Finding features of quadratic functions | Mathematics II | High School Math | Khan Academy

So I have three different functions here. I know they're all called f, but we'll just assume they are different functions. For each of these, I want to do three things. I want to find the zeros, and so the zeros are the input values that make the value of the function equal to zero. So here would be the T values that make F of t equal 0. Here would be the X values that make the function equal zero.

So I want to find the zeros. I also want to find the coordinates of the vertex, and I want to find the equation of the line of symmetry. The line of symmetry, and in particular, to make it a little bit more specific, the vertical line of symmetry, which will actually be the only line of symmetry for these three.

So pause the video and see if you could figure out the zeros, the vertex, and the line of symmetry. I'm assuming you just did that, and now I'm going to attempt to do it. If at any point you get inspired, pause the video again and keep on working on it. The best way to learn this stuff is to do it yourself.

So let's see. So let's first find the zeros. To find the zeros, we can set t - 5² - 9 equal to 0. So we could say, for what T's does t - 5² - 9 equal 0? Let's see. To solve this, we could add nine to both sides, and so we could say if we add nine to both sides, the left-hand side is just T - 5², the right-hand side is going to be nine.

So if t - 5 squared is 9, that means that t - 5 could be equal to the positive square root of 9 or t - 5 could equal the negative square root of 9. To solve for T, we could add five to both sides, so we get T is equal to 8 or t is equal to, if we add five to both sides here, T is equal to 2.

Just like that, we have found the zeros for this function because if T is equal to 8 or 2, the function is going to be zero. F of 8 is zero, and F of 2 is going to be zero. Now let's find the vertex, the coordinates of the vertex. The coordinates of the vertex, so the x-coordinate of the vertex—sorry, I should say the T coordinate of the vertex since the input variable here is T.

The T coordinate of the vertex is going to be halfway in between the zeros. It's going to be halfway between where the parabola, in this case, is going to intersect the T-axis. I keep saying x-axis, the T-axis for this case. So halfway between 8 and 2? Well, it's going to be the average of them: (8 + 2) / 2, that's 10 / 2, that's 5.

So the T coordinate is five, and five is three away from eight and three away from two. When T is equal to five, what is f of t, or what is f of five? Well, when T is equal to 5, 5 - 5² is just zero, and then F of five is just going to be -9.

This form of a function is actually called vertex form because it's very easy to pick out the vertex. It's very easy to realize, like, okay, look, for this particular one, we're going to hit a minimum point when this part of the expression is equal to zero because this thing, at the lowest value it can take on, is zero because you're squaring it. It can never take on a negative value, and it takes on zero when T is equal to five.

If this part is zero, then F of five is going to be -9. So just like that, we have established the vertex. Now we actually have a lot of information if we wanted to draw it. So if we want to draw this function—and I'll just do a very quick sketch of it—whoops! So a very quick sketch of it. So that is our T-axis, not our x-axis. I have to keep reminding myself, and that is my... let's call that my y-axis.

We're going to graph y is equal to F of T. Well, we know the vertex is at the point (5, -9). So this is T is equal to 5, and Y is equal to -9, so that's the vertex right over there. Then we know we have zeros at T = 8 and T = 2. So let me make that a little bit clearer. T = 8 and T = 2, those are the two zeros. So just like that, we can graph F of T, or we can graph y is equal to F of T.

So y is equal to F of T is going to look something like... let me draw something like that. That's the graph of y is equal to F of T. Now the last thing that I said is the line of symmetry. Well, the line of symmetry is going to be the vertical line that goes through the vertex. So the equation of that line of symmetry is going to be T is equal to 5, and it's really just the T coordinate of the vertex that defines the line of symmetry.

Let's do the other two right over here. So what are the zeros? Well, if you set this equal to zero, if we say x + 2 * (x + 4) is equal to 0, well, that's going to happen if x + 2 is equal to 0 or x + 4 is equal to 0. This is going to happen if we subtract two from both sides when X is -2 and if we subtract four from both sides or when X is equal to -4.

As we said, the vertex, the x-coordinate of the vertex is going to be halfway in between these, so it's going to be (-2 + -4) / 2. So that would be -6 / 2, which is just -3. When X is -3, f of x is going to be, let's see, it's going to be -1 * 1. Right? -3 + 2 is 1, neg, and so times one, so it's just going to be negative 1. There you have it.

The line of symmetry is going to be the vertical line X is equal to -3, and once again we can graph that really fast. So let me... this is my y-axis, see everything is happening for negative X's, so I'll draw it a little bit more skewed this way. This is my x-axis, and we see that we have zeros at x = -2 and x = -4.

So NE 1, 2, 3, 4. So we have zeros; we have zeros there: x = -2, let me be careful… -2 and -4. The vertex is at (-3, -1), so -3, -1. Make sure we see that. This is -1 right over here, this is -2, this is -4.

We can sketch out what the graph of y is equal to F of x is going to look like. It's going to look something like that; that is y is equal to F of x. Let's do one more. Hopefully, we're getting the hang of this. So here, to solve x² + 6x + 8 is equal to 0, it will be useful to factor this.

This can be written as... and if you have trouble doing this, I encourage you to watch videos on factoring polynomials. What adds up to six and when you take their product is eight? Well, four and two: 4 + 2 is 6, and 4 * 2 is 8. So is equal to zero.

Then this is actually the exact same thing as what we have blue right over here. These are actually the exact same function; they're just written in different forms. The solutions are going to be the exact solutions that we just saw right over here, and the graph is going to be the same thing that we have right over there. So same vertex, same line of symmetry, same zeros. These functions were just written in different ways.