![]() ![]() So now that we have some basic graphs that we're familiar with, we're going to use those in order to look at some transformations, and the first transformation we want to consider is a translation, which is going to be a shifting. So again, these are functions that we want to be familiar with, so that we can use them in order to do some problems. And so it looks kind of like a parabola- half of a parabola on its side. The domain will be all values of x greater than or equal 0, and the range is all values of y greater than or equal to 0. Then we look at the square root function, which is going to have the equation f of x equals the square root of x. The range would be all values of y greater than or equal to 0. ![]() So again, the distinctive V-shape of the absolute value function. As you substitute in to the absolute value, then your y value ends up being positive, also. The part that's in quadrant II is actually if I take my negative xs and now make them positive. The first part that's in the quadrant is the same as y equals x. It's actually two- you think of it as a linear function that's been separated. This is going to have the distinctive V shape. Let's now look at f of x equals the absolute value of x. The domain is all real numbers, and so is the range. And we can see that it increases as we move from left to right. Notice that we have a section that's in quadrant I and a section that's in quadrant III. Another function that we want to be familiar with is f of x equals x cubed, also knows as the cubing function. We have a range from 0 to infinity, including 0, and we can see where it increases and decreases. The other thing to point out is that we have a domain of all real numbers. Its vertex will be at the origin, and we have this symmetry to the y-axis in this format. Now, this is going to have the shape of a parabola. Let's start off with f of x equals x squared. ![]() That is, you should be familiar with them, but it's worth the time at the outset to make sure we're familiar with these different functions that we're going to use in this session. To begin with, let's review some functions that you should already have at your beck and call. We'll look at reflections, compressions and stretches, and then combinations of transformations. We'll look at vertical and horizontal shifts. ![]() Now, in particular, we're going to look at some specific types of transformations you can do on functions in order to change their graphs. During this session, we're going to look at a topic called transformations. This is my y equals the square root of –x plus 4 and it’s the reflection of this graph which is y equals the square root of x plus 4.Hi, my name is Rebecca Mueller. So I have something like this, very predictable. And then I have (-3, 1), and then I have (0, 2). This is the table for x and root (-x plus 4). As far as the y values go, because we let u equal –x, we really just need the values of root u plus 4, which are these values. So -4 becomes 4, -3 becomes 3 and 0 stays 0. All we have to do is take our u values and change their sign. What we’re going to do here is we’re going to let u equal to –x, and therefore x equals –u. Now remember our reflection is y equals square root of –x plus 4. Let’s just use these three points to graph a reflection. And how can we make this 2? If u is 0 we’ll get 2. How can we make this 1? If u is -3, we’ll get 1 and the square root of 1 is 1. Now let’s think of values for u that will make this u plus 4 a perfect square. So we’ll start with -4.And you get the square root of -4 plus 4, square root of zero which is zero. So x is going to have to be -4 or larger. Now, keep in mind that this function is only going to be defined when x plus 4 is greater than or equal to zero. But I will call this u and root u plus 4. First, I could graph this function using transformations but it’s such an easy function that I’m going to do without this time. Let’s graph this function and this function together on a coordinate system. So y equals square root of –x plus 4 is our reflection across the y axis. Remember, all you need to do to get the equation of the reflection across the y axis, is replace x with –x. What’s the equation of its reflection across the y axis? First, let’s consider the function y equals the square root of x plus 4. Let’s graph another reflection across the y axis. ![]()
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