Comparing Linear and Exponential Growth

This lesson teaches how to determine which of two growth patterns, one linear and one exponential, will exceed the other at a specific point in time. It explains the differences between linear and exponential growth, using salary growth examples to illustrate the concepts. The lesson guides students through writing function equations and evaluating them for specific years to see which growth pattern leads. It concludes with a summary of how to determine growth pattern dominance.

What is the main objective of this lesson?

• To calculate the initial value of a function.

• To learn how to write function equations.

• To determine which growth pattern exceeds the other at a specific time.

• To understand the history of growth patterns.

How is linear growth characterized?

• By subtracting a constant number from the quantity.

• By adding a constant positive number to the quantity each time period.

• By multiplying the quantity by a constant factor.

• By dividing the quantity by a constant factor.

What happens to exponential growth in the long run?

• It becomes equal to linear growth.

• It eventually exceeds linear growth.

• It decreases over time.

• It always remains below linear growth.

In the salary example, what is the initial salary for option A?

• $3,000

• $1,000

• $20,000

• $50,000

Which option represents exponential growth in the salary example?

• Option A

• Option B

• Both options

• Neither option

What is the formula for the linear function in the salary example?

• y = 20,000 + 1,000x

• y = 50,000 * 1.03^x

• y = 20,000 * 1.03^x

• y = 50,000 + 1,000x

How do you determine which growth pattern is ahead at a specific time?

• By guessing based on initial values.

• By looking at the starting year.

• By writing and evaluating function equations at that time.

• By comparing the growth factors only.

In the 20th year, which salary option yields a higher amount?

• Option B

• Option A

• Neither option

• Both are equal

What is the approximate salary for option B in the 20th year?

• $1,000

• $50,000

• $20,000

• $36,122.22

What is the key takeaway from the example calculation?

• Exponential growth eventually surpasses linear growth given enough time.

• Exponential growth can surpass linear growth in the short term.

• Linear growth always surpasses exponential growth.

• Initial values do not affect growth patterns.