# Unit 1: Relationships between Quantities and Reasoning with Equations Lesson 13- writing Exponential Equations in Two Variables Objectives

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Unit 1: Relationships between Quantities and Reasoning with Equations

Lesson 13- Writing Exponential Equations in Two Variables

Objectives:

• I can write an exponential equation with two variables to model a situation, solve it, and graph it.

• I can determine if a model is an example of exponential growth or exponential decay.

Warm-Up: Review.

Solve.

a) 42x + 1 = 17

b) 3x + 19 = 100

Graph.

 x 0 1 2 3 y 1 2 4 8

Is it linear?

Exponential Equations:

An exponential equation is an equation in which the independent variable is an exponent. In the exponential equation y = abx, y is the dependent variable, x, is the independent variable, and a and b are constants. The base, b, can be any positive real number other than 1.

Is it exponential?
 y = 32x Why or why not? y = 7x3 Why or why not?

In y = abx, the constant a represents the starting value of a quantity being measured, such as the number of living things in an area. The base b shows how that quantity changes as the variable x changes.

The graph of an exponential equation is not a straight line. It is a curve that is either always increasing or always decreasing.

 Exponential Growth Graph For the equation y = abx, b > 1 and a > 0. Exponential Decay Graph For the equation y = abx, 0 < b < 1 and a > 0.

Is it exponential growth, exponential decay, or neither?
 y = 2x y = 3  0.5x y = y = 54x y = y =

Graphing Exponential Equations:

We will need to make an input output table to graph exponential equations.

Graph.

y =

 x y

y =

 x y

We can use the information we have learned about exponentials to write an equation that models an exponential growth or exponential decay problem.

Let’s try a few…

A colony of bacteria starts out with 150 cells and triples in population every hour. Write an equation that models the number of cells, y, after x hours. Then, graph the equation.

Is this model an example of exponential growth or exponential decay? How can you tell?

The half-life of a substance is the time it takes for half of that substance to break down or decay. The half-life of fermium-253 is 3 days. Write a equation for the amount remaining from a sample of 700 grams of fermium-253 after x-days. Make a graph that models the decay of fermium-253.

Every three days, the amount of fermium is cut in half. Use this information to complete the table below and then compare it to your graph.

 Days 0 3 6 9 Fermium-253 (in grams)

Compound interest is calculated by using the exponential function A = P(1 + )nt , where A is the accumulated amount after t years, P is the principal (the amount invested), r is the annual interest rate expressed as a decimal, and n is the number of times the interest is compounded per year.
Write an equation to find the amount that is accumulated when \$500 is invested in an account with a 3% annual interest rate, compounded monthly. Make a graph that shows how the account grows over time.

How could you determine how much money would be in the account after 18 months?

You try! Exit Ticket.
 A population of rabbits quadruples each year. If the population started out with 24 rabbits, write an equation that represents the total population after x years. Then, graph the equation. How many rabbits would there be after 4 years if none of the rabbits died?

Name: _________________________

Unit 1 Lesson 13: Writing Exponential Equations with Two Variables
Write an equation for each situation.
 1) Sarah bought a car for \$18,500. According to her insurance company, the value of the car depreciates 5% each year. What will the value of the car be x years after Sarah purchased it? 2) A colony of bacteria doubles in population every 24 hours. If there were 20 bacteria initally, how many bacteria will there be after x days? 3) Caitlin put \$300 in a bank account with 4% annual interest compounded monthly. Write an equation to represent the total amount of money in Caitlin’s account after t months.

Write the equation that models the situation and then graph the equation.

4) A colony of bacteria starts out with 300 cells and doubles in population every hour. Write an equation that models the number of cells, y, after x hours. Then, graph the equation.

5) Write an equation to find the amount of money that is accumulated in a bank account when \$1000 is invested in an account with a 2.5% annual interest rate, compounded monthly. Make a graph that shows how the account grows over time.

How much money will be in the account after 2 years?

6) The half-life of Zinc-71 is 2.4 minutes. If Chris had 100.0 g at the beginning, how many grams would be left after 7.2 minutes has elapsed? Graph the half-life.
 Minutes 0 2.4 4.8 7.2 Zinc-71 (in grams)

Is this an example of exponential growth or exponential decay? How do you know?

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