Asymptotes - Algebra 2

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 .

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

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.

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:

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 .

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:

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.

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 .

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

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.

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:

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 .

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:

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.

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 .

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

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.

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:

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 .