 This lesson is about parent functions or the library of functions, and we will also work on relations. In previous math classes, you have studied these, but I want to review them with you as we go through calculus. We will need every single one of these functions and relations that we will be going over in our review. So the first function we're going to work on is y equals x squared. Everybody knows what it looks like, but let's see how we grab it so we get a pretty accurate graph. What we need is a domain and range table. So we make our domain range table, which is x, y. And we substitute values for x, and then we compute the values for y. And some of the values we will substitute for x are zero, one, negative one, two, and negative two. If we substitute in zero for x, of course we get zero. If we substitute a one in for x and square, we get a one. If we substitute a negative one in for x and square it, we will get a positive one. If we substitute a two in for x, we will get four. And if we substitute a negative two and square it, we will get another positive four. So we will graph these points on our Cartesian coordinate system, and don't forget it is set up with x is going in this horizontal direction, and y is going into the vertical direction. So the first point we want to plot is zero, zero, which is here. Then the next one is one, one, which is one to the right, one up, that's there. The next point on our list is negative one, one. So this goes one to the left and one up. The next point is two, four, two to the right, four up. And the last point is negative two, positive four. So now that we have our five points graphed, we will connect them into the nice parabola for y is equal to x squared. And as you know, it does come out to be a nice u. And when we think about this graph, we also think about it as a graph that decreases and then increases. So not only are we plotting points, we're thinking about what the graph is doing. Our second function is y equals log base two of x. Now this was a little bit more difficult, and some of you might have struggled in previous classes in doing this. But remember, change this to two to the y is equal to x. And use y as your independent variable and x as your dependent variable. And again, make your domain range table. Only we're going to do it a little backwards. We're still going to put it x, y, but we're going to fill in for y first. So the first value we always try to put in is zero, and then we'll put in a one and then a negative one, a two and a negative two. Okay, let's do that substitution. Now remember, we are substituting in for y, not x. So we'll say two to the zero is what? Well, two to the zero is one. Next, we'll substitute in the one and get two to the one, which we know to be two. If we put in negative one, we get two to the negative one, which is one half. And if we substitute two in for y, we get two squared, which is four. And if we substitute in the negative two for y, we get one, four. We are now going to plot these points. So we'll make our graph again and plot those points. One zero lies here. Two one is there. One half, negative one. Well, it's one half to the right and one down. So it's here. Four two goes four to the right and two up. And then one fourth, negative two means one fourth to the right and two down. You see, as we plot these points, they come closer and closer to the y-axis. So we know there is some sort of an asymptote in here. So when we create our graph, we will not let it touch the y-axis, but go up like that towards the right. And as you look at this graph, you see that it is always increasing. And that's very important, again, when you look at your graphs to visualize what they do. The next function is y equals arc sine x. The inverse trig function. I know most people don't like graphing this function, but it isn't that difficult to do. First, remember that the domain is negative one to one. And the range is negative pi over two to pi over two. Now from that, we are again just going to make that domain range table and just do a few points. Since the domain starts at negative one, we know that if x is equal to negative one here, then sine of negative pi over two is negative one. So our y is the negative pi over two. And that's where we begin our range anyway. Then we have zero and we know sine of zero is zero or the arc sine of zero is zero. So that's a zero. If we put in a positive one for our x, we know that the y value will be pi over two. Now to graph those three, we'll plot them first and then connect the dots. We have negative one, negative pi over two. We have a zero zero and then we have a one pi over two. To connect these dots, if you remember what this looks like, which is really important, and it's a little curve that starts at negative pi over two and ends at pi over two on the y's and of course negative one and one on the x's. And again, this graph is always increasing but it only increases between the negative one and one and negative pi over two and pi over two. And that's what you have to remember when you're graphing your arc sine function. The next graph we have is not a function. It's a relation. And it is x squared over four plus y squared over nine equals one. If you recall what has happened in your previous classes, you might recognize this that this is an ellipse. And we have a very special way of graphing our conic sections. What we looked for on an ellipse is the major axis and the points that end the ellipse on the major axis, which is this axis here because nine is bigger than four so that makes the y the major axis. And we will be plotting points zero plus or minus three. And then we have a minor axis and that is our x axis and we will be plotting points plus or minus two, zero. So let's try that. Make our coordinate system. On the y's we're going to do plus three and minus three and plus two for the x and minus two for the x and you can see our ellipse forming just by those points and what we will do because we just want this to be a quick graph is to connect those. And once we have connected it, we have our ellipse. This ends this lesson. I hope you can do the homework on it. So good luck with the homework.