Writing a quadratic function from a graph

0:00- [Tutor] We're told here is 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. Alright, 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?

0:24In 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 minus M times X minus 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 zero, zero times everything else is zero.

0:50Or if X is equal to N, N minus N is zero. So everything else is going to be equal to zero. And 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 could see that right over here.

1:07When X is equal to four, the value of our function is zero. We have an X intercept there. And we also see when X is equal to negative eight, that's the other X intercept. And the function F of negative eight is also equal to zero. So now we could say that F of X is equal to A times, and then we have here X minus negative eight times X minus four.

1:43Now 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 and then we could say, well, what value of A would get us the right Y intercept?

2:01So that F of zero is equal to eight. So let's just multiply this all out. So this is going to be equal to, I'll leave the A here. Now X times X is X squared, X times negative four is negative four X. And then let's see, we're subtracting a negative eight. So let's just make this the same thing as adding a positive eight.

2:28And so then we have eight times X is eight X and then positive eight times negative four is negative 32. And so this is equal to A times X squared minus four X. Sorry, we're going to have minus four X plus eight X. So if we combine these two terms, we're gonna have plus four X, eight X minus four X is positive four X and then we have minus 32.

2:59Now we can multiply everything by A, this is going to be equal to A, X squared plus four AX and then minus 32A. This is what F of X is going to be equal to. Now what would F of zero be equal to? Well, if I take F of zero, all of these terms with an X in it, A times zero squared is zero, four A times zero is zero.

3:25So we would just be left with this negative 32A, negative 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 negative 32A. But we also know when F is or when X is equal to zero.

3:43F of zero is eight. We see that from the graph. So this is equal to eight. And so we could use this to solve four A. Divide both sides by negative 32. Let's do that. Negative 32, negative 32, and we get A is equal to eight over 32 is the same thing as one over four.

4:06So this is negative one fourth. So now we can substitute that back into this expression or this equation I guess is we could say, and we get F of X is equal to AX squared. We now know that A is negative one fourth, so negative one fourth X squared. And then we have plus four times A times X, well four times negative one fourth is negative one.

4:34So this is all going to become negative one times X. So this is gonna be minus X. And then last but not least, negative 32 times negative one fourth is going to be equal to positive eight. And we can verify again that when X is equals, when X equals zero, our function is indeed equal to eight, but we are done.

4:59We have written this equation now in standard form. We started with general factor 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.