In this video, we're going to talk about how we can find an exponential function going through two points. We're going to use the points two, six and five, 48. And we're going to build an exponential model that is y equals a, b. to the x.
Now, what we're gonna do is we're gonna set up a model for each point. You know, this is an x value, this is a y value. So if I plug those in, I get six equals a, b.
to the two. And I'll do the same thing with this point. 48 equals A, B to the five.
And I write them right on top of each other. And you're gonna see why in a second. It doesn't matter which one I put put a quote on top.
In fact, ours might have been easier to do the other way, but that's okay. We're gonna get the same answer. Doesn't matter.
What we have now is we've created two equations with two unknowns, A and B. So we can solve this via substitution or elimination. Now, we've seen with linear equations, we can do elimination by adding or subtracting.
With exponential, because we have a common ratio that makes a function exponential, we do elimination with dividing. So I am gonna divide both sides of the equation. So I have six over 48, which reduces down to 1 8th. And over here, look what happens. Here's my elimination, because the a's cancel.
Now I have b squared over b to the fifth. Remember two things of the same base, their exponents subtract, so that's b to the negative third. So we have enough information now to solve for b. Now, what we would do to solve this is we could, one thing we can do is we can raise both sides to the negative 1 third. So we're gonna raise both sides to the negative 1 third, because over here what that does because that makes that just b, because negative three times 1 3rd is one.
Over here, this is what we can do by hand, but if we had something more difficult that we couldn't, we could either leave our b this way, or we could calculate it with a calculator. Now the negative exponent means we're taking the reciprocal, so this is the same thing as eight to the one, positive 1 3rd, and the cube root of eight is two. We have one that's simplified kind of nice in this case. But that may not always be the case. Our base could be anything.
So let's see what we have for a model now. We have y equals a times two to the x. So we still need to find our y-intercept, basically our initial value.
Now with linear functions, what we do is we find our slope, then we plug in a point to get our y-intercept. Nothing different here. We're gonna plug in either one of our points for x and y.
So I'm gonna use this point. So I plug six in for y. equals a times two squared. So I got six equals a times four. Divide both sides by four, I get a equals six over four.
or three halves. So I now have my exponential model that goes through those two points. Y equals three halves times two to the x.
And we've done this algebraically. You could check this using exponential regression in your calculator, or if you want, graph it and make sure it goes through those two points.