Solving and Graphing Exponential Equations - Algebra 2

When solving exponential equations, we need to ensure that we have the same base. When that happens, our equations our based on the exponents. We don't have the same base in this case, but we do know that

therefore

With the same base, we can now write

Subtract on both sides.

Divide on both sides.

When solving exponential equations, we need to ensure that we have the same base. When that happens, our equations our based on the exponents. We don't have the same base in this case, but we do know that

therefore

With the same base, we now have

Subtract on both sides.

Divide on both sides.

When solving exponential equations, we need to ensure that we have the same base. When that happens, our equations our based on the exponents. We don't have the same base in this case, but we do know that

By having a base of , this will make solving equations easier.

Apply power rule of exponents.

With the same base, we now can write

Add and subtract on both sides.

Divide on both sides.

When solving exponential equations, we need to ensure that we have the same base. When that happens, our equations our based on the exponents. We don't have the same base in this case, but we do know that

therefore

Apply power rule of exponents.

With the same base, we can now write

Add and subtract on both sides.

Divide on both sides.

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:

a)

This is exponential decay since the base, , is between and .

b)

This is exponential growth since the base, , is greater than .

Set and solve for

We need not work further. It is impossible to raise a positive number 2 to any real power to obtain a negative number. Therefore, the equation has no solution, and the graph of has no -intercept.

An exponential function of the form

has as its one and only asymptote the horizontal line .

Since we define as

,

then ,

and the only asymptote is the line of the equation .

For , our base is greater than so we have exponential growth, meaning the function is increasing. Also, when , we know that since . The only graph that fits these conditions is .

For , we have exponential growth again but when , . This is shown on graph .

For , we have exponential decay so the graph must be decreasing. Also, when , . This is shown on graph .

We need to determine the constants and . Since in 2010 (when ), then and

To get , we find that when , . Then .

Using a calculator, , so .

Then our model equation for the fish population is

To determine the constant , we look at the graph to find the initial value of , (when ) and find it to be . We can then plug this into our equation and we get . Since , we find that .

To find , we use the fact that when , is one half of the initial value . Plugging this into our equation with now known gives us . To solve for , we make use the fact that the natural log is the inverse function of , so that

.

We can write our equation as and take the natural log of both sides to get:

or .

Then .

Our model equation is .

The -intercept of any graph describes the -value of the point on the graph with a -value of .

Thus, to find the -intercept substitute .

In this case, you will get,

.

The -intercept of a graph is the point on the graph where the -value is .

Thus, to find the -intercept, substitute and solve for .

Thus, we get: