![]() The graph of $-y=f(x)$ is the reflection of $y=f(x)$ about the x-axis.Ĭaption Left: Horizontally reflecting the graph of $y=x 1$.The graph of $y=f(-x)$ is the reflection of $y=f(x)$ about the y-axis.To dilate a graph toward or away from the y-axis, you divide $x$ by a positive factor $a$. Just like translations, you can dilate horizontally or vertically. However, this is a challenge for those who always seek reasons! Dilationĭilation means to stretch or contract a graph. You certainly do not have to memorise this. If you find this too confusing, please do not worry! It is a very confusing concept (I took several years to finally understand it). caption Translating a graph by moving the origin. If you want to move your graph up, you instead move the coordinate plane down. Here comes the secret: you are not moving the graph, but you are changing the perspective by moving the coordinate plane itself. Still, if you want to move $y=3x$ to the right by three units, you have to subtract three from $x$! For example, if you want to move a point, say $A(0,1)$, to the right by three units, you have to add three to the x-coordinates (so it becomes $A\rq(3, 1)$). And that's it.Some of you might be thinking that this rule is counter-intuitive. ![]() Just remember, any time you take a function and you replace its x with a -x, you reflect the graph around the y axis. So as predicted, it's a reflection it's a reflection of our parent graph y equals 2 to the x. I have 1 comma one half, I have 0 1, so passes through this point and -1 2. ![]() Now what about y equals 2 to the -x? Let me choose another colour. 1 one half, 0 1 and 1 2 and I've got my recognizable 2 to the x graph that looks like this. And so I'm just going to plot these two functions. But if -x=u then really I just have the 2 to the u values here so these values just get copied over. So -1 becomes 1, 0 stays the same and 1 becomes -1. ![]() So if I let u equal -x and x=-u and all I have to do is change the sign of these values. So those are nice and easy and then to make the transformation, I'm going to make the change of variables -x=u. 2 to the negative 1 is a half, 2 to the 0 is 1, 2 to the 1 is 2. 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. 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. So I want to graph y equals 2 to the x and y equals y equals 2 to the -x together. Now to see this, let's graph the two of them together. 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. So let's consider an example y=2 to the negative x. So you replace the x with minus x and that will reflect the graph across the y 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. 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. Now recall how to reflect the graph y=f of x across the x axis.
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