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

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Write and exponential function y=ax^b for a graph that includes (2,-12) and (4,-48)      Log On


   

Question 611462: Write and exponential function y=ax^b for a graph that includes (2,-12) and (4,-48)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
y+=+ax%5Eb
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:
-12+=+a%2A%282%29%5Eb and -48+=+a%2A%284%29%5Eb
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 2%5Eb we get:
-12%2F2%5Eb+=+a

Now we substitute this expression in for the "a" in the other (second) equation:
-48+=+%28-12%2F2%5Eb%29%2A4%5Eb
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":
-48+=+%28-12%2F2%5Eb%29%2A%284%5Eb%2F1%29
-48+=+%28-12%2A4%5Eb%29%2F2%5Eb
Divide both sides by -12:
4+=+4%5Eb%2F2%5Eb
Use the p%5Ek%2Fq%5Ek+=+%28p%2Fq%29%5Ek rule for exponents:
4+=+%284%2F2%29%5Eb
Simplify
4+=+2%5Eb
It should be obvious that "b" = 2.

Now we go back to
-12%2F2%5Eb+=+a
and use the "b" we found to find the "a":
-12%2F2%5E%282%29+=+a
-12%2F4+=+a
-3+=+a

Now that we finally have an "a" and a "b" we can write the exponential function:
y+=+-3x%5E2