is forms a horizontal asymptote.

### Example Question #1 : Solving Exponential Equations

Solve the equation for .

**Possible Answers:**

**Correct answer:**

Begin by recognizing that both sides of the equation have a root term of .

Using the power rule, we can set the exponents equal to each other.

### Example Question #1 : Solving Exponential Equations

Solve the equation for .

**Possible Answers:**

**Correct answer:**

Begin by recognizing that both sides of the equation have the same root term, .

We can use the power rule to combine exponents.

Set the exponents equal to each other.

### Example Question #3 : Solving Exponential Equations

In 2009, the population of fish in a pond was 1,034. In 2013, it was 1,711.

Write an exponential growth function of the form that could be used to model , the population of fish, in terms of , the number of years since 2009.

**Possible Answers:**

**Correct answer:**

Solve for the values of *a *and *b*:

In 2009, and (zero years since 2009). Plug this into the exponential equation form:

*. *Solve for to get * .*

In 2013, and . Therefore,

or *. *Solve for to get

.

Then the exponential growth function is

.

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