Write an exponential function y=ab^x including points (-2, 36) and (1, 4/3)

Answer 1

Here we are given two points to come up with an exponential function.

We are provided the basic formula: y=ab^x.

Let's start by taking one of our points and plugging it in, so we can eventually get rid of one of the variables. I'm going to use the (1,4/3) in this case, since 1 is easier to work with in an exponential function.

(4/3)=ab^(1), where 4/3 is the y value in the point and 1 is the x value in the point. Now we can work either the a or the b to one side. In the end it shouldn't matter which one you use. I'm going to use algebra to solve for a.

4/3=ab

4/(3b)=a

Now that we have found a, we can sub that in in the original equation.

y=(4/(3b))b^x

Now we use our second point, following a similar process to before.

36=(4/(3b))b^-2

Using properties of exponents, the -2 exponent can become positive and move to the denominator.

36=4/(3b*b²)

The b's can be combined to make a power of 3.

36=4/(3b³)

Now that this equation is nicer, we can use algebra and solve for b.

1. 36*3b³=4
2. 3b³=4/36
3. 3b³=1/9
4. b³=1/27
5. b=1/3

Now we have solved for b, we can refer to the equation in italics that we worked out earlier.

y=(4/(3b))b^x

We can plug b in now. I've used a variety of parenthesis to make the equation a bit easier to make out in plain text form.

y= {4/[3*(1/3)]}*(1/3)^x

We can now simplify...

y= (4/1)(1/3)^x

y=4(1/3)^x should be the answer you come up with. You can punch this equation into a graphing calculator to verify that it does in fact pass through (-2,36) and (1,4/3).

If you have any questions, don't hesitate to ask!