 So let's expand our graphing repertoire by looking at what are called quadratic graphs. So the simplest non-linear function is f of x equals x squared. So let's consider the graph of y equals f of x. We'll make some observations. First, it's helpful to find a point on the graph. Since the equation has the form y equals stuff, we can calculate the value of y when we choose a value of x. So let's choose x equals zero and compute our value for y. y equals f of x equals means replaceable. f of x equals x squared, so we'll replace it. Equals means replaceable. x equals zero, so we'll replace it. And at this point there's nothing else to replace. Zero squared is zero, so our y value is zero. So the point where x equals zero, y equals zero, is on the graph of y equals f of x. We might consider the domain and range as well. So f of x equals x squared, and we can square any value of x. Then the domain is all real numbers. But since f of x equals x squared, the possible values of our function are non-negative numbers. Equals means replaceable since y equals f of x. This also says that the possible values of y are non-negative numbers. And this means our graph consists of points x, y, where x is some real number, but y is going to be in the interval from zero included all the way to infinity. Finally, let's consider symmetry. If x, y is the point on the graph, then we know that y is equal to f of x, which is equal to x squared. So first we'll check for symmetry about the x axis. We'll check to see if x negative y is on the graph. So we'll replace our y value with negative y and leave our x value alone. And now the question is whether or not this is a true statement. To determine that, equals means replaceable. This statement here says that wherever we see x squared, we can replace it with y. And so the question is whether minus y is the same as y. So this statement is false, so x negative y is not on the graph, and so the graph is not symmetric about the x axis. How about symmetry about the y axis? So we'll check to see if minus x, y is on the graph. And with a little bit of algebra, we find y may be equal to x squared. But we know this is true, so negative x, y is on the graph, and the graph is symmetric about the y axis. We'll add a little detail to this. Because it's symmetric about the y axis, we say that the y axis is a line of symmetry. Equivalently, the y axis is the same thing as the line x equals zero, so we say that the line x equals zero is a line of symmetry. Let's find a few more points. If x is equal to one, equals means replaceable. y is equal to f of x, f of x is equal to x squared, x is equal to one, and so that tells me that y is equal to one, so x equals one, y equals one is a point on the graph. But because the graph is symmetric about the y axis, we also know that minus one, one is also on the graph. If x equals two, y equals f of x, f of x is x squared, x is two, and so y is equal to four. So x equals two, y equals four is a point on the graph, and so by symmetry, we also know that negative two four is on the graph. So let's try x equals three. If x equals three, then y equals nine, so x equals three, y equals nine is a point on the graph, and by symmetry, we know that negative three nine is also on the graph. Let's graph these points. It's not entirely clear how we should try to draw a graph between the points, but if we were to draw many many many many many more points, we could run them together to form a smooth curve. We'd get something like this, and this is the graph of what's called a parabola. Now there's some more details we can add to this. Notice that as we move from left to right on the graph, our y values, our vertical distances, decrease for a while. We say the graph is falling, but eventually we get to a lowest point on the graph, which we might refer to as a turning point, after which the y values start to increase again, we say that the graph is rising, and this turning point has a special name. The turning point of the parabola is called the vertex, and for the graph of y equals x squared, the vertex is at zero zero. So remember, the graph of y equals x squared is a parabola with a line of symmetry x equals zero, and a vertex at zero zero. We can subject this graph to any geometric transformation, and what's important to recognize is whatever the geometric transformation is, it will also affect the line of symmetry and the vertex. So any translation will translate the line of symmetry and vertex. Any reflection will reflect the line of symmetry and vertex. For example, let's find the equation for the graph produced by shifting the graph of y equals x squared upward by three units, and rightward by two units. Then let's draw the graph and find the vertex and line of symmetry. So we'll perform our transformations. First, we'll shift the graph upwards, so y will become y minus three, and then we'll shift the graph rightwards by two units, so x will become x minus two, and since our equation was given in the form y equals stuff, we should write our answer in this form as well. To graph, we'll apply these transformations to our parabola. First, we'll shift it three units vertically, and then write two units. The vertex and the line of symmetry undergo the same transformations, so the vertex is going to be shifted three units upward, and two units rightward to the point three two, and the line of symmetry will be shifted three units upward, which doesn't affect it because it's a vertical line, and two units rightward to the line x equals two.