Therefore, any exponential function will have a horizontal asymptote at 0 as x goes to negative infinity. Functions of the form a x are always strictly positive. Horizontal Asymptoteįor what x-value does the function 2 x=0? This means that, unless the graph has a vertical or horizontal shift, the y-intercept of an exponential function is 1. No matter what real number we use for a, a 0 will always be equal to 1. From this, we can make various transformations, including shifting the graph to the left and the right, reflecting it, and stretching it.Ĭonsider any function a x. In particular, it is important to learn the shape of the parent function. Graphing functions of the form a x, where the base, a, is a real number greater than 0, is similar to graphing other functions. This topic will include information about: Graphing exponential functions is sometimes more involved than graphing quadratic or cubic functions because there are infinitely many parent functions to work with.īefore learning to graph exponential functions, it is a good idea to review coordinate geometry and exponents generally. Graphing exponential functions allows us to model functions of the form a x on the Cartesian plane when a is a real number greater than 0.Ĭommon examples of exponential functions include 2 x, e x, and 10 x. And that's it.Graphing Exponential Functions – Explanation and Examples 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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