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Analyzing graphs of exponential functions: negative initial value

CCSS Math: HSF.IF.B.4

Video transcript

- [Voiceover] So we have a graph here of the function f of x, and I'm telling you right now that f of x is going to be an exponential function. It looks like one, but it's even nicer when someone tells you that. And our goal in this video is to figure out at what x value, so when does f of x? At what x value is f of x going to be equal to negative 1/25. And you might be tempted to just eyeball it over here, but when f of x is negative 1/25, that's like right below the x-axis. If I try to eyeball it, it would be very difficult. It's very difficult to tell what value that is. It might be at three, it might be at four. I am not sure. Well, I don't want to just eyeball it, just guess it. Instead, I'm gonna actually find an expression that defines f of x because they've given us some information here. And then I can just solve for x, so let's do that. Well, since we know that f of x is an exponential function, we know it's going to take the form f of x is equal to our initial value a times our common ratio r to the xth power. Well, the initial value is straightforward enough. That's going to be the value that the function takes on when x is equal to zero. And you can even see it here, if x is equal to zero, the r to the x would just be one. And so f of zero will just be equal to a. And so what is f of zero? Well, when x is equal to zero, which essentially we're saying, where does it intersect the y-axis? We see f of zero is negative 25. So a is going to be negative 25. When x is zero, the r to the x is just one. So f of zero is going to be negative 25. We see that right over there. Now to figure out the common ratio, there's a couple of ways you could think about it. The common ratio is the ratio between two successive values that are separated by one. What do I mean by that? Well, you could view it as the ratio between f of one and f of zero. That's going to be the common ratio. Or the ratio between f of two and f of one. That is going to be the common ratio. Well, lucky for us, we know f of zero is negative 25. And we know that f of one, when x is equal to one, y, or f of x, or f of one is equal to negative five. And so just like that we're able to figure out that our common ratio r is negative five over negative 25, which is the same thing as 1/5. Divide a negative by a negative, you get a positive. So you get a five over 25, which is 1/5. We can write an expression that defines f of x. f of x is going to be equal to negative 25 times 1/5 to the x power. And so let's go back to our question. When is this going to be equal to negative 1/25? So when is this equal to negative 1/25? Well, let's just set them equal to each other. There's a siren outside. I don't know if you hear it. I'll power through. So let's see at what x value does this expression equal negative 1/25? Let's see, we can multiply. Well, actually, we want to solve for x. So let's see, let's divide both sides by negative 25. And so we are going to get 1/5 to the xth power is equal to, if we divide both sides by negative 25, this negative 25 is gonna go away. And on the right-hand side, we're going to have, divide a negative by a negative is going to be positive. It's going to be 1/625. And 1/5 to the xth power. This is the same thing as one to the xth power over five to the xth power is equal to 1/625. Well, one to the xth power is just going to be equal to one. It doesn't matter that we have this to the xth power over here. I thought I was erasing that with a black color. There you go, that's a black color right over there. So we can see that five to the xth power needs to be equal to 625. So let me write that over here. Whoops, didn't change my color. Five to the xth power needs to be equal to 625. Now the best way I could think of doing this is let's just think about our powers of five. So five to the first power is five. Five squared is 25. Five to the third is 125. Five to the fourth, we'll multiply that by five, you're going to get 625. So x is going to be 4 cause five to the fourth power is 625. So we can now say that f of four is equal to negative 1/25. And once again, you can verify that. You can verify that right over here. 1/5 to the fourth power is gonna be 1/625. Negative 25 over positive 625 is going to be negative 1/25. So hopefully that clears things up a little bit.