![]() ![]() ![]() To make the house shift 3 units to the right, we need to subtract 3 from the input. Horizontal shifts are often less intuitive than vertical shifts. The x-axis corresponds to the input of the function, so if we wanted to shift the house to the right or left, we would need to change what is inside the function parentheses (the function argument). Since function outputs correspond to y-axis values, we can shift a function up by adding or or down by subtracting from the whole function. If we subtracted \(4\) from every output of the function, we would get \(H(x)-4\), which corresponds to the little blue house shifted downward. The house has a point near the chimney at the coordinate \((1,1)\). The graph below shows an example "function", \(H(x)\), that draws a little red house at the origin. This section presents a simplified visual example of several ways to transoform functions: translation, compression, expansion, and reflection. And that's it.So far we have worked with basic linear, quadratic, radical, exponential, and logarithmic functions, but these functions often appear in different forms. 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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