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Given the points (1,5) and (2,7) determine the (a) linear function, (b) exponential function and (c) power function they determine.

(a) $2 x+3$(b) $\frac{25}{7}\left(\frac{7}{5}\right)^{x}$(c) $y=5 x^{0.485427}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 2

Exponential Functions

Missouri State University

Idaho State University

Lectures

03:26

Given the points (2,6) and…

01:38

Find a formula for the exp…

02:20

Write an exponential funct…

00:29

Find an exponential functi…

02:52

Determine if the given fun…

01:29

Determine whether the equa…

07:11

(A) Find the linear functi…

01:48

04:29

So if we want to find the equations for these three functions, given these two points, um, let's go and start with the line and then work our way to the right. So first we need to find our slope. And remember, Slope is supposed to be y tu minus y one over x two minus x one. So this is X one y one x two y two. So this is going to be seven minus five over two minus one, which is just going to give us to over one or two. So we know em is to and then for us to solve for B, we could just go ahead and used point slope form. So would be why minus y one is used to em x minus X one. And it doesn't matter which point we really use. We could plug either one it, so I'll just plug in one in five b Y minus five is equal to two x minus one. Distribute the to add the five. So we get Y is equal to X minus two plus five. So I get why is you get a two X plus three. So this will be our line now, In order for us to get our other equations, we'll do pretty much the same thing. So let's first just go out and plug in one and five. Um, so if I plug in 15 I guess five is equal to, um, a Times B to the first power. And if I plug in to seven, that's going to give seven is equal to a E squared. So now we could solve for a But what I like to do instead is just divide each of these equations, so I'm just going to bite them and then notice we get 7. 50 is equal to the A's. Cancel out that we have beast. Word over B, which is just be so be, is seven tips. And then if we come over here and plug that in, we get Y is equal to a 7/5 raised to the X and now to solve for a we can just plug in one of those points so I'll just plug in one. Um, and doing that would give us five is equal to a race 71 and then that just 7 50 multiplied by five. Divide by seven, so that would give A is equal to 25 7. So we end up with y is equal to 25 7 and then seven Phipps, raise the X. And so this is our exponential equation. And now over here to solve for this one, uh, we'll do the same thing that we did in part. So we plug in 15 So that is going to give. Why is he going to or not? Why five five is equal to x one to the P and, well, one to the P. It's just going to be one. So that just tells us five decay. And now we can go ahead and plug that in and then just use the second point to solve for peace. We'd have seven is five times to raise the Pete, so we would go ahead and divide by five. So we get 7/5 as you go to to to the P. And now, since this isn't a proper power of two, what we can do instead is just take any longer rhythm on each side. I just like to use the natural algorithm. So that's what I'm gonna do. Um and this is going to give us. We're actually very outlets do log base two. So this would be log base two on each side. And the reason why I want to do this is because log base two in the exponential to our inverse is so the right side is just p. So that's what we have for Pete and that we already have ks five. So now we can just say, Okay, why is equal to five times two raised to the wog face? I hope not to x x rays to the blog base too. Oh, um 7/5. And then this is our power function.

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