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

So we're asked which function has the greater Y intercept.

The Y intercept is the y-coordinate when X is equal to zero. So F of 0, when X is equal to zero, the function is equal to, let's see, F of 0 is going to be equal to 0 - 0 + 4, is going to be equal to 4. So this function right over here has a y-intercept of four. It would intersect the Y axis right over there.

While the function that we're comparing it to, G of X, we're looking at its graph. Y is equal to G of x, and its Y intercept is right over here at Y is equal to 3. So which function has a greater Y intercept? Well, it's going to be f of x. F of x has a greater Y intercept than G of X does.

Let's do a few more of these where we're comparing different functions, one of them that has a visual depiction and one of them where we're just given the equation. How many roots do the functions have in common? Well, G of X, we can see it. Their roots are x = -1 and x = 2. So at these two functions, at most, are going to have two roots in common because G of X only has two roots.

There's a couple of ways we could tackle it. We could just try to find F's roots, or we could plug in either one of these values and see if it makes the function equal to zero. I'll do the first way; I'll try to factor this. So let's see, what two numbers, if I add them, do I get one? Because that's the coefficient here, or implicitly there.

And if I take their product, I get -6. Well, they're going to have to have different signs since their product is negative. So let's see, -3 and positive 2. No, actually the other way around because it's positive 1. So positive 3 and -2. So this is equal to x + 3 * x - 2. So f of x is going to have zeros when x is equal to -3 or x is equal to 2.

These are the two zeros; if x is equal to 3, this expression becomes 0. 0 * anything is 0. If x is 2, this expression becomes 0, and 0 * anything is 0. So F of -3 is zero and F of two is zero. These are the zeros of that function.

So let's see which of these are in common. Well, -3 is out here; that's not in common. X = 2 is in common, so they only have one common zero right over there. So how many roots do the functions have in common? One.

All right, let's do one more of these, and they ask us, do the functions have the same concavity? The way I think, or one way to think about concavity, is whether it's opening upwards or opening downwards. So this is often viewed as concave upwards and this is viewed as concave downwards—concave downwards.

The key realization is, well, you know, if you just look at this blue, if you look at G of X right over here, it is concave downwards. So the question is, would this be concave downwards or upwards? The key here is the coefficient on the second-degree term, on the square x term. If the coefficient is positive, you're going to be concave upwards.

As X gets suitably far away from zero, this term is going to overpower everything else and it's going to become positive. So, as X gets further and further away, or not even further away from zero, as X gets further and further away from the vertex, this term dominates everything else and we get more and more positive values.

That's why if your coefficient is positive, you're going to have a concave upwards graph. So if this is concave upwards, this one is clearly concave downwards. They do not have the same concavity. So, no, if this was -4x^2 - 108, then it would be concave downwards and we would say yes.

Anyway, hopefully, you found that interesting.