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The Reflection y = f(-x) - Problem 1 2,587 views

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Let’s graph another reflection across the y axis. First, let’s consider the function y equals the square root of x plus 4. What’s the equation of its 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. So y equals square root of –x plus 4 is our reflection across the y axis.

Let’s graph this function and this function together on a coordinate system. First, I could graph this function using transformations but it’s such an easy function that I’m going to do without this time. But I will call this u and root u plus 4.

Now, keep in mind that this function is only going to be defined when x plus 4 is greater than or equal to zero. So x is going to have to be -4 or larger. So we’ll start with -4.And you get the square root of -4 plus 4, square root of zero which is zero.

Now let’s think of values for u that will make this u plus 4 a perfect square. Like 1. How can we make this 1? If u is -3, we’ll get 1 and the square root of 1 is 1. And how can we make this 2? If u is 0 we’ll get 2. And so 0, 0 plus 4, square root of 4, 2. Let’s just use these three points to graph a reflection. Now remember our reflection is y equals square root of –x plus 4.

What we’re going to do here is we’re going to let u equal to –x, and therefore x equals –u. All we have to do is take our u values and change their sign. So -4 becomes 4, -3 becomes 3 and 0 stays 0. 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 we just copy them over. This is the table for x and root (-x plus 4).

Let’s graph these two sets of data. I’ll graph this one first. (-4, 0) is this point. And then I have (-3, 1), and then I have (0, 2). The graph looks something like this. I’ll change colors. Now let’s graph this table of data. I’ve got (4, 0), (3, 1) and (0, 2). (4, 0) is here, (3, 1) is here and (0, 2) is here. So I have something like this, very predictable. 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.