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Comparing features of quadratic functions

Sal compares the y-intercepts, the zeros, and the concavity of quadratic functions given graphically and algebraically.

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Video transcript

- [Voiceover] So we're asked which function has the greater y-intercept? So, the y-intercept is the y-coordinate when x is equal to zero. So, f of zero, when x is equal to zero, the function is equal to, let's see, f of zero is going to be equal to zero minus zero, plus four, is going to be equal to four. So this function right over here, it has a y-intercept of four. So 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, its y-intercept is right over here, at y is equal to three. 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 their roots. The roots are, x equals negative one and x is equal to two. So these two functions, at most, are going to have two roots in common, because this 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 a zero. I'll do the first way, I'll try to factor this. So let's see, what two numbers, if I add them I get one, 'cause that's the coefficient here, or implicitly there. And if I take the product, I get negative six. Well, their gonna have to have different signs since their product is negative. So, let's see, negative three and positive two. No, actually, the other way around 'cause it's positive one. So positive three, and negative two. So this is equal to x plus three, times x minus two. So f of x is going to have zeros when x is equal to negative three. X is equal to negative three. Or, x is equal to two. These are the two zeros. If x is equal to negative three, this expression becomes zero. Zero times anything is zero. If x equals two, this expression becomes zero, and zero times anything is zero. So f of negative three 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, negative three is out here, that's not in common. X equals two 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?" And 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. And the key realization is, well, 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?" And the key here is the coefficient on the second degree term, on the x squared term. If the coefficient is positive, you're going to be concave upwards, because 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, we're not even further away from zero, as x gets further and further away from the vertex, as x gets further and further away from the vertex, this term dominates everything else, and we get more and more positive values. And so that's why if your coefficient is positive, you're going to have concave upwards, a concave upwards graph. And so if this is concave upwards, this one is clearly concave downwards. They do not have the same concavity, so no. If this was negative four x squared minus 108, then it would be concave downwards and we would say yes. Anyway, hopefully you found that interesting.