This lesson continues to examine quantities that change exponentially, focusing on a quantity that decays or decreases. Students are alerted that sometimes people use the terms exponential growth and exponential decay to distinguish between situations where the growth factor is greater than or less than 1. Additionally, students learn that when the growth factor is less than 1 (but still positive), people sometimes refer to it as the decay factor.
The opening activity encourages students to view a quantity that decreases by a factor of itself using multiplication rather that subtraction. If a computer costs \$500 and loses \(\frac15\) of its value each year, after one year we could write this, in dollars, as \(500 - \left(\frac15\right)\boldcdot 500\). But if we write this using multiplication, as \(500 \boldcdot \left(\frac45\right)\), then we are in a better position to see that after 2 years its value in dollars will be \(500 \boldcdot \left(\frac45\right) \boldcdot \left(\frac45\right)\) and, after \(t\) years, the value will be \(500 \boldcdot \left(\frac45\right)^t\). In other words, exponents are a particularly useful way to express repeated loss by a factor of the original amount. Students will carry this understanding into future lessons that deal with repeated percentage change situations.
In the second activity, students apply this idea to write an equation for the value of a car after \(t\) years, assuming that it decreases by the same factor each year.
Using an exponent to express repeated decrease by the same factor is a good example of a generalization based on repeated calculation (MP8). Writing \(500 \boldcdot \left(\frac45\right)^t\) expresses the computation of repeatedly decreasing by \(\frac15\), \(t\) times.
- Comprehend that the term "exponential growth" describes a quantity that changes by a growth factor that is greater than 1, and the term “exponential decay” describes a quantity that changes by a growth factor that is less than 1 but greater than 0.
- Use only multiplication to represent “decreasing a quantity by some fraction of itself.”
- Write an expression or an equation to represent a situation where a quantity decays exponentially.
Let’s look at exponential decay.
- I can use only multiplication to represent "decreasing a quantity by a fraction of itself."
- I can write an expression or equation to represent a quantity that decays exponentially.
- I know the meanings of “exponential growth” and “exponential decay.”
In an exponential function, the output is multiplied by the same factor every time the input increases by one. The multiplier is called the growth factor.