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What is Exponential Growth?

What is exponential growth?

Summary

  1. f(x)=a•bx is the formula for an exponential function
  2. a•bx is called an exponential function because x is an exponent
  3. If a>0 and b>1, then the exponential function will show exponential growth
  4. Exponential growth can be written as as f(t)=a•(1+r)t, where r and a must be greater than 0
  5. r>0 is very important for exponential growth! If 'r' is less than 0, then we'd have exponential decay!
  6. (1+r) is called the growth factor

Notes

    1. f(x)=a•bx is the formula for an exponential function
    2. a•bx is called an exponential function because x is an exponent
    3. If a>0 and b>1, then the exponential function will show exponential growth
    1. We can write a function with exponential growth as f(t)=a•(1+r)t
    2. 'r' and 'a' must be greater than 0
    3. 'r' is the rate of increase of the function, 'a' is the initial amount, and 't' is time
    4. (1+r) is called the growth factor
    5. If 'r' was less than 0, then a•(1+r)t would represent exponential decay instead of exponential growth!
    1. When you see an exponential growth function, get in the habit of figuring out what 'a' and 'b' or 'r' are equal to. That will help you understand exponential functions!
    1. f(x)=4•3x looks like an exponential function f(x)=a•bx, with a=4 and b=3
    2. To write 4•3x as exponential growth, we need to get it to look like a•(1+r)x
    3. Remember, both 'r' and 'a' must be greater than 0 for this to be an example of exponential growth
    4. 3x = (1+2)x
    1. f(x)=100•(1.3)x looks like an exponential function f(x)=a•bx, with a=100 and b=1.3
    2. To write 100•(1.3)x as exponential growth, we need to get it to look like a•(1+r)x
    3. Remember, both 'r' and 'a' must be greater than 0 for this to be an example of exponential growth
    4. (1.3)x = (1+0.3)x
    1. f(x)=4•(0.8)x looks like an exponential function f(x)=a•bx, with a=4 and b=0.8
    2. To write 4•(0.8)x as exponential growth, we need to get it to look like a•(1+r)x
    3. Remember, both 'r' and 'a' must be greater than 0 for this to be an example of exponential growth
    4. (0.8)x = (1+(-0.2))x, which means r<0!