Contenuto principale

## Caratteristiche e forme delle funzioni quadratiche

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# Comparing features of quadratic functions

## Trascrizione del video

- [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.