SOLUTION: Write and exponential function y=ax^b for a graph that includes (2,-12) and (4,-48)

Question 611462: Write and exponential function y=ax^b for a graph that includes (2,-12) and (4,-48)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

If the two given points are supposed to be on the graph of the equation we find, then they must fit the equation. Before we even know what the "a" and "b" will be, we can go ahead an substitute in the coordinates:
and
What we have is a system of two equations with two variables. We should be able to use this system to solve for those variables. This will tell us what "a" and "b" should be for our equation.

The Substitution Method is a method for solving systems that can work for any system. We will use it here. We'll solve one equation, either one, for a variable, either one. I'm going to solve the first equation for "a". Dividing both sides by we get:

Now we substitute this expression in for the "a" in the other (second) equation:

We now have an equation with just a single variable. We should be able to solve for this variable. One way to to do is to manipulate the expression on the right so it has just one "b":

Divide both sides by -12:

Use the rule for exponents:

Simplify

It should be obvious that "b" = 2.

Now we go back to

and use the "b" we found to find the "a":

Now that we finally have an "a" and a "b" we can write the exponential function:

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