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### Course: Algebra (all content)>Unit 11

Lesson 21: Distinguishing between linear and exponential growth (Algebra 2 level)

# Comparing growth of exponential & quadratic models

Sal discusses two functions that model the shipment rate of cars. One function is quadratic and the other is exponential. Which one will eventually exceed the other? Created by Sal Khan.

## Want to join the conversation?

• How could we have proved that Japan will have received more cars after the 7th month?

I tried doing it in many ways, but I always got to a point where I couldn't solve the problem without a graphing calculator, here's what I tried to find the values for which j(t) will be greater than v(t).

2^t - 2t^2 > 0
2 - ((2t^2)^(1/t)) > 0
-((2t^2)^(1/t)) > -2
((2t^2)^(1/t)) < 2

Then I inputed ((2t^2)^(1/t)) and looked what values were less than 2 to know when j(t) will be greater.

Please show me how to solve it manually.
• Wow what a question! Sounds simple, but it is actually extremely advanced. So, you can find the exact moment Japan passes Vietnam by setting `j(t) = v(t)` and solving. So `2^t = 2*t^2`. Might look simple, and, well, you can use a calculator to approximate that and find `6.31972235584...` months. However, expressing the answer precisely involves something called the Lambert W function, which I'd never heard of until just now. You can see the precise expression of the value by looking at the third of the "Real solutions" on http://www.wolframalpha.com/input/?i=2%5Et+%3D+2*t%5E2 - this is a transcendental number that's not easy to even express in a formula. I'll have to read a lot more to about Lambert W now. :)
• How come, the quadratic graph isn't a parabola or has a parabolic shape?
• It is hard to see the shape because of the window settings. Rest assured, the shape of the curve is parabolic.
• If the graph of cars shipped to Vietnam is quadratic, then why does it look like a straight, linear line when it is graphed on the calculator?
• It sort of looks like a straight line because the scale had to be so large to see both of them together. Here's a much larger graph of both equations: You can clearly see the red quadratic graph is curved, but the blue exponential curve just explodes so much faster: http://tinyurl.com/ojkrdgr - let me know if that explains it.
• On the graphing calculator, the quadratic function looks linear. Can anyone explain this?
• It depends on multiple problems that can exist in your case. One, the graping plane may be zoomed in a little too much or ,two, the function may be incorrectly put in.
• How does the quadratic equation have a linear graphing even though it has an exponent?
• It is definitely curved, it's just that the exponential curves up so much faster that it doesn't look like the quadratic is curving at all. In fact (and you can see this later in calculus), the slope of an exponential increases (or decreases) exponentially, and the slope of a quadratic increases (or decreases) linearly. So the difference in how the slope changes is even more dramatic between the two functions. And a final point that is also very important in calculus, is that any smooth curve, if you zoom in far enough, looks like a straight line.
• the number of cars shipped to japan is modeled by function = j(t) = 2^t where 't' is time ..
how do i comprehend 2^t in real world e.g. if it was 2(t) and after 5 months i can see 2*5 = 10,
but i don't understand how i express 2^t ??

thanks
(1 vote)
• It is 2 to the t th power. So, j(5) is actually 2^5, which is 32, not 10.
• i put 2(7)^7 and got 1647086. he mathed wrong some where. why?
(1 vote)
• I think you mixed up the 2 functions. Sal's work is correct.
If you are calculating v(7), then you need 2(7)^2.
If you are calculating j(7), then you need 2^7
Neither function creates 2(7)^7.
Hope this helps.
(1 vote)
• Does the equation form a parabola? Why or Why Not?
(1 vote)
• Is this part of algebra 2?
(1 vote)
• my teacher uses a common rate/ % box and i still haven't understood the subject and ive been trying for a week and have a test this Tuesday.... is there anyone who can help me understand?
(1 vote)