Rewriting a quadratic function from vertex form to standard form
About this video
The video rewrites the function f(x) = -2(x - 5)^2 + 7 from vertex form into standard form ax^2 + bx + c. It expands (x - 5)^2 to x^2 - 10x + 25, distributes -2 carefully with the signs, and adds 7 to get f(x) = -2x^2 + 20x - 43.
You'll be able to
- describe standard form as f(x) = ax^2 + bx + c.
- expand the square of a binomial such as (x - 5)^2.
- distribute a negative number and combine constants to rewrite a vertex-form function in standard form.
Key words
- vertex form
- A quadratic written like a(x - h)^2 + k, which shows the vertex.
- standard form
- A quadratic written as ax^2 + bx + c, where a, b and c are constants.
- binomial
- An expression with two terms, like x - 5.
- distribute
- Multiply a number by every term inside the brackets.
Transcript
- [Instructor] So what I have right over here is the equation of a function in vertex form. What I wanna do is rewrite it so it is in standard form, so pause this video and have a go at that before we do it together. All right, let's just remind ourselves what standard form looks like.
Standard form would be a function f of x in the form, some constant, a, times our second degree term, plus another constant, b, times our first degree term, plus another constant, c. So how do we put this thing into this form? Well, we essentially will expand things out.
So we could start with x minus five squared. So we're gonna multiply x minus five times X minus five, then we're going to take that, multiply it by -2 and then add seven. So first of all, what is x minus five times x minus five? Some of y'all might already have a lot of practice doing this by taking the square of binomials, but I'll give a little bit of review.
We have x times x here, which is going to be x squared, and then you have -5 times x, which would be a -5x. I could write that out -5x. And then you have another -5 times, you have another -5 times this x, which is going to be another -5x. And then, you're going to have a -5 times a -5 is plus 25, or we could write this as x squared minus 10x plus 25.
And so, that's what I have boxed off here. Let me put that in parentheses, and then, I'm going to multiply that by -2, and then I'm going to add seven. So all of this is going to be equal to f of x. So what's this going to be equal to? Well, let's take our -2 and distribute it onto each of these terms, so -2 times x squared.
We have to be very careful here, make sure we get our signs right, is -2x squared.
- 2 times -10x, negative times a negative is a positive, positive 20x and then, -2 times positive 25 is -50, or we're gonna subtract 50 there. And then, last but not least, we want to add seven, and so all of this, f of x, is going to be equal to -2x squared plus 20x.
And then, if I take -50 or -50 plus seven, it's -43, or seven minus 50 is -43. And we're done, we have rewritten the function in standard form.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.