Formulating quadratic and square-root equations using graphs - Part 1
About this video
The video plots the points (1, -1), (2, -3) and (3, -1) of a quadratic f(x) and uses the symmetry of the points to spot the vertex (2, -3). It writes vertex form y = A(x - 2)^2 - 3, substitutes (1, -1) to find A = 2, and gets f(x) = 2(x - 2)^2 - 3.
You'll be able to
- identify the vertex of a parabola from symmetric points in a table or graph.
- write a quadratic in vertex form y = A(x - h)^2 + k.
- use one extra point to find the value of A.
Key words
- vertex
- The turning point of a parabola, its lowest or highest point.
- vertex form
- y = A(x - h)^2 + k, where (h, k) is the vertex.
- symmetry
- A parabola is a mirror image on either side of a vertical line through its vertex.
Transcript
- [Instructor] We're told the table shows some points on a quadratic function, F of X. All right, there we go. Graph F of X and use the graph to write its equation. All right, so let me just graph it. Maybe I'll graph it in this white area here. So let me make an X axis there in purple.
It's a nice change of pace. So that's our X axis, that is our Y axis, and all of these are negative values, so actually, let me make my X axis a little bit higher, since I'm operating in the third and fourth quadrants, primarily, at least the part that I'm gonna graph.
All right, that is my Y axis. Let's see, I'm going one, two, three, so, and actually, I could put my Y axis here. I'm primarily operating in the fourth quadrant, so this is Y, and then we have one, two, three, maybe I'll go to four, so X is one, two, three, four, and then I should go up to -3 in the Y direction, negative one, negative two, and then negative three, and now let's plot the points.
I have one, comma, negative one. One, comma, negative one is right over there, this is negative one right over there, and then I have two, comma, negative three, two, comma, negative one, negative one, negative two, negative three. Right over there, that is negative three, and then last but not least, I have three, comma, negative one.
Three, comma, negative one is right over there. Now, there's a couple of things. They're telling us this is a quadratic, and we wanna write the graph, and if we know what the vertex is in one other point, then we're going to be able to, or we need to be able to write its equation, I should say, and if we know the vertex in one other point, we're gonna be able to know the equation.
Now, to identify the vertex, this point right over here, two, comma, negative three, looks like the vertex, and the reason why we can feel good about it is we have this symmetry about it. You move one to the right of that, you have the same Y value as if you go to one to the left of it, so you can see, as you would expect around the vertex, you can see this symmetry happening, so that's why I can feel good that this right over here is indeed the vertex, and I can graph it so that that looks like the vertex.
It's gonna look a little something like this. I don't know where the actual zeros are, so forgive me if I'm not intersecting the X axis at exactly the right place, but it's gonna look something like that, and now I can use the information here, I can use the vertex and one other point, in order to write the equation.
We know, in general vertex form, that Y is equal to some constant times X minus the X value of the vertex squared plus the Y value of the vertex. Well, we know what H and K are, they're two, comma, negative three, so we know that Y is equal to A times X minus two.
Let me do that in that blue color, just so you would see where I'm getting things from, X minus two squared and then plus K, but K is a negative three, so I'm just going to write minus three there. So how do we solve for A? Well, we can just input one of the other points here, and one, comma, negative one's a little bit simpler, so let's do that, so when X is one, Y is negative one, and now we can just solve for A.
What happens inside, I'll go to a new color here, so if I have one minus two, that's negative one squared, so all of this just becomes one, which is a nice simplification, and so we have negative one is equal to A times one, which is just A minus three. We can add three to both sides, so add three to both sides, and what we're going to get is, and I'll just go up here, you're going to get negative one plus three is two, and then that's going to be equal to A, and so now we know enough to write the equation.
We can write Y is equal to two times X minus two squared minus three, and if you want, you could verify that these points indeed do sit on, or if all of these points satisfy this equation right over here, and so we are done, and we could also write this as F of X is equal to all of this stuff.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.