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.

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There are different types of transformations and their graphs, one of which is a math reflection across the y-axis. If we get the same function from a **math reflection**, it is a symmetrical function, specifically even. A math reflection flips a graph over the y-axis, and is of the form y = f(-x). Other important transformations include vertical shifts, horizontal shifts and horizontal compression.

Let's talk about reflections. Now recall how to reflect the graph y=f of x across the x axis. All you have to do is put a minus sign in front of the f of x right? Y=-f of x flips the graph across the x axis. But how do you reflect it across the y axis? Well instead of flipping the y values, you want to flip the x values. So you replace the x with minus x and that will reflect the graph across the y axis.

So let's consider an example y=2 to the negative x. This is a reflection of what parent function? Well it's y equals to the x right? This will be a reflection of y equals to the x. Now to see this, let's graph the two of them together. So I want to graph y equals 2 to the x and y equals y equals 2 to the -x together. We call the y equals 2 to the x is one of our parent functions and has this shape sort of an upward sweeping curve passes through the point 0 1, and it's got a horizontal asymptote on the x axis y=0.

Let's plot a few points. We've got u and 2 to the u. I'm going to change variables to make it easier to transform and I'm going to pick easy values of u like -1 0 and 1 to evaluate 2 to the u. 2 to the negative 1 is a half, 2 to the 0 is 1, 2 to the 1 is 2. So those are nice and easy and then to make the transformation, I'm going to make the change of variables -x=u. So if I let u equal -x and x=-u and all I have to do is change the sign of these values. So -1 becomes 1, 0 stays the same and 1 becomes -1. But if -x=u then really I just have the 2 to the u values here so these values just get copied over. And so I'm just going to plot these two functions. First 2 to the x. -1 one half, 0 1 and 1 2 and I've got my recognizable 2 to the x graph that looks like this.

Now what about y equals 2 to the -x? Let me choose another colour. I have 1 comma one half, I have 0 1, so passes through this point and -1 2. So it's going to look like this. So as predicted, it's a reflection it's a reflection of our parent graph y equals 2 to the x. This is y equals 2 to the negative x.

Just remember, any time you take a function and you replace its x with a -x, you reflect the graph around the y axis. And that's it.