Comparing features of quadratic functions

so we're asked which function has the greater y-intercept so the y-intercept is the y coordinate when X is equal to 0 so f of 0 when X is equal to 0 the function is equal to you'll see F of 0 is going to be equal to 0 minus 0 plus 4 is going to be equal to 4 so this function right over here it has a y-intercept of 4 so it would intersect the y axis right over there well 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 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 then G of X does let's do a few more of these where we're comparing different functions one of them that's this in a visual has a visual depiction and one of them were just given the equation how many roots do the functions have in common well G of X we can see it there roots the roots are x equals negative 1 and x is equal to 2 so at 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 tap 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 I get one because that's the coefficient here or implicitly there and if I take their product I get negative 6 well they're going to have to have different signs since their product is negative so let's see negative 3 and positive 2 no they actually the other way around because it's positive 1 so positive 3 and negative 2 so this is equal to X plus 3 times X minus 2 so f of X is going to have zeros when X is equal to negative 3 X is equal to negative 3 or X is equal to 2 these are the two zeros if X is equal to negative 3 this this expression becomes 0 0 times anything is 0 if x equals 2 with this expression become zero and zero times anything is zero so F of negative 3 is zero and F of 2 is here these are the zeros of that function so let's see which of these are in common well negative 3 is out here that's not in common x equals 2 is in common so they only have one common zero right over there so how many or 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 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 concave downwards and the key realization is well you know what 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 on the square x squared term if the coefficient is positive you're going to be concave upwards because as X gets suitably far away from 0 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 0 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 this is concave upwards this one is clearly concave downwards they do not have the same concavity so no and this was negative 4x squared minus 108 then it would be concave downwards and we would say yes anyway hopefully you found that interesting