this is s.a.t math section 12. sat math section 12 deals with quadratics quadratics are extremely important for the sat they are extremely common you will see several quadratics questions pop up on every single test and quadratics are tested in many many different ways there are many variations of quadratics questions you will need to know a lot of in-depth detail about quadratics you will need to know how to graph them you will need to know the various forms of a quadratic function of course you will need to know how to spot a quadratic function though so that is where we are going to start we will start with what is a quadratic function so first of all just uh i guess sort of a school type definition here we've got a quadratic function known as a polynomial of degree two has this standard form f of x equals a x squared plus b x plus c almost all of you sophomores juniors seniors at least should be very familiar with that form of a quadratic function as you've been dealing with it for several years by this point however keep in mind that quadratics again they come in many different sizes and shapes all of these equations in fact represent quadratic functions quadratic relationships um in all of these you will notice that you have two variables so for instance you've got a y and an x here here you've got a w and a z even when you see stuff like p of n and n those really are two different variables if you were graphing this p of n function on the x y coordinate plane the n would be your x and the p of n would be your y we should know that f of x notation p of n notation those are really just other ways of talking about the y coordinate when these functions are graphed so all of these equations here have two variables y x w z p of n and n and in every case what we see is that only one of the two variables is squared the other variable is to the first of course the uh to the first power that one power is unwritten but we have one variable to the first and the other variable squared even in cases like this example here h of a if we were to dis distribute the a outside those parentheses there to the terms inside parentheses we would get a squared plus 2a and again the a would be squared h of a which like i said before is like a y or you could just call that an h that is to the first so any time we have an equation or a relationship between two variables where one variable is squared and one variable is to the first that is a quadratic relationship you need to know how to spot when you are dealing with a quadratic relationship now that we have a general sense of how to spot a quadratic situation we do need to know the three specific forms of a quadratic function standard form we are going to write y equals a x squared plus b x plus c again that's the form that i think most students are probably most familiar with that form keep in mind that y is interchangeable with f of x we could say f of x equals all of this stuff over here on the right we could say y equals i'm just going to keep this notation here with y just to make it a little bit more efficient the vertex form of a quadratic function is going to be y equals a x minus h squared plus k keep in mind that occasionally this form in school or in a in a school textbook might be written like this those are obviously exactly the same except that in one case that k has been subtracted from both sides and now ends up over there on the left as a minus k and of course in this case we've added k to both sides and ended up with the positive k over there on the right intercept or factored form that is going to be y equals a parentheses x minus p times x minus q with these forms it is not only imperative that you know these forms that you recognize these forms it will also be extremely imperative that you know why these forms are called what they're called for instance why this is called vertex form why this is called interceptor factored form it will also be important that we know what information we can pull from each of these three forms of a quadratic we're going to go over that in this section in quite a bit of detail what do the graphs of quadratic functions look like again by the time you are a junior or senior or even sophomore you should know that all quadratics when they are graphed will have this typical u shaped curve called a parabola so we can see here f of x and p of n these are two of the functions we saw at the top of the page that were examples of quadratic functions these two functions f and p are graphed and we can see that they both come out to be these typical u-shaped curves that all quadratic functions look like when they are graphed on the x y plane another important part of quadratic functions and their graphs specifically parabolas is knowing what the important features or important points on a parabola are and we will run through those right now points a over here so these two you might even want to pause the video and run through these most of you should know what these points are called already a those two points are of course the points at which the parabola hits the x-axis so those are known as our x-intercepts x-intercepts by the way those are also synonymous with the zeros or roots of the parabola again we will be talking in a lot of detail about the relationship between intercepts and zeros and roots later on in this section b point b of course is where the parabola hits the y axis so b is known as the y intercept points c and d are both what are called vertices or singular vertex each of those is a vertex however point c we're going to go ahead and write down vertex but in parentheses we're going to write minimum point c does represent a vertex and and again all of vertex is is where the parabola goes from being decreasing to increasing or increasing to decreasing it's sort of the turning point however what is most important about points c and d let's actually write down vertex maximum over here what is of vital importance about points c and d is that we understand that points z and d represent those points where the function itself hits its minimum or maximum value we have to remember that the points on the parabola more specifically the y values of those points so point c would have whatever this y value is point d let's say if we use this y axis over here point d would have this y value looks like it's the same as the y intercept of this parabola over here but whatever it is we need to understand that those y values would represent values of the function values of the function that produced the parabola and in the case of point c we would reach the minimum value of whatever function this parabola represents and we would reach in point d the maximum value of whatever function this parabola on the right represents so again vertices are the turning points but they can be either minimums or maximums depending upon the shape or orientation of the parabola parabola opens down or is like a frown then we have a maximum if the parabola opens up if it's a smiley face then we have a minimum and finally we have these vertical lines e you can see that those are two vertical lines that go through the vertices of these parabolas one through point c one through point d those are known as the i'm just going to do this singularly axis of symmetry we should understand that parabolas are always symmetric with respect to the axis of symmetry with respect to the vertical line that goes through the vertex of the parabola what that means it's a couple things that it means the two legs of the parabola so one leg to the left of the axis of symmetry one leg to the right of the axis of symmetry those legs are mirror images of each other across the mirror that is the axis of symmetry we also should understand that if we were to graph two points that have the same y value so let's say these two points have the same y value in other words they're on the same horizontal line the two points are equidistant from that axis of symmetry whatever the distance is from the axis of symmetry to this point on the left is the same distance would be from the axis of symmetry to the point on the right so those are several features of or points on a parabola that will play an important role throughout our discussion of quadratics and graphing quadratics you definitely should know all that information