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The Reflection y = f(-x) - Problem 2 2,446 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 about the y axis. Here we have the function y equals -1/2, quantity x plus 1 times the quantity x minus 3. The first question is what is the equation of the reflection about the y axis? Remember, all you needed to do to get the reflection is to replace x with –x. So the reflection is going to be y equals -1/2(-x plus 1)(-x minus 3).

Let’s graph these two equations together. Instead of using a parent function, I’m actually going to use this as my parent function. It’s pretty easy to graph a quadratic equation, so I’m going to graph this one as my parent function. Let’s call this f(x). I'll have u and f(u). You’ll notice that it’s got x intercepts at -1 and at 3. That means that at -1 and 3, the output’s going to be zero. That means, because this is a quadratic function, the graph’s going to be a parabola. So halfway between these two points, we’re going to have a vertex. That’s an important point to graph. Halfway between -1 and 3 is 1. If you’re not sure about that though you could just average the two numbers. -1 plus 3 is 2, over 2 is 1.

Let’s plot 1, -1/2, 2 times -2, is -4, -1/2 times -4 is 2. That’s a pretty good start. I’m noticing that between -1 and 1 we have zero. So that will give me a y intercept. That will be important to graph. So when I plug in zero I get 1 times -3 which is -3 times -1/2, 3/2. Let’s graph these points and get our graph of the original function. We have (-1, 0), (3, 0) (1, 2) and (0, 3/2). So our original function looks something like this.

Now let’s graph the reflection. I changed the letter from x to u so that I could make the substitution u equals –x. What I’m going to graph here is x and then -1/2, -x plus 1, -x minus 3, the reflection of this guy. So you can see that what I need to graph here is f(-x), so I need to make x equals to the opposite of u. So these I have to multiply by -1. That means from -1 I get 1, from 1 I get -1. 3 becomes -3 and zero stays put.

Now this is -1/2u plus 1, u minus 3, that’s exactly f(u). These values are going to stay exactly the same. So I plot these points; (1, 0) (-3, 0) (1, 0) (-3, 0) (-1, 2) and (0, 3/2) again is the y intercept. This is my reflection. This is the graph of y equals -1/2, (-x plus 1) (–x minus 3). And the first is the graph of y equals -1/2(x plus 1)(x minus 3). It’s really easy to get a reflection across the y axis. All you have to do is replace x with –x.

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