If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Finding features of quadratic functions

## Video transcript

so I have three different functions here I know they're all called F but we're let's just assume they are different functions and 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 equals zero 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 finally I want to find the equation of the line of symmetry line of symmetry and in particular to make it a little more specific the vertical line of symmetry which would like to be the only line of symmetry for for these three so pause the video and see if you could figure out the zeros the vertex in the line of symmetry and I'm assuming you just did that and now I'm going to attempt to do it if any point you get inspired a 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 so to find the zeros we can set T minus five squared minus nine equal to zero so we could say T for what T's does t minus five squared minus nine equals zero 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 minus five squared the right hand side is going to be nine and so if t minus five squared is nine that means that t minus five could be equal to the positive square root of nine or t minus five could equal the negative square root of nine and to solve for t we could add five to both sides so we get T is equal to eight or T is equal to if we add five to both sides here T is equal to two and just like that we have found the zeros for this function because if T is equal to eight or two the function is going to be zero f of 8 is zero and F of and F of two 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 of size 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 is going to be halfway in between where the where the the parabola in this case is going to intersect the x axis or the T axis that 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 plus 2 over 2 that's 10 over 2 that's 5 so the T coordinate is 5 and 5 is 3 away from 8 and 3 away from 2 and when T is equal to 5 what is f of T or what is f of 5 well when T is equal to 5 5 minus 5 squared is just 0 and then F of so f of 5 is just going to be negative 9 and this form of a function this is actually called vertex form because it's very easy to pick out the vertex it's very easy to realize like ok look for this particular one we're going to hit a minimum point when this part of the expression is equal to 0 because this thing at the lowest value can take on is 0 because you're squaring it it can never take on a negative value and it takes on 0 when T is equal to 5 and with that's if this part is 0 then the F of 5 is going to be negative 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 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 after I keep reminding myself and that is my let's call that my Y axis and we're going to graph y is equal to f of T well we know the vertex is at the point 5 comma negative 9 so this is T is equal to 5 and Y is equal to negative 9 so that's the vertex right over there and then we know we have zeros at T equals 8 and T equals 2 so T equals 8 and T equals 2 let me make that a little bit and T equals 2 those are the two zeros so eight and two two and so just like that we can graph f of T or we could graph y is equal to f of T so the y is equal to f of t is going to look something like 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 five 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 plus two times X plus four is equal to zero well that's going to happen if X plus two is equal to zero or X plus four is equal to zero this is going to happen if we subtract two from both sides when X is negative two and if we subtract four from both sides or when X is equal to negative four 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 negative two plus negative four over two so that would be negative six over two which is just negative three negative 3 and when X is negative three f of X is going to be let's see it's going to be negative one times x one right negative through negative three plus two is negative one negative and so times one so it's just going to be negative one there you have it and the line of symmetry is going to be the vertical line X is equal to negative three and once again we can 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 equals negative 2 and x is equal to negative 4 so negative 1 2 3 4 so we have zeros we have zeros there negative 2 let me be careful negative 2 and negative 4 and the vertex is at negative 3 comma negative 1 so negative 3 comma negative 1 make sure we see that so this is negative 1 right over here negative 1 this is negative 2 this is negative 4 and so we can sketch out sketch out what the graph of y is equal to f of X is going to look like it's going to look something 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 x squared plus 6x plus 8 is equal to 0 it will be useful to factor this and so 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 6 and when you take their product as 8 well 4 + 2 4 plus 2 is 6 and 4 times 2 is 8 so is equal to 0 and then this is actually the exact same thing is what we have in blue right over here these are actually the exact same function they're just written in different forms and so 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 this these functions were just written in different ways