Unit 11: Exponential Growth and Decay
Exponential functions multiply by the same factor every step, which is why they eventually blow past any straight line no matter its slope.
5 topics · 15 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Exponential versus Linear Growth
- How it worksHow do a linear function's constant addition and an exponential function's constant multiplication race apart on the same axes?
- Worked exampleCompare f(x) = 2x + 3 and g(x) = 2x at x = 1, x = 5, and x = 10.
- Common mistakeWhy doesn't a bigger starting value guarantee a function stays ahead if its growth is only linear?
Your turn1 of 2
Let f(x) = 5x + 6 and g(x) = 3x. Write h(x) = g(x) - f(x), as a function of x.Writing Exponential Growth Functions
- How it worksHow do a starting amount and a growth rate combine into an exponential growth function?
- Worked exampleFind the exponential growth function for a population of 500 that grows 8 percent per year, and use it to find the population after 6 years.
- Going furtherWhat happens to an exponential growth function's graph when the growth rate doubles?
Your turn1 of 2
A population of 687 grows 9 percent per year. Write the population P(t) as a function of the number of years t.Writing Exponential Decay Functions
- How it worksHow does a decay rate between 0 and 1 make an exponential function shrink instead of grow?
- Worked exampleFind the exponential decay function for a $20000 car that loses 15 percent of its value each year, and use it to find its value after 5 years.
- Common mistakeWhy doesn't a 15 percent yearly loss mean the car is worthless after about 7 years?
Your turn1 of 2
A car worth 25446 dollars loses 16 percent of its value each year. Write its value V(t) as a function of the number of years t.Compound Interest
- How it worksHow does the compound interest formula turn a principal, a rate, and a number of years into a final balance?
- Worked exampleFind the balance after 10 years on $2000 invested at 5 percent annual interest, compounded yearly.
- Ties togetherHow does compound interest connect to the general exponential growth function?
Your turn1 of 2
Find the balance A(t) after t years on 3786 dollars invested at 6 percent annual interest, compounded yearly, as a function of t.Graphing Exponential Functions
- How it worksHow does changing the base of an exponential function change how sharply its graph curves upward?
- Worked exampleGraph the function y = 3 times 2x and identify its horizontal asymptote.
- Going furtherWhat happens to an exponential function's graph when its base is between 0 and 1 instead of greater than 1?
Your turn1 of 2
An exponential function has initial value 5 and increases by a factor of 3 for every increase of 1 in x. Write the function y(x).