Hey everyone, we've worked with some basic graphs of functions and here we're going to focus on one specific type of function, the quadratic function. Now, it might seem like a lot of information at first and this graph behind me might look a little bit intimidating, but don't worry, I'm going to walk you through absolutely everything that you need to know about the graphs of quadratic functions here and you'll be an expert in no time. So let's go ahead and jump right in. A quadratic function is a polynomial of degree 2 that has a standard form of f(x) = ax2 + bx + c. Now, all of these functions here are examples of quadratic functions and you can see that a, b, and c can be any real number whether it be a fraction, a negative number, or even 0 so long as a is not 0 and our largest exponent is still 2, making it a quadratic function.
Now here we're going to focus on the graphs of quadratic functions so let's go ahead and take a look here. We've worked with the square function before f(x) = x2 and you may remember that this square function was a parabola. Now this is actually going to be the same for all quadratic functions. They are all going to have this curved parabolic shape. Whether it is right-side-up, upside-down, or located anywhere on our coordinate plane, our quadratic functions will always be that same shape of a parabola.
So we're going to look at the different elements of a parabola here and some are going to be ones that we're familiar with like the x intercept or the y intercept but we're also going to work with some that are specific to parabolas, like the vertex or the axis of symmetry. So let's go ahead and get started with our vertex. The vertex of a parabola is either going to be the lowest point or the highest point depending on whether our parabola is opening upwards or downwards. So for our square function here, we see that our vertex is right at that origin point and we're always going to write our vertex as an ordered pair, so my vertex is simply (0, 0). Now, for my other function here, it is not at the origin, it is actually at the point (-2, 1), so my vertex is (-2, 1).
Now the other thing that I want to consider with my vertex is whether I'm dealing with a minimum or a maximum point. So again, looking at my square function here, this is the lowest point on my graph of that quadratic function. So it is a minimum point. It's all the way at the bottom and nothing goes below it on that parabola. So not only is my vertex here (0, 0), it also represents a minimum.
Now for my other function, my vertex is all the way at the top, which tells me that I am not dealing with a minimum. I'm actually dealing with a maximum point here. So here, my vertex is (-2, 1), and it is a maximum. So let's go ahead and look at something that we are more familiar with, our x intercept. Now, the x intercept of a graph is anywhere that our graph crosses our x-axis, not our y-axis, our x-axis.
So we're going to look for the points that cross that x-axis. And between my two functions here, I have 3 different points that cross my x-axis that represent my x intercepts. So let's take a look at our square function. Now, there's only one point that crosses the x-axis and it is at my origin, so my x intercept is simply 0. Now looking at my other function, there are 2 points that cross the x-axis, so I need to account for both of them, (-3, 0) and (-1, 0), as my x intercepts.
Now you'll only ever have 1 or 2 x intercepts, never less or more than that. So let's move on to our y intercept. Now the y intercept is where a graph crosses the y-axis, this time the y-axis. So let's look at our square function first. We have this point.
This represents our y intercept. Now our y intercept here is again at the origin, simply at 0. And for my other graph, my other function, my y intercept is down here at -3, so it's simply -3. Now we've looked at our intercepts and our vertex. Now we want to look at our axis of symmetry.
And now our axis of symmetry is something that is going to be specific to parabolas, and it represents the line that divides our parabola perfectly in half. It is symmetric about that line. So our axis of symmetry is going to perfectly cut it in half. So looking at my square function, it's going to go straight through the middle and it's actually always going to go straight through our vertex. So here, my axis of symmetry is simply the line x = 0, a vertical line through my vertex.
Now looking at our other function here, if I draw a line that divides this perfectly in half, you're often going to see this actually written with a dotted line but I've just highlighted them here so that it's easier to see. But here again, we're going straight through that vertex point at x = -2. So it's always going to be a straight vertical line through our vertex and that represents our axis of symmetry. If I were to fold my parabola in half at that line, it's going to perfectly match up because it is symmetric about that line. Now something else to consider whenever we're looking at graphs of functions is always the domain and the range.
So first looking at our domain, the domain of all quadratic functions is actually always going to be the same and it's always going to be negative infinity to infinity, which you might also recognize as being all real numbers. So for both my square function and for my other function here, they are both all real numbers, negative infinity to infinity. Now our range is going to depend on whether we have a minimum or a maximum point. So looking first at our square function, we know that we're dealing with a minimum point here and that looking at that, my y values can go anywhere from that minimum and all the way up beyond that. So whenever I'm dealing with a minimum, my range is always going to go from y minimum, whatever that minimum point is, all the way up to infinity.
So in this case, since I start down at 0, my range is simply 0 to infinity. Now when I'm dealing with a maximum point like I have here on my other graph, it's always going to go from my maximum point down to negative infinity. So in interval notation, I would write that as negative infinity up to my maximum point, y maximum. So in this case, it goes from negative infinity up to my y point at y = 1, and that's my domain and range. Now we want to look at one last thing on our graphs here.
Something that you're going to often be asked when dealing with parabolas is to tell them the interval for which your parabola is increasing and decreasing. Now this is just a fancy way of saying for what values of x is our graph going up and for what values of x is it going down. So let's just take a look at our second function here that's facing downward and decide where it's increasing and decreasing. Now looking at the first side of this, going down here all the way up to my vertex point, we see that our graph is going up. It is increasing.
So the interval for which it is increasing is simply from negative infinity up to that vertex point at -2. And our vertex is actually always going to divide the intervals for which we're increasing or decreasing. So the other side, from -2 all the way to infinity, we're going to be decreasing because we see our graph is going down. So this is just simply the opposite, -2 to infinity. So that's all the stuff we need to know about parabolas.
One more thing that I want to mention here is that you might have noticed that my equation here doesn't quite look like it's in standard form. And that's actually because we're often going to see quadratic functions written in what's called a vertex form, which is what that is right there. And it's actually going to help us to graph with ease. It's going to help us to easily graph these quadratic functions, which is what is coming up next. So thanks for watching.
I'll see you in the next video.