5.3 Writing Expontential Equations
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Last updated 10 months ago
26 questions
Note from the author:
OBJECTIVES & STANDARDS
Math Objectives
- Recognize and define the parts of an exponential function
- Write exponential functions to model a given situation
- Evaluate and solve exponential functions
Common Core Math Standards
- Link to all CCSS Math
- CCSS.PRACTICE.MP4
- CCSS.PRACTICE.MP7
- CCSS.HSF.LE.A.1.C
- CCSS.HSM
Personal Finance Objectives
- Analyze return on an investment given starting investment, rate of return and time periods
National Standards for Personal Financial Education
Investing
- 2b: Compare nominal annual rates of return over time on different types of investments, including cash flow and price changes
DISTRIBUTION & PLANNING
Distribute to students
- Student Activity Packet
- Application Problems
OBJECTIVES & STANDARDS
Math Objectives
- Recognize and define the parts of an exponential function
- Write exponential functions to model a given situation
- Evaluate and solve exponential functions
Common Core Math Standards
- Link to all CCSS Math
- CCSS.PRACTICE.MP4
- CCSS.PRACTICE.MP7
- CCSS.HSF.LE.A.1.C
- CCSS.HSM
Personal Finance Objectives
- Analyze return on an investment given starting investment, rate of return and time periods
National Standards for Personal Financial Education
Investing
- 2b: Compare nominal annual rates of return over time on different types of investments, including cash flow and price changes
DISTRIBUTION & PLANNING
Distribute to students
- Student Activity Packet
- Application Problems
Intro/Warm-Up
DISCUSSION PROMPTS: Repeated Doubling
Answer each question using any appropriate math notation.
1
If you were asked to start with the number 5 and double it three times, how would you write this problem using math notation?_______
1
If you were asked to start with the number 5 and double it nine times, how would you write this problem using math notation?_______
1
If you were asked to start with the number 5 and double it one hundred times, how would you write this problem using math notation?_______
Learn It
An exponential function is a function in the form y = ab^x where
- a is the starting value of the function
- b is the number that is repeatedly multiplied, commonly referred to as the growth or decay factor. Assuming positive numbers, then the following is true:
- If b > 1, then the function will grow. This is also referred to as exponential growth
- If b is between 0 and 1 (fractions and decimals) then the function will shrink. This is also referred to as exponential decay
- If b = 1, then the function will not grow or decay because multiplying by 1 does not change the starting value.
- x is the number of times that you will be multiplying by b
*Remember your order of operations!!!!
PEMDAS
4
Given the exponential function
- State the starting value:_______
- State the growth/decay factor:_______
- Is the function growing or decaying?_______
- Describe in words what this function is doing. Refer to the language in the intro activity._______
- What is the result if x = 3?_______
5
Given the exponential function
State the starting value:_______
State the growth/decay factor:_______
Is the function growing or decaying?_______
Describe in words what this function is doing. Refer to the language in the intro activity._______
What is the result if x = 5?_______
Practice It
Use what you have learned about the parts of exponential functions to write an equation for each of the following situations and answer the questions.
2
Everytime Pinocchio lies, his nose doubles in size. His nose is 1.5 inches long before he has told any lies.
a. Write an equation that represents this situation where x is the number of lies and y is the size of Pinochhio’s nose after x lies._______
b. Use your equation to calculate how long Pinocchio’s nose will be after 6 lies._______ inches
2
You make a $10,000 investment that has historically doubled every 7 years.
- Write an equation that represents this situation where x is the number of years and y is the investment value after x years._______
- Use your equation to calculate the value of your investments after 15 years.$_______
As with other types of equations, you can solve exponential functions for any variable. In this section, we will solve for the exact initial value but only estimate the exponent since solving for exponents is beyond what we have learned so far.
1
Sari makes an investment with the goal of having $20,000 after 30 years. If the growth factor is 1.12, how much money will she need to invest to reach her goal?_______
1
Charlotte has $12,000 to invest and would like this investment to grow to $18,000. If the growth factor is 1.09, how many years will it take to reach her goal? Use guess and check to find an approximate answer_______ years
Application Problems
You are given $10,000 that will triple every year for the next 5 years.
1
Write an equation that a represents this situation where y represents the value of the investment after 5 years._______
1
Your equation has three numbers: starting value, growth factor, and the exponent. You are allowed to double one of these numbers. Which one do you think will result in the highest outcome? Explain your reasoning.
Your equation has three numbers: starting value, growth factor, and the exponent. You are allowed to double one of these numbers. Which one do you think will result in the highest outcome? Explain your reasoning.
3
Let’s find out which one gives the highest outcome!
Double the starting value
_______
Double the growth factor
_______
Double the exponent
_______
1
Which double resulted in the highest return on your investment?
Which double resulted in the highest return on your investment?
1
Think about the change that resulted in the highest return on your investment. What does that represent in the real world context of the original situation?
Think about the change that resulted in the highest return on your investment. What does that represent in the real world context of the original situation?
1
After performing this activity, if you had to give advice to someone making an investment that results in exponential growth, what would you tell them?
After performing this activity, if you had to give advice to someone making an investment that results in exponential growth, what would you tell them?
Level 2:
The world record for folding a piece of paper in half over and over again is 12 folds by Britney Gallivan in 2018.
Every time she folded her paper in half, the thickness doubled! Let’s pretend that we can beat Britney’s record and fold a piece of paper as many times as we want.
1
Start by writing an equation that represents starting with a 0.1 mm thick piece of paper that is folded in half x number of times._______
1
Based on your equation, approximately how thick was the piece of paper Britney folded?_______
0
Complete the table using your equation. As the numbers get larger, you may need to convert millimeters to centimeters, meters, or kilometers. Be sure to include units with your numbers!
Complete the table using your equation. As the numbers get larger, you may need to convert millimeters to centimeters, meters, or kilometers. Be sure to include units with your numbers!
0
Choose any three rows of your table and find an object that exists in the world that is approximately the same size. For example, if your measurement was 7 mm, you could say that an iPhone 13 is about 7mm thick. Write your answers in the space provided below.
Choose any three rows of your table and find an object that exists in the world that is approximately the same size. For example, if your measurement was 7 mm, you could say that an iPhone 13 is about 7mm thick. Write your answers in the space provided below.
0
Imagine that there was an investment that performed similarly by starting small and doubling after a certain amount of time has passed. What is the best advice that you would give to someone that was holding this investment?
Imagine that there was an investment that performed similarly by starting small and doubling after a certain amount of time has passed. What is the best advice that you would give to someone that was holding this investment?
Level 3:
We can represent the motion of a tennis ball bouncing using math. Here’s a graph that lets you see different bounces from different heights. You can move the sliders to adjust the height of the ball and how high each additional bounce will be.
In this video, the initial height of the ball is 36 inches and each bounce is represented in the table below.
Let’s create a decay model to represent the ball’s height after a certain number of bounces.
0
Calculate the decay factor for each bounce by dividing the second height by the first height, then the third by the second, until you reach the bottom of the table. Make a list of those decay factors in the space below to three decimal places.
Calculate the decay factor for each bounce by dividing the second height by the first height, then the third by the second, until you reach the bottom of the table. Make a list of those decay factors in the space below to three decimal places.
0
What is the average of the 5 decay factors?
0
Use your average decay factor to write an exponential function where x is the number of bounces and y is the height of the ball after x bounces._______
0
Test your function by substituting a value on your table for x and seeing if it is close to the actual height measured in the video. Does your equation appear to be a good representation of the ball’s height?
Test your function by substituting a value on your table for x and seeing if it is close to the actual height measured in the video. Does your equation appear to be a good representation of the ball’s height?
0
Use your equation to make a prediction about the height of the ball after 7 bounces.
Use your equation to make a prediction about the height of the ball after 7 bounces.
0
Your purchasing power after an increase in inflation works in a similar way to a bouncing ball. If your purchasing power decreased by 2%, 5%, 3.3% and 4.1% over the last 4 years, write an equation to represent the change in purchasing power of $100 over the past 4 years. (Note that your decay factor will be 1 - r for each year’s percentage change)_______