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. The image will be the same if you first reflect over the x-axis and then the y-axis or reflect over the y. 1 one half, 0 1 and 1 2 and I've got my recognizable 2 to the x graph that looks like this. Note that the order of reflections does not matter. 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. See Problem 1c) below.Let's talk about reflections. The argument x of f( x) is replaced by − x. And every point that was on the left gets reflected to the right. Conceptually, a reflection is basically a flip of a shape over the line of. Every point that was to the right of the origin gets reflected to the left. A reflection is a kind of transformation. Making the output negative reflects the graph. A vertical reflection reflects a graph vertically across the. We can also reflect the graph of a function over the x-axis (y 0), the y-axis(x 0), or the line y x. If points reflect across the y-axis, their y-coordinates remain unchanged but their x-coordinates change into their opposites. Every y-value is the negative of the original f( x).įig. Another transformation that can be applied to a function is a reflection over the x or y-axis. Its reflection about the x-axis is y = − f( x). Similarly when we reflect a point (p,q) over the y-axis the y-coordinate stays the same but the x-coordinate changes signs so the image is (-p,q). Only the roots, −1 and 3, are invariant.Īgain, Fig. And every point below the x-axis gets reflected above the x-axis. Every point that was above the x-axis gets reflected to below the x-axis. Sometimes the line of symmetry will be a random line or it can be represented by the x. The distance from the origin to ( a, b) is equal to the distance from the origin to (− a, − b).į( x) = x 2 − 2 x − 3 = ( x + 1)( x − 3).įig. Learn how to reflect points and a figure over a line of symmetry. If we reflect ( a, b) about the x-axis, then it is reflected to the fourth quadrant point ( a, − b).įinally, if we reflect ( a, b) through the origin, then it is reflected to the third quadrant point (− a, − b). When you graph the 2 lines on the same axes, it looks like this: Note that if you reflect the blue graph (y 3x + 2) in the x -axis, you get the green graph (y 3x 2) (as shown by the red arrows). It is reflected to the second quadrant point (− a, b). C ONSIDER THE FIRST QUADRANT point ( a, b), and let us reflect it about the y-axis.
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