Hi, and welcome. My name is Anthony Vurella. And today, I'd like to talk about the graph of an exponential equation. So we'll take a look at the graph of y equals b to the x, talk about y-intercepts and end the behavior to the graph. So here we have a graph of y equals 2 raised to the power of x.
And one point that I'd like to display on our graph right away is the point 0, 1. This is the y-intercept of the graph. It occurs at x equals 0. And so when x equals 0, 2 raised to the power of 0 equals 1, because anything raised to the power of 0 is 1.
Well, now let's take a look at the exponential nature of this graph. So as x continues to get larger, we see that y tends towards positive infinity. And as x continues to get more and more negative, so approaching negative infinity, y is actually getting smaller, but it's approaching 0. It will never equals 0, and it will never fall below 0 for the graph of y equals 2 raised the power of x.
Now I'd like to compare this to a slightly different graph here this is y equals 3 to the x So our base is still a positive number. It's a little bit larger than 2. So we have 3 to the x. We still have that point 0, 1, because anything raised to the power of 0 equals 1.
But taking a look at the end behavior, it's a little bit more dramatic. So as x approaches positive infinity, y still approaches positive infinity. But it does so at a quicker rate, because our y values are tripling every time x is increased by 1 versus doubling every time x is increased by 1.
And on the other side of the graph, it's still a more dramatic difference. Y approaches 0 but does so quicker, because y is being cut into thirds every time x is decreased by 1 whereas over here, y was being cut in half every time x was decreased by 1. So when our base is a positive value of greater than 1, as x tends towards positive infinity, so does y. But as x tends towards negative infinity, y tends towards 0.
So we saw from our examples before that this y-intercept occurred at x equals 0, y equals 1. And this was because no matter what our base is, if we raise it to the power of 0-- remember x is always 0 at the y-intercept. Our y value is always 1. So if our base is 5, raise that to the power of 0, we get 1.
If our base is 8.2, raise that to the power of 0 and we get 1. So if we take any base and raise that to the power of 0, we're going to get 1. Now, there are some exponential graphs that have a y-intercept not at y equals 1.
So when do those occur? Well, we might have the exponential y equals a times b to the x. So if you think about a being multiplied by b raised to the 0 power. The exponent of 0 only applies to b. So b to the 0 equals 1. But when we multiply it by a, we get a.
So if we have y equals a times b to the x, our y-intercept is going to be at 0, a. Another way to think about this without an a written explicitly out in front, that would be an implied a value of 1.
Now I'd like to talk about some other features of our exponential graph. So we already talked about y equals b to the x when b is greater than 1. We have as x approaches positive infinity, so does y. And as x approaches negative infinity, our curve approaches 0.
Well, here is the graph of y equals b to the negative x. This is also when b is greater than 1. Here we have a reflection about the y-axis from our graph before. So we still have this y-intercept here at 0, 1. But as x approaches positive infinity, y tends towards 0. And as x approaches negative infinity, y approaches positive infinity.
Now an interesting note about these two graphs is that if b was a number between 0 and 1, it would actually look like this. One example would be y equals 2 to the negative x would have a graph that looks like this. But if we had y equals 1/2 to the positive x, a graph would look exactly like this. There's that reciprocal relationship between 1 half and 2 and the positive exponent and the negative exponent.
Well, now I'd like to show some graphs where we have an a value being multiplied out in front that is a negative value. So this would be a reflection about the x-axis from our graph before. So this was y equals b to the x. Here we have y equals negative a times b to the x.
So here are y-intercept would be 0, negative 1, because a would be negative 1 if we're assuming a is negative 1. As x approaches positive infinity, y approaches negative infinity as opposed to positive infinity before. But as x approaches negative infinity, our curve still tends towards 0, just from the other side of our graph here.
And we can combine those two elements. And we can have y equals negative a times b to the negative x So when we started with a y equals b to the x, we could reflect about both axes and get a curve that looks like this whereas x approaches positive infinity, y approaches 0, but from below instead of above. And as x approaches negative infinity, y approaches negative infinity as well.
So let's review our notes on the graph of an exponential equation. Well when b is greater than 1-- so the base is greater than 1-- as x tends towards positive infinity, y tends towards positive infinity as well. But as x tends towards negative infinity, y tends towards 0. So here's a graph of that behavior.
We talked about the y-intercept occurring at 0, 1 if we just have y equals b to the x. If there is an a value that's being multiplied out in front, then the y-intercept would be a, 0. And we also looked at variations of the graph for different positive and negative values of a and x.
Our exponential graph kind look like this as well. So thanks for watching this tutorial on the graph of an exponential equation. Hope to see you next time.
