Lesson 4

Representing Functions at Rational Inputs

  • Let’s find how quantities are growing or decaying over fractional intervals of time.

Problem 1

A bacteria population is tripling every hour. By what factor does the population change in \(\frac12\) hour? Select all that apply.











Problem 2

A medication has a half-life of 4 hours after it enters the bloodstream. A nurse administers a dose of 225 milligrams to a patient at noon.

  1. Write an expression to represent the amount of medication, in milligrams, in the patient’s body at:
    1. 1 p.m. on the same day
    2. 7 p.m. on the same day
  2. The expression \(225 \boldcdot \left(\frac12\right)^{\frac52}\) represents the amount of medicine in the body some time after it is administered. What is that time?

Problem 3

The number of employees in a company has been growing exponentially by 10% each year. By what factor does the number of employees change:

  1. Each month?
  2. Every 3 months?
  3. Every 20 months?

Problem 4

The value of a truck decreases exponentially since its purchase. The two points on the graph shows the truck’s initial value and its value a decade afterward.

Graph of exponential decay function that includes the points 0 comma 40,000 and 1 comma 24,000.
  1. Express the car’s value, in dollars, as a function of time \(d\), in decades, since purchase.
  2. Write an expression to represent the car’s value 4 years after purchase.
  3. By what factor is the value of the car changing each year? Show your reasoning.

Problem 5

The value of a stock increases by 8% each year. 

  1. Explain why the stock value does not increase by 80% each decade.
  2. Does the value increase by more or less than 80% each decade?

Problem 6

Decide if each statement is true or false.

  1. \(50^\frac12 = 25\)
  2. \(\sqrt{30}\) is a solution to \(y^2 = 30\).
  3. \(243^{\frac13}\) is equivalent to \(\sqrt[3]{243}\).
  4. \(\sqrt{20}\) is a solution to \(m^4 = 20\).
(From Unit 4, Lesson 3.)

Problem 7

Lin is saving $300 per year in an account that pays 4.5% interest per year, compounded annually. About how much money will she have 20 years after she started?









(From Unit 2, Lesson 26.)