Now, that's true for the y equals x squared, but in general, if we want to think about any parabola, notice that this point, which is in this case at the origin, is going to be that point that's going to lie along the axis of symmetry. And we have a range of values in this case, which is going to be only the non-negative values. So our domain is going to be from negative infinity to positive infinity. Some other things to point out are that we end up having a domain of all real numbers, because we can square any number we'd like to. You need to give the equation of the line for that. So don't just say that your axis of symmetry is 0. Remember that when you're talking about an axis, you're talking about a straight line. So we end up with an axis of symmetry, and that equation is x equals 0. And so our axis of symmetry in this case is going to be the y-axis, whose equation is going to be x equals 0. So this is symmetric to our y-axis, which means that we have this line of symmetry that's going directly in between the two halves- the two symmetric halves- and that's going to be called our axis of symmetry. When that occurs, we end up having a symmetry to the y-axis. So we had f of negative x was equal to f of x. One is even from just using these values, you notice that when we substituted in the value of x, we came up with some f of x, but when we substituted in the value of negative x somewhere over here, we also came up with the value of f of x- that is, f of negative x was actually the same value as f of x. For now, we're just going to go with "a parabola is a U-shaped graph." Notice that in this parabola, we have a couple of things that we can point out. Now, if you end up having conic sections somewhere in your future, you'll end up with a more geometric definition for what a parabola is. It's going to be a U-shaped shaped graph, and that has a particular term that's associated with it- it's called a parabola. So we end up with what is the characteristic shape of a quadratic function. And then as we move to the left, notice we have the same kind of shapes going on. And in doing so, notice that this section down by the origin, we're going to round it off like this and then go up toward the right. Now we're going to connect the dots and assume that we have a general smooth shape between them. So again, kind of going up and about right there. When x was equal to negative 3, I ended up with 9. When x was equal to negative 2, I ended up with four. When I look at my negative values for x, when I went to x equals to negative 1, I ended up at 1. And lining it up, that looks about right. And then moving over three, we need to go up to 9. And then go over two units and go up to 2, 3, 4. Then moving over one unit and then up one unit, I have the 0.1 comma 1. So on the xy plane, let's just start at the origin of plot 00. Negative 1 squared is 1, 0 squared is 0, 1 squared is 1, 2 squared is 4, 3 squared is 9. Negative 2 times itself will end up giving us a value of 4. So negative 3 squared is the same as negative 3 times negative 3, which is going to give us a positive 9. So squaring all of our x values will give us our y-coordinates. Let's just plot values where x goes from negative 3 to positive 3 along integer values. In particular, if I just draw in a quick xy plane, and I'm going to just plot some points, so making a T-chart. We're going to do a little bit more detailed look at that graph now. Now, you may be familiar with this graph from having seen it in a previous section. So the most basic quadratic function is going to have the equation y equals x squared, or f of x equals x squared. We'll also look at methods for finding the vertex, and finally we'll look at optimization problems. We'll look at its intercepts and its vertex. We will consider the axis of symmetry for a quadratic function. We'll look at properties of a quadratic function, namely, we'll look at domain and range, and then we'll have the parabolic shape of the graph. In specifics, let's look at our subtopics. Now we're going to look at the two variable format where it's equal to f of x, a function of x. So just like we had looked at quadratic equations where we had ax squared plus bx plus c equals 0. A quadratic function is a function of the form f of x equals ax squared plus bx plus c, where a, b, and c are real numbers and our a value does not equal 0. Those are also known as second degree functions. We're now going to go up one degree and look at quadratic functions. Now, you should already be familiar with solving linear and quadratic equations, and in the previous session, we looked at linear functions.
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