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.