 Let's consider the graph of a basic exponential function like y equals 2 to power x. We'll find some points x, y on the graph of y equals 2 to power x. The y-intercept will be where x is equal to 0, in which case y is equal to... and so 0, 1 is on the graph of y equals 2 to power x. If we let x equals 1, we find y is... 2, and so 1, 2 is on the graph of y equals 2 to the x. If we let x equals 2, y is... 4, and so 2, 4 is on the graph. How about some negative values? If we let x equal negative 1, then y is equal to... 1 half. So negative 1, 1 half is on the graph. If we let x equal negative 2, y is... 1 fourth, so negative 2, 1 fourth is on the graph. And we could plot a lot more points, but let's also look at some end behavior. Notice that if x gets very large and positive, 2 to power x will also get very large. And that means that the right side of the graph is going to shoot off to infinity. On the other hand, if x gets very large and negative, 2 to power x will get very close to 0. So as we go to the left, our graph approaches the x-axis. And this means that y equals 2 to the power x will have an asymptote of y equals 0. Now we can consider many transformations of the graph of y equals 2 to the power x. To begin with, suppose we compress y equals 2 to the x horizontally by a factor of k. So this will give us the graph of y equals 2 to the power kx. Now remember our rules of exponents. If I have something raised to a product, I can split that product. So this 2 to the power kx, I can rewrite that as 2 to the k to the power x. And 2 to the power k is just some real number a. So this becomes the graph of y equals a to the x. Now remember we compress this horizontally. So that means k is going to be greater than 1. Now if k is greater than 1, then 2 to the power k is greater than 2 to the 1. But 2 to the power k is what we were calling a. So that says a is greater than 2. And so if a is greater than 2, the graph of y equals a to the x will be similar to the graph of y equals 2 to the x, but compressed horizontally. We could also stretch the graph of y equals 2 to the x by a factor of k. And so this gives us the graph of y equals 2 to the power x divided by k. Again I could split the exponent up into 1 divided by k times x. And 2 to the power 1 over k is some number a. Now again since k is greater than 1, we know that 2 to the power 1 over k is less than 2 to the 1. And 2 to the power 1 over k is what we called a. So we know that a is less than 2. But since 1 over k is greater than 0, because k is positive, then we know that 2 to the 1 over k is greater than 2 to the 0. So we know that a is greater than 1. And so if 1 is less than a is less than 2, the graph of y equals a to the x is a horizontal stretch of the graph of y equals 2 to the x. We could also reflect the graph of y equals 2 to the x across the y axis. And so remember how this affects the equation of the graph. This gives us the graph of y equals 2 to the power negative x. And again, we could split that exponent negative 1 times x. But 2 to the power negative 1 is 1 half. Now let's put all these things together. By horizontal stretches or compressions, we find that if a is greater than 1, the graph of y equals a to the power x resembles the graph of y equals 2 to the x stretched or compressed horizontally. If a is between 0 and 1, the graph of y equals a to the power x resembles the graph of y equals 1 half to the power x stretched or compressed horizontally. And so in some sense, all graphs of exponential functions look like the graph of y equals 2 to the power x stretched or compressed horizontally and possibly reflected across the y axis. So for example, let's describe the transformations needed to produce the graph of y equals 4 to the x from the graph of y equals 2 to the x, and then let's graph it. So we note that y equals 4 to the x, while 4 is 2 to the second, and so y is 2 to the power 2x. And so this tells us the graph of y equals 4 to the x can be produced from the graph of y equals 2 to the x by a horizontal compression by a factor of 2. Or we could try to find the graph of y equals 1 eighth to the x. And so we can apply the rules of exponents, y equals 1 eighth to the power x, while 1 eighth is really the same as 1 over 2 to the third, which is the same as 2 to the power negative 3. And so this would be the graph of y equals 2 to the power negative 3x. And so we can take the graph of y equals 2 to the x, then reflect it across the y axis to produce the graph of y equals 2 to the power minus x, then compress it by a factor of 3 to produce the graph of y equals 2 to the power negative 3x.