Define the quadratic function ƒ having x-intercepts (2, 0) and (5...

Question

Define the quadratic function ƒ having x-intercepts (2, 0) and (5, 0) and y-intercept
(0, 5).

Accepted Answer

Hey, everyone in this problem, we're asked to find a quadratic function that satisfies the following conditions. We have X intercepts four comma zero and seven comma zero and A Y intercept at zero comma four or given four answer choices. Option A F of X is equal to X squared divided by two minus 11 X divided by two plus 14. Option B F of X is equal to X squared divided by two minus 11 X divided by two minus 14. Option C F of X is equal to X squared divided by two plus 11, X divided by two plus 14. And option D F of X is equal to X squared minus X plus 14. Now we're asked to find a quadratic function and we're given the Y intercepts. So we're called that a general quadratic function written in factored form can be written as Y is equal to a multiplied by X minus P multiplied by X minus Q. Now I've written it in factored form because we're given the X intercepts. OK. So if Y is equal to zero, then either X minus P is equal to zero or X minus Q is equal to zero. And we know that those have to equal four. And so, and we know that P and Q are the roots of this polynomial. And so right off the bat, we can write this as Y is equal to a multiplied by X minus four, multiplied by X minus seven. And again, if you imagine setting Y equals to zero zero, on the left hand side, I mean, the right hand side must equal zero. We have factors multiplied together. So if either one of them is zero, then the entire expression will be zero. If we set X minus four, equal to zero, we get X is equal to four which corresponds with the first X intercept. And if we were to set X minus seven equal to zero, we would get X equals seven which corresponds to the second X intercept. OK. So when we write our equation in factored form like this, we can just go straight to those zeros P and Q and that's why now what we need to do is use our Y intercept to find this value of A. So let's substitute zero comma 14. OK. So the Y value is going to be 14. This is going to equal to a multiplied by zero minus four, multiplied by zero minus seven. So we get 14 and S equal to a multiplied by negative four, multiplied by negative seven. This tells us that 14 is equal to 28 A and we can divide both sides by 28. We get that A is equal to one half. So A is equal to one half, which means we can write our function as Y is equal to one half multiplied by X minus four, multiplied by X minus seven. Now, we've figured out everything we need here. We've got our A values, we've got these P and Q values from our zeros. If we look at the answer choices though, we can see that these are expanded and they aren't in factor form. So we need to go ahead and expand this function. So let's do that. We get Y is equal to one half multiplied. No, you can use your foil method or any other method you'd like. And we have X multiplied by X will give us X squared X multiplied by negative seven will give us negative seven, X, negative four, multiplied by X gives us negative four, X and negative four multiplied by negative seven gives us plus 28. OK. So we have Y is equal to one half multiplied by X squared minus 11 X plus 28. OK. Just simplifying those like terms and we can distribute that one half. We get X is equal to or sorry, Y is equal to one half X squared minus 11 X over two plus 14. And that is the quadratic function we were looking for if we compare this with our answer choices. OK? Again, we found the correct answer is that Y or F of X is equal to X squared divided by two minus 11, X divided by two plus 14, which corresponds with answer choice. A. Now, before I wrap this one up, I want to make one quick note, we were able to use this form of the function. This factored form Y equals a multiplied by X minus P multiplied by X minus Q because we were given the X intercepts, OK. If you were just given two random points, you would still be able to do this problem. But you wouldn't be able to start with your factored form or it wouldn't be as easy to start with your factored form. You'd want to start with just the general equation of a quadratic Y equals A X squared plus B X plus C. OK? You'd need to substitute those points in and then use the substitution method between two different equations. OK. So it does get messier. And so that's why we used this form of the equation because we were given the X intercepts. Thanks everyone for watching. I hope this video helped see you in the next one.

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Define the quadratic function ƒ having x-intercepts (2, 0) and (5, 0) and y-intercept (0, 5).