Comparing exponential functions with the same initial value
About this video
Three decaying exponential functions l, m and n start at the same value but have factors 0.7, 0.9 and 0.5. The video matches each to its graph: the smallest factor (0.5, halving each time) decreases fastest, the largest (0.9) decreases slowest, and 0.7 is in the middle. It checks n(1) = 0.5 x 175 = 87.5 on the graph.
You'll be able to
- identify the decay factor in an exponential function.
- tell which exponential function decreases fastest by comparing decay factors.
- check a match by evaluating a function at a point.
Key words
- decay factor
- A number between 0 and 1 that you multiply by each time x goes up by 1, making the value shrink.
- initial value
- The value of the function when x = 0; here all three share it.
Transcript
- [Instructor] So we are given three of these functions, l of x, m of x and n of x. And they are plotted here, but we don't know which one's which, and they ask us which curve on the graph corresponds to which function equation. So why don't you pause this video, and have a go at that before we go through it together.
Okay, so there's a couple of ways to approach it. You can see that all of these are exponential functions as x increases by an interval, we're multiplying by the same amount for every successive x. For example, if x is equal to one, you're multiplying your initial value by 0.7.
When x equals two, you're multiplying that by 0.7 again, so on and so forth. So one way to think about it is which one is going to be decreasing fastest? And which is going to be decreasing the slowest? Well, the one that decreases the fastest is the one that's essentially multiplying by the smallest factor every time.
And that right over here is n of x. As x increases by one, you're essentially halving it every time. So n of x is going to be the one that decreases the fastest. So n of x is C. So this is n of x right over here is c. And we can even validate that. When x equals one, we're gonna take 0.5 times 175.
When x equals one, 0.5 times 175 is indeed 80, what is it? 87.5. So it's going to be, and that looks indeed right where it is. And similarly, which one, which one's going to decrease the slowest? Well that's m of x because it's multiplying by 0.9, which is higher than 0.7.
So the slowest declining one is m of x. So that is this right over here, m of x. And the one in the middle of is l of x. And, once again, you can try to validate it by substituting some numbers, but you just have to think about which one's declining the fastest, and which one is declining the slowest.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.