Today we're going to talk about graphs of exponential equations. So we're going to start by looking at the characteristics of an exponential equation and how the graphs of an exponential equation will change as we make changes to the equations.
So let's look at some examples of exponential graphs. So let's start with graphing the equation y is equal to 2 to the x. Again, we know this in an exponential equation because we have a variable in our exponent.
So if I want to graph this, I'm going to start by evaluating the equation at x is equal to 0 to find the corresponding y value. When x is 0, this becomes two to the zero, which is 1, so y is 1. So my first point for this graph will be at x is 0, y is 1.
If I substitute 1 in for x, 2 to the 1 is 2, so when x is 1, y is 2. If I substitute 2 in for x, that gives me to the second which is 4. So when x is 2, y is 4. And when x is 3, y is going to be 8. So graphing a few points here, we get a general shape of our graph on the positive side. We can see that it's growing exponentially, increasing at a faster and faster rate.
On the negative side, we know that our y values are going to tend closer and closer to 0 as x goes towards negative infinity.
Let's try graphing a second exponential equation and compare it. Let's try y is equal to 3 to the x. So we'll graph this the same way. We'll start by evaluating x is equal to 0. 3 to the 0 is also just 1. So they share this point 0, 1. When x is 1, this becomes 3 to the 1, which is 3. So when x is 1, y is 3. When x is 2 that becomes 3 to the second power, which is 9. So when x is 2, y is 9.
So here, we're already about to go off of our graph. So we'll go ahead and connect these points. We see that when our base increased from 2 to 3, it affects the graph by having a sharper increase. So this exponential equation y is equal to 3 to the x is increasing at a faster rate than y is equal to 2 to the x.
And it also is going to decrease faster. It's going to get closer and closer to 0 as x goes to negative infinity.
So let's look at the y-intercept of the graph of an exponential equation. So the first graph, y equals 2 to the x, is the same as the one from our previous example, still graphed in green. And as we said before, we can see that the y-intercept is 1, and that's because when we substitute 0 in for x, then this evaluates to be 2 to the 0, or just 1.
So any exponential equation that has an a value of 1 or the number in front of the base is 1 or there's no number there, those equations are going to have a graph with the y-intercept of 1. And again, that's because 2 or any value for our base to the 0 power is always going to be 1.
If we have an exponential equation that does have a number in front of our base, so we do have a value for a-- like in this example, our value for a is 3-- then the y-intercept is not going to be 1. It's going to be whatever number is in front of your base, so in this case, 3. So we can see from our graph that the y-intercept is 3.
And for any exponential equation the y-intercept is always going to be the value of the number in front of our base. And that again is because whenever you have a 0 for our exponent, whenever x is 0 which is where our y-intercept is, anything raised to the 0 power is just going to be 1. So this will be 1, and we'll multiply that by this number, which will give us the value of, in this case 3, or whatever number is in front of the base. So the y-intercept will be the value of the number in front of our base.
So let's look at a couple more characteristics of exponential equations and their graphs. The first is having a positive versus a negative exponent. So I have this equation y is equal to 2 times 2 to the positive x, and that's graphed in black.
And I'm comparing it to the equation y is equal to 2 times 2 to the negative x, so same equation, but with the negative exponent. And that graph is in green. So we can see that since negative x and positive x are opposite, their graphs have opposite effects. So with our original graph, as x is tending towards positive infinity, y goes to positive infinity. And as x goes to negative infinity, y approaches 0.
But we have the opposite effect when our exponent is negative. As x approaches positive infinity, now our y is approaching 0. And as x is approaching negative infinity, then our y is approaching positive infinity. And it's important to remember that we can represent our negative exponents as a fraction with a positive exponent in the denominator.
So the second characteristic we'll look at is comparing the equations and graphs of exponential equations when the value of a, our number in front of the base, is positive or negative.
So we still have our original exponential equation y is equal to 2 times 2 to the x graphed in black. And we're comparing that to the equation and graph y is equal to negative 2 times 2 to the x. So we're comparing when we have a positive or negative a value.
And in our graphs, we can see that it looks like the graph is reflected over the x-axis. And in our graphs we can see that because our a value is negative, as x is approaching positive infinity, our y values are approaching negative infinity. It's decreasing instead of increasing. And we also notice that as x is approaching negative infinity, our y values are still approaching 0 as they were with a graph of our equation with a positive a value.
So finally let's look at the characteristics of exponential equations that have both a negative a value and a negative exponent. So I have still my original graph, an equation y is equal to 2 times 2 to the x comparing that this time to y is equal to negative 2 times 2 to the negative x. So this equation has both a negative a value and a negative exponent.
And we can see in our graph that it has the characteristics of our exponential equations with a negative a value and a negative exponent. That makes sense. So when our x value is approaching positive infinity, we see that our y values are approaching 0. And as x is approaching negative infinity, we see that our y values are also approaching negative infinity.
So let's go over our key points from today. An equation in the form y equals a times b to the x is an exponential equation. The y-intercept of an exponential function graph is equal to the value of a in the equation. And a negative exponent reflects the graph over the y-axis. while a negative a coefficient reflects the graph over the x-axis.
So I hope that these key points and examples helped you understand a little bit more about graphs of exponential equations. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.
