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The Reflection y = f(-x) - Concept

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.

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.

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