yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Analyzing graphs of exponential functions: negative initial value | High School Math | Khan Academy


3m read
·Nov 11, 2024

So we have a graph here of the function ( f(x) ) and I'm telling you right now that ( f(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—when does ( f(x) ) equal -125?

You might be tempted to just eyeball it over here, but when ( f(x) ) is -125, that's like right below the x-axis. So if I tried to eyeball it, it would be very difficult. It's very difficult to tell what value that is; it might be at 3, it might be at 4, I am not sure. So instead of actually—well, maybe I don't want to just eyeball it and guess it—instead, I'm going to actually find an expression that defines ( f(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(x) ) is an exponential function, we know it's going to take the form ( f(x) = a \cdot r^x ). Well, the initial value is straightforward enough; that's going to be the value that the function takes on when ( x ) is equal to 0. You could even see here if ( x = 0 ), the ( r^x ) would just be 1, and so ( f(0) ) will just be equal to ( a ).

And so what is ( f(0) )? Well, when ( x = 0 ), this essentially—we're saying where does it intersect? Where does it intersect the y-axis? We see ( f(0) = -25 ), so ( a = -25 ). When ( x ) is 0, the ( r^x ) is just 1, so ( f(0) ) is going to be -25; we see that right over there.

Now to figure out the common ratio, there are 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(1) ) and ( f(0) ); that's going to be the common ratio, or the ratio between ( f(2) ) and ( f(1) )—that is going to be the common ratio.

Well, lucky for us, we know ( f(0) = -25 ), and we know that ( f(1) = -5 ). So just like that, we're able to figure out that our common ratio ( r ) is ( -5 / -25 ), which is the same thing as ( 1/5 ). Divide a negative by negative; you get a positive. So you're going ( 5 / 25 ), which is ( 1/5 ).

So now we can write an expression that defines ( f(x) ). ( f(x) ) is going to be equal to ( -25 \cdot (1/5)^x ). And so let's go back to our question: When is this going to be equal to -125?

So when does this equal -125? Well, let's just set them equal to each other. So let—there's a siren outside, I don't know if you hear it—so negative; I'll power through. Alright, negative. So let's see, at what ( x ) value does this expression equal -125?

Let's see, we can multiply—well, actually we want to solve for ( x ). So let's see, let's divide both sides by -25 and so we are going to get ( (1/5)^x = (-125) / (-25) ). This -25 is going to go away, and on the right-hand side, we're going to have—dividing negative by negative, it's going to be positive—it's going to be ( 1 / 5 ).

And ( (1/5)^x ) is the same thing as ( 1^x / 5^x ) is equal to ( 1 / 5 ). So we can see that ( 5^x ) needs to be equal to 625.

So let me write that over here. ( 5^x = 625 ). Now, the best way I could think of doing this is let's just think about our powers of 5. So ( 5^1 = 5 ), ( 5^2 = 25 ), ( 5^3 = 125 ), ( 5^4 = 625 ). So ( x ) is going to be 4, because ( 5^4 = 625 ).

So we can now say that ( f(4) ) is equal to -125. Once again, you can verify that; you can verify that right over here: ( (1/5)^4 = 1 / 625 ). ( -25 / 625 ) is going to be -125.

So hopefully that clears things up a little bit.

More Articles

View All
What Do You Need to Do to Survive Coronavirus? | Ask Mr. Wonderful #22 Kevin O'Leary
[Music] Hi, everybody. I started to do some work on this week’s Ask Mr. Wonderful and went into the database to start listening to the questions from all around the world. Clearly, this is an extraordinary situation because everybody is concerned about t…
HE'S FOLLOWING YOU! -- DONG!
Hey, Vsauce. Michael here, and if you need more DONG, don’t worry, because I’m bringing you more things you could do online now, guys. “JaydenMarkAnderson” recommended Clubcreate, where you can make your own remixes right inside your browser. As we liste…
Tiger Sharks' Superpowered Jaws | SharkFest | National Geographic
Tiger sharks are one of the largest predatory sharks on the planet. They feed off an extensive menu: whales, birds, even other sharks. But there’s one delicacy that takes more effort than others. Turtles! So how much jaw power does it take to crunch throu…
Refraction in a glass of water | Waves | Middle school physics | Khan Academy
So, something very interesting is clearly going on when we look at this pencil dipped in this cup of water. We would expect if maybe there was no water in this glass that we would just see the pencil continue straight down in a line that looks something l…
Food and energy in organisms | Middle school biology | Khan Academy
Hey, quick question for you. You ever look at a person’s baby pictures and wonder how people go from being small to, well, big? I mean, yes, I get it; people grow up, but here I’m thinking more on the level of the atoms and molecules that make up the body…
Digital SAT Prep for School Districts - Khan Academy Districts
Hello and welcome to driving digital SAT success with Khan Academy! As teachers and students are navigating through the new digital SAT assessment this spring, we know how important it is to ensure your students are ready for the big day. My name is Eliza…