Lesson 2
Representations of Growth and Decay
- Let’s revisit ways to represent exponential change.
Problem 1
In 1990, the value of a home is $170,000. Since then, its value has increased 5% per year.
- What is the approximate value of the home in the year 1993?
- Write an equation, in function notation, to represent the value of the home as a function of time in years since 1990, \(t\).
- Will the value of the home be more than $500,000 in 2020 (assuming that the trend continues)? Show your reasoning.
Problem 2
The graph shows a wolf population which has been growing exponentially.
- What was the population when it was first measured?
- By what factor did the population grow in the first year?
- Write an equation relating the wolf population, \(w\), and the number of years since it was measured, \(t\).
Problem 3
Here is the graph of an exponential function \(f\).
Find an equation defining \(f\). Explain your reasoning.
Problem 4
The equation \(f(t) = 24,\!500 \boldcdot (0.88)^t\) represents the value of a car, in dollars, \(t\) years after it was purchased.
- What do the numbers 24,500 and 0.88 mean?
- What does \(f(9)\) represent?
- Sketch a graph that represents the function \(f\) and shows \(f(0), \) \(f(1),\) and \(f(2)\).
Problem 5
The first two terms of an exponential sequence are 18 and 6. What are the next 3 terms of this sequence?
Problem 6
A bacteria population has been doubling each day for the last 5 days. It is currently 100,000. What was the bacterial population 5 days ago? Explain how you know.
Problem 7
Select all expressions that are equivalent to \(27^{\frac13}\).
9
3
\(\sqrt{27}\)
\(\sqrt[3]{27}\)
\(\sqrt[3]{3^3}\)
\(\frac{1}{27}\)
\(\frac{1}{27^3}\)