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 .

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 

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. 

Multiply the numerator and the denominator by the conjugate of the denominator, which is . Then take advantage of the distributive properties and the difference of squares pattern:

Recall that we are looking for a number that, when multiplied by itself, yields . We look for the perfect squares surrounding  and we find  and . Thus, we know that our number must be between  and   is much closer to  and thus it will be very close to  but still less, namely .