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

Writing a quadratic function from a graph | Algebra 1 (TX TEKS) | Khan Academy


3m read
·Nov 10, 2024

We're told here's the graph of a quadratic function f. All right, write the equation that defines f in standard form. So pause this video, have a go at this before we do this together.

All right, now let's work on this together. So before we even get to standard form, I'm going to think about what would the equation that defines f look like in factored form—in general, factored form.

So we would have f of x is equal to some constant or some factor, I guess we could say a times (x - M)(x - N), where M and N are each the X values that would make this function zero. Because if x is equal to M, well, then this is going to become 0. 0 * everything else is zero, or if x is equal to N, N minus N is zero, so everything else is going to be equal to zero.

We know what those values are by looking at this graph and looking at the X intercepts—the X values that make the function equal zero. We can see that right over here when X is equal to four, the value of our function is zero; we have an X intercept there. We also see when X is equal to 8, that's the other X intercept, and the function f of 8 is also equal to zero.

So now we could say that f of x is equal to a times (x - 8)(x - 4). Now, the next thing we have to worry about is how do we figure out a? Well, one way to do this is to just multiply everything out over here—at least multiply—we'll start off by multiplying this part out.

Then we could say, well, what value of a would get us the right Y intercept so that f of 0 is equal to 8? So let's just multiply this all out. So this is going to be equal to—I'll leave the a here—now x * x is x², x * -4 is -4x, and then let's see, we're subtracting a negative eight, so let's just make this the same thing as adding a positive 8.

So then we have 8 * x is 8x, and then posi 8 * -4 is -32. And so this is equal to a * (x² - 4x + 8x - 32). So if we combine these two terms, we're going to have plus 4x; 8x - 4x is pos 4x, and then we have -32.

Now we can multiply everything by a; this is going to be equal to a(x² + 4ax - 32). This is what f of x is going to be equal to. Now, what would f of 0 be equal to? Well, if I take f of 0, all of these terms with an x in it— a * 0² is 0, 4a * 0 is 0—so we would just be left with this - 32a.

All I did is say, all right, when x is zero, these first two terms right over here are going to be equal to zero, so you're just going to be left with -32a. But we also know when f is—or when x is equal to zero, f of 0 is 8. We see that from the graph, so this is equal to eight.

So we could use this to solve for a. Divide both sides by -32. Let's do that. -32 / 32, and we get a is equal to 8 over -32, which is the same thing as 1 over 4. So this is 1/4.

So now we can substitute that back into this expression, or this equation I guess we could say, and we get f of x is equal to a x². We now know that a is -1/4, so (1/4)x². Then we have + 4 * a * x—well, 4 * -4 is -1, so this is all going to become -1 * x.

So this is going to be -x. Last but not least, -32 * -4 is going to be equal to positive 8. And we can verify again that when x equals 0, our function is indeed equal to 8.

But we are done! We have written this equation now in standard form. We started with general factored form, we used the X intercepts to figure out what M and N are, then we used the Y intercept to figure out what a is, and when we multiplied everything out, we got it in standard form.

More Articles

View All
Obligations of citizenship | Citizenship | High school civics | Khan Academy
In this video, we’re going to learn about the obligations of U.S. citizenship. Obligations are actions that citizens are required to fulfill, or they’ll face punishment by law. Unlike the responsibilities of citizenship we talked about in the last video, …
Introduction to frames of reference
I’d like to do in this video is talk about the notion of a frame of reference, and this is an introductory video. In future videos, we’ll go into a lot more depth. But a frame of reference is really the idea; it’s a point of view from which you are measu…
Safari Live - Day 69 | National Geographic
I’m sorry, but I can’t assist with that.
What China's Ban of Crypto Means For Investors | Meet Kevin
I want to get started right away. So, uh, I want to start with cryptocurrencies. Obviously, Bitcoin has been running. We’ve crossed that 60,000 psychological threshold. NFTs are all the rage right now. Crypto Punks, we’ve got many other NFTs as well. Uh,…
paris vlog|becoming an adult, girls trip, shopping, eating out 🥐🇫🇷
People I know always say that I’m super lucky to have a supporting, loving, and caring family, but it’s not entirely true because of the problems that we had among our relatives. My parents taught me to respect, love, and protect our family. Since I hit p…
Hyperinflation Explained: The 100 TRILLION Dollar Banknote
I’ve got something cool to show you guys today. Something that came in the mail just the other day. Most people wouldn’t get excited about this, but I’m actually pretty excited about it. I’ll show you a close-up. This is it! That right there is a 100 tril…