Graphing Exponential Functions - Math

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:

The -intercept of any function describes the point where .

Substituting this in to our funciton, we get:

Exponential decay describes a function that decreases by a factor every time increases by .

These can be recognizable by those functions with a base which is between and .

The general equation for exponential decay is,

where the base is represented by and .

Thus, we are looking for a fractional base.

The only function that has a fractional base is,

.

The -intercept of a function is where . Thus, we are looking for the -value which makes .

If we try to solve this equation for we get an error.

To bring the exponent down we will need to take the natural log of both sides.

Since the natural log of zero does not exist, there is no exponent which makes this equation true.

Thus, there is no -intercept for this function.

Exponential functions with a base greater than one are models of exponential growth. Thus, we know that our function will increase and not decrease. Remembering the graph of an exponential function, we can determine that the graph will begin gradually, almost like a flat line. Then, as increases, begins to increase very quickly.

To find the -intercept, set in the equation and solve.

To find the -intercept(s) of , we need to set the numerator equal to zero.

That means .

The best way to solve for a funky equation like this is to graph it in your calculator and calculate the roots. The result is .

To find the -intercept(s) of , we need to set the numerator equal to zero and solve.

First, notice that can be factored into . Now set that equal to zero: .

Since we have two sets in parentheses, there are two separate values that can cause our equation to equal zero: one where and one where .

Solve for each value:

and

.

Therefore there are two -interecpts: and .

To find the -intercept(s) of , set the value equal to zero and solve.

To find the -intercept, set in the equation and solve.

To find the -intercept(s) of , we need to set the numerator equal to zero.

That means .

The best way to solve for a funky equation like this is to graph it in your calculator and calculate the roots. The result is .

To find the -intercept(s) of , we need to set the numerator equal to zero and solve.

First, notice that can be factored into . Now set that equal to zero: .

Since we have two sets in parentheses, there are two separate values that can cause our equation to equal zero: one where and one where .

Solve for each value:

and

.

Therefore there are two -interecpts: and .

To find the -intercept(s) of , set the value equal to zero and solve.