Identifying key features of exponential functions | Algebra 1 (TX TEKS) | Khan Academy

We're told to consider the exponential function f where f of x is equal to 3 * 12 to the power of x. Now they ask us several questions about the y-intercept of 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.

All right, so first, what is the y-intercept of f? One way to think about it is what y value, if you were to graph it, if you were to say that y equals f of x. Another way to think about it is, what values does a function take on when x is equal to zero? So another way to think about it: f of 0 is going to be equal to 3 * 12 to the 0. 12 to the 0 power is just one, so it's just 3 * 1, which is 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 y-axis at that point.

What 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, the thing that you're taking the exponent of; and in this situation, that is 1/2. So our common ratio is 1/2.

Now, what is the equation of the asymptote of f? 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, let's think about this scenario here. If x becomes very positive, if I were to say take f of, I don't know, 20, that's 3 * 12 to the 20th power. You might realize if I took 1/2 and I multiplied it 20 times, you're going to get a very, very, very small number. It's going to be approaching zero but not quite getting to zero. You multiply it by three, it'll be three times bigger, but it's still going to get very small.

And this is just when x equals 20. If x equals 30, 40, or 100, you're going to get closer and closer to zero because you're taking a number between zero and one, and when every time you multiply, you're getting smaller and smaller and smaller. So if you take it to the 20th, 30th, or 100th power, you're getting closer and closer to zero without actually equaling zero. So as x gets bigger, our function is approaching y equals zero.

So we could say y equals zero. If we go the other way, if we said f of -20, this is the same thing as 3 * 12 to the -20, or we could say this is the same thing as 3 * we could take the reciprocal here and get rid of this negative on the exponent, 3 * 2 over 1, or I could even just say 2 to the 20th power. I don't even need this parenthesis; well, the parentheses are good still.

Now this is going to 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. But as x becomes more and more positive, it looks like our function is approaching y equals 0; it's getting closer and closer to zero 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. Pause 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 zero, this is all going to be 1 times a negative, so this is -1. So, not a y-intercept of 4.5; rule that one out.

So g of 0, right over here, is going to be 4.5 * 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 * 4.5 to the 0 power, which is 3 * 1, which is equal to 3. So that is not a y-intercept of 4.5, so I'll rule that one out as well.