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. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. Suppose we wish to work out the equation of f(x) = x 2−3x+2 after a translation of 3 units to the left parallel to the x axis as shown in the first image in this post.Let's talk about reflections. Another transformation that can be applied to a function is a reflection over the x- or y-axis. So f(x) becomes f( –x) so we replace x by –xĪnd we have ( –x ) 2−5( –x)+4 = x 2 +5x+4 ![]() To reflect in the y axis the transformation is f( –x) To work out the equation of f(x) = x 2−5x+4 after reflection in the y axis. To reflect in the x axis the transformation is –f(x) Suppose we wish to work out the equation of f(x) = x 2−5x+4 after reflection in the x axis. Note that you can work out the equation of the transformed graphs. Graph of sin x translated parallel to y axis.Or try translating this graph of f(x) = sin x parallel to the x axis.Translations parallel to the x axis.as illustrated above. ![]() In this post we will look at reflections and translations.Įxperiment with the following Desmos pages to examine these transformations, f(x) + a, f(x + b), -f(x) and f(-x) where a and b are integers.
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