 So what else can we do with the graph of y equals x squared? If we take the graph of y equals x squared and apply a vertical stretch with or without a reflection across the x-axis, then a horizontal and or vertical shift will get a new graph with the equation of the form y equals c times x minus h squared plus k. This is called the vertex form of a quadratic polynomial because from it we're able to easily find the vertex and the line of symmetry. And the transformations we applied to the graph of y equals x squared to obtain the graph of y equals c x minus h squared plus k. For example, let's consider the graph y equals minus 1 third x minus 4 squared plus 7. Let's see if we can describe the transformations and then find the vertex and line of symmetry and then sketch the graph. In this process, if possible, do reflections and stretches first, then translations. So we might notice that we have this factor minus 1 third, so let's see if we can do some sort of vertical stretch or possibly a reflection. So let's stretch the graph vertically by a factor of 1 third. If we do that, the equation of the new graph will be obtained by replacing y with y over 1 third. And if we solve this equation for y, our equation is starting to look like the equation that we want. Again, if it's not written down, it didn't happen. So we might say that we let the graph of y equals x squared undergo the transformations, first being stretched by a factor of 1 third to become the graph of y equals 1 third x squared. Now the minus suggests that there is some sort of a reflection across the x-axis. So if we reflect the graph across the x-axis, the equation of the new graph will be obtained by replacing y with minus y, which we solve for y. And now we have a coefficient of minus 1 third, which is what we'd like to see. And again, if it's not written down, it didn't happen. So our next transformation was reflecting across the x-axis to become the graph of y equals minus 1 third x squared. We also have an x minus 4, and this suggests a horizontal translation. If we shift the graph to the right by 4 units, the equation of the new graph will be obtained by replacing x with x minus 4. And if it's not written down, you can pretty much get away with anything. So if you're a criminal or a politician, don't write things down. But if you're a good human being or a mathematician, write stuff down. And so we've shifted our graph to the right by 4 units to become the graph of y equals minus 1 third x minus 4 squared. Finally, we have a vertical translation. If we shift the graph downward by 7 units, the equation of the new graph will be obtained by replacing y with y plus 7. So if we do that, then solve for y. We don't get what we want. Oh well, no worries. It doesn't matter what you write at first. What matters is what you write at last. So if the transformation downward didn't give us what we want, let's try shifting upward. If we shift the graph upward by 7 units, the equation of the new graph will be obtained by replacing y with y minus 7, and we'll get, which we'll solve for y. And that is exactly the equation that we want. And since we are neither a criminal nor a politician, we'll write this down. We have shifted upward by 7 units to become the graph of y equals minus 1 third x minus 4 squared plus 7. Now we've written down all of our transformations, and so we can take our graph of y equals x squared and apply the transformations. So first, we'll stretch the graph by a factor of 1 third. That means we'll compress it by a factor of 1 third. We'll reflect it across the x-axis. We'll shift to the right by 4 units, then shift upward by 7 units. The vertex and the line of symmetry undergo the same transformations. The position of the vertex won't be affected by stretching or reflecting, but it will be shifted to the right by 4 and up by 7 to the point 4-7. The line of symmetry won't be affected by a vertical stretch or a reflection or a vertical translation, but it will be affected by the horizontal translation. So let's do the time warp and take a step to the right for units. And this puts the line of symmetry at x equals 4. What about a quadratic equation like this? If the equation were in vertex form, it would be easy to read off the transformations. But it's not in vertex form. Well, that's OK. We'll put this equation into vertex form by completing the square. So first, we'll get rid of that constant term by adding 15 to both sides. We can factor a 4 out of the terms on the right-hand side, so let's do that. We'll complete the square inside the parentheses by adding 1 and then paying for it by subtracting 1. The first three terms form a perfect square, so we'll write it that way. We'll expand this out slightly. And since we started with an equation of the form y equals, we should end with an equation of the form y equals, so we'll subtract 15. And we get our vertex form of the equation. So we can obtain the graph of this equation by taking the graph of y equals x squared and stretching it vertically by a factor of 4, which gives us the graph whose equation is, which we'll solve for y. We don't seem to have any sort of reflection, so we'll shift our graph horizontally to the right and we'll get the graph whose equation is. If we then shift vertically downward by 19 units, we get the graph whose equation is. Since we've written down all of our transformations, we can graph our new equation. We'll start with a graph of y equals x squared. We'll stretch it vertically by a factor of 4. We'll shift horizontally one unit to the right, and then we'll shift downward by 19 units. And if we do that, our vertex is going to be affected by the horizontal and vertical shifts, so the vertex will go to the right by 1 and down by 19, and that will take it to the point 1 negative 19. The only transformation that affects the line of symmetry will be the horizontal shift, and so our line of symmetry will go from x equals 0 to x equals 1. Finally, we see that the vertex is the lowest point on the graph. There is no highest point. In a seemingly unrelated matter, let's consider the problem of finding the maximum or minimum values of a function if they exist. And the connection is that for this function, because it's a quadratic polynomial, the graph of y equals f of x will be the graph of a parabola, and the vertex will either be the highest or lowest point on the graph. So let's put our equation into vertex form. We'll start with our equation and then complete the square. So let's identify the transformations we need. If we start with the graph of y equals x squared, we'll obtain the graph we want by reflecting across the x-axis, shift the graph to the right by two units, then shift the graph upward by 19 units. And that means that our vertex will be at 219, and this will be the highest point on the graph. There is no lowest point. We want to find the maximum value of f of x equals means interchangeable, and so this means we want to find the maximum value of y, which will be 19. It's also useful to identify where this occurs, and so we might say that this maximum value will occur at x equals 2.