Comparing quadratic and exponential functions

0:00- [Presenter] We are told that f of x is equal to x squared + 6x + 5, and g of x is equal to 3 times 2 to the x. The table shows several values of each function and the growth factors, each output divided by the previous output, all right? Now they're giving us a question.

0:18I haven't showed it to you yet. "At what whole number of x "does g of x first become greater than f of x?" Pause this video. Have a go at that. All right. So when x = 1, and actually before I even look at the table to answer the question, let's just think a little bit about each of these functions.

0:36This first function over here, this is a quadratic. How do I know it is a quadratic? Well, it's a polynomial where the highest degree term is a second-degree term, right over here. So that's what makes that a quadratic. And this one over here is an exponential, exponential.

0:54It has the form some constant times something else to the x power something that might have, something involving the x power over here. So now if we wanted to answer the question, we see at x = 1, f of x is greater than g of x. At x = 2, f of x is greater than g of x.

1:13X = 3, f of x is greater than g of x. X = 4, f of x is less than g of x. So even though it looks like g of x would have passed up f of x before 4, some place in between x = 3 and x = 4, they're saying, "At what whole number value "does g of x first become greater than f of x?"

1:38So it looks like it happens at x = 4. X = 4 is my answer there. And now let's do the second part of this question. They are now asking, "Which statement best describes how the two functions grow "as x increases?" So the first choice says, "Both functions grow "by multiplying by the same factor each time."

2:00Well, that is true of an exponential. Every time you increase x by a whole number, you are essentially multiplying by 2 again, and you can see that right over here. If you take g of 2 divided by g of 1, that's what's happening here, this is g of 2 divided by g of 1, you get 2, and that keeps happening.

2:24Or another way of thinking, if you just look at g of x, for each successive x, we multiply by 2, we multiply by 2, we multiply by 2, we multiply by 2. And that makes sense 'cause 2 to the xth power. So that's true of g of x, but it's definitely not true of f of x.

2:42If you look at the second term divided by the first term, it's 1.75, but it doesn't stay 1.75. Then it's roughly 1.52. So that growth factor changes. So this is not going to be true. It would've been true for g of x but not for both of them. Both functions grow by adding the same amount each time.

3:04Well, that's definitely not the case. We can see that here. Here, from 12 to 21, we add 9. Then from 21 to 32, we add 11. So we already know that's not true there. And from 6 to 12 we add 6, and then from 12 to 24 we add 12. So we're definitely not adding the same amount every time.

3:24"The exponential's function growth factor decreases, "and the quadratic growth factor stays the same." Well the growth factor is what you essentially multiply each successive term, or, essentially, if you take, say, the second term divided by the first term or the third term divided by the second term, and we see the exponential, the growth factor is constant.

3:46It's a constant 2. So the exponential function's growth factor does not decrease. So hopefully D is our answer. Let's see. "The quadratic function's growth factor decreases." We see that. The quadratic function's growth factor does indeed decrease. "And the exponential function's growth factor "stays the same."

4:07Yep, that's what we've already talked about. So we like choice D. Another insight to know, especially if we're trying to compare quadratic to exponential functions is, and it depends on where you start, et cetera, but, in many cases, a quadratic, there are certain cases where an exponential can always be higher than a quadratic, but, in many cases, a quadratic might start at a higher level, but the exponential will pass it up later on, and then, beyond that point, the exponential will just keep growing faster and faster and faster, and you can actually even see that in the growth factor.

4:38It stays at a constant 2 while the quadratic, the growth factor decreases. If you think about it visually, a quadratic, maybe if I just think about where the quadratic is increasing, it might look something like this, while an exponential might look something like this.

4:58That's my shaky-hands attempt to do it. So at some point it's going to start, it's really going to be increasing much faster than a quadratic.