Identifying key features of exponential functions

0:00- We're told, consider the exponential function f, where f of x is equal to three times 1/2 to the x power. And now they ask us several questions about the y-interceptive f, the common ratio of f, and what is the equation of the asymptote of f. So pause this video, and see if you can figure these out before we do them together.

0:19All right, so first, what is the y-intercept of f? So one way to think about it is what y value if you were to graph it. If you were to say that y is equal to f of x, or another way to think about it, what values does the function take on when x is equal to 0?

0:37So another way to think about it, f of 0 is going to be equal to 3 times 1/2 to the 0. 1/2 to the 0 power is just one. So it's just 3 times 1, it's equal to 3. So the y-intercept of f is 3. Why do we call that the y-intercept? Because if you were to graph y equals f of x, when x equals 0, whatever the value of the function is, it's going to be intersecting the x, sorry, it's gonna be intersecting the y-axis at that point.

1:10What is the common ratio of f? So if we're dealing with an exponential function like this, it's the thing that keeps repeatedly getting multiplied, or another way to think of, the thing that you're taking the exponent of. And in this situation that is 1/2. And so our common ratio is 1/2.

1:30Now what is the equation of the asymptote of f? So one way to think about an asymptote is, does f approach but not quite reach some value as x gets very large or as x becomes very, very negative. As it becomes very positive or it becomes very negative? So let's think about this scenario here.

1:50If x becomes very positive, so if I were to say take f of, I don't know, 20, that's 3 times 1/2 to the 20th power. You might realize if I took 1/2, and I multiplied it 20 times, you're gonna get a very, very, very small number. It's going to be approaching 0. but not quite getting to 0.

2:13You multiply it by 3, it'll be 3 times bigger, but it's still gonna get very small. And this is just when x equals 20. If x equals 30, 40, 100, you're going to get closer and closer to 0, 'cause you're taking a number between 0 and 1. And when every time you multiply it, you're getting smaller and smaller and smaller.

2:32So if you take it to the 20th power, 30th power, 100th power, you're getting closer and closer to 0 without actually equaling 0. So as x gets bigger, our function is approaching y is equal to 0. So we could say, y is equal to 0. If we go the other way, if we said f of negative 20.

2:54This is the same thing as 3 times 1/2 to the negative 20. Or we could say this is the same thing as 3 times, we could take the reciprocal here and get rid of this negative on the exponent, 3 times 2 over 1. Or I could even just say 2 to the 20th power. I don't even need this parentheses.

3:10Well, the parentheses are good still. Now this is gonna be a very large number, so it's not really approaching anything. Some people would say it's approaching infinity, but it's not really... As x gets more and more negative, there doesn't seem to be an asymptote there.

3:23But as x becomes more and more positive, it looks like our function is approaching y equals 0. It's getting closer and closer to 0 without quite reaching it. Let's do another example here. So here, we are asked, "Which exponential function has a y-intercept of 4.5?

3:45Pause this video and try to figure that out. So as I said, the y-intercept is the value that the function takes on when x equals 0. So let's just try it out here. F of 0 in this situation. When x is 0, this is all gonna be 1 times a negative. So this is negative 1, so not a y-intercept of 4.5, rule that one out.

4:08So g of 0 right over here is gonna be 4.5 times 2 to the 0, which is 1, which is equal to 4.5. I like this one. I will fill it in. Now let's just double check this one. H of 0 is equal to 3 times 4.5 to the 0 power 3 times 1, which is equal to 3. So that is not a y-intercept of 4.5.

4:31So I'll rule that one out as well.