This is the line about which the parabola is symmetrical.

That is the graph is a mirror image on either side of this axis of symmetry.

Now in general we can graph a parabola by plotting its vertex and a few points on

either side of it. But how do we find the vertex? Well, we

have the following vertex form for the equation of a quadratic function.

f(x) = a(x - h)^2 + k, where (h, k) is the vertex.

And if a > 0 , then the parabola will open upward like the second figure here.

And if a<0, then the parabola will open downward like the fist figure.

But how do we get a quadratic function that's in standard form like ours here,

into vertex form? Well, we complete the square, so let's do that.

We have f(x) = x^2 - 8x + 15. Now remember that the first step in

completing the square is to make sure that the coefficient of the square term

is 1, which it is here. And then we take 1/2 the coefficient of x

Which in this case is, -8. Which gives us -4.

And then we square this number. (-4)^2 is 16.

And then we add and subtract this number. That is, this is equal to x^2 - 8x + 16 -

16 and then we still have the + 15. Now these first three terms will form a

perfect square, namely (x - 4)^2. And then -16 + 15 is -1.

And so we've put our function, into vertex form.

So we have f(x) = (x - 4)^2 - 1. And therefore, our a = 1, our h, is equal

to 4, and our k, is equal to -1. And since a here, is greater to zero, our

parabola will be opening upward, and our vertex is at 4 -1.

So let's plot our vertex. Let's say that this is the y axis, and

this is the x axis. And let's say that this is 1, 2, 3, 4,

and that this is -1. And our vertex is here at (4, -1).

Now let's find some other points in our parabola.

So x and y. When x is equal to 2, for example, we can

plug that value into this form here, and we get that y is equal to what? 2 - 4 is

-2. -2^2 is 4 and 4 - 1 is 3.

So when x is 2, y will be 3. What about when x is 3? We get 3 - 4 =

-1, (-1 )^2 = 1. And then 1 - 1 = 0.

So when x = 3, y = 0. And when x = 5, we have 5 - 4 = 1, 1^2 =

1. And 1 - 1 = 0.

And then when x is equal to 6, we have 6 - 4, which is 2.

2^2 is 4. And 4 - 1 is 3.

So let's plot these points. So this is 1, 2, 3.

So (2, 3) will be here. (3, 0) will be here.

(5, 0) will be here. And (6, 3) will be here.

And our parabola will look like this. So this is the sketch that we're looking

for, however, let's notice a few things. Let's draw our line of symmetry here,

that is x is equal to 4. And doing this we can see that this graph

is symmetric about that axis. Or the mirror image on either side of it.

And so looking back over here in the table, we really only needed to determine

these 2 points here. That is, this point over here on the

graph (2, 3) and this point over here (3, 0).

Because we get the other 2 by symmetry, don't we? (5, 0), and (6, 3).

So using this axis of symmetry, is very useful when sketching parabolas.

Thank you, and we'll see you next time. [SOUND]