Videos and lessons with examples and solutions to help High School students learn how to distinguish between situations that can be modeled with linear functions and with exponential functions.

Related Topics:Common Core (Functions)

Common Core Mathematics

Example:

Archimedes is draining his bathtub. Every 2 minutes that pass, 7 gallons of water are draines.

Which of the following functions can represent the number of gallons of water in the tub, W, as a function of the minutes that have passed, T?

W = 2/7 T

W = 100 + 7/2 T

W = 50 - 7/2 T

W = 2/7 T + 20

Example:

Entrance to the paintball court costs $5 and paint balls are paid separately. A ticket for entrance with 5 balls, for example, costs $8.

Select all that apply.

The relationship is proportional.

Entrance with 10 balls costs $16.

When the number of balls increases by 11, the price increases by $6.60.

When the x-axis represents the number of paint balls, the slope of the graph of the relationship is 5/3.

This lesson demonstrates how linear functions can be applied to the real world. It discusses strategies for figuring out word problems.

1. Bathtub Problem: You pull the plug from the bathtub. After 40 seconds, there are 13 gallons of water left in the tub. One minute after you pull the plug, there are 10 gallons left. Assume that the number of gallons varies linearly with the time since the plug was pulled.

a. Write the particular equation expressing the number of gallons (g) left in the tub in terms of the number of seconds (s) since you pulled the plug.

b. How many gallons would ve left after 20 seconds? 30 seconds?

c. At what time will there be 7 gallons left in the tub?

d. Find the y-intercept (gallon-intercept). What does this number represent in the real world?

e. Find the x-intercept (time-intercept). What does this number represent in the real world?

f. Plot the graph of this linear function. Use a suitable domain.

g. What is the slope? What does this number represent?

2. Thermal Expansion Problem: Bridges on highways often have expansion joints, which are small gaps between one bridge section and the next. The gaps are put there so the bridge will have room to expand when the weather gets hot. Suppose a bridge has a gap of 1.3 cm when the temperature is 22°C, and the gap narrows to 0.9 cm when the temperature warms to 30° C. Assume the gap varies linearly with the temperature.

a. Write the particular equation for gap width (w) in terms of temperature (t).

b. How wide would the gap be at 35°? At -10° C?

c. AT what temperature would the gap close completely? What mathematical name is given to this temperature?

d. What is the width-intercept? What does this tell you in the real world?

How to solve word problems involving exponential functions?

Examples:

1. Write an exponential function to model the situation. Tell what each variable represents. A price of $125 increases 4% each year.

2. Write an exponential function to model the situation. Then find the value of the function after 5 years to the nearest whole number. A population of 290 animals that increases at an annual rate of 9%.

3. Write an exponential function to model the situation. Then find the value of the function after 5 years to the nearest whole number. A population of 300 animals that increases at an annual rate of -22%.

Examples:

1. Suppose a radioactive substance decays at a rate of 3.5% per hour. What percent of the substance is left after 6 hours?

2. Nadia owns a chain of fast food restaurants that operated 200 stores in 199. If the rate of increase is 8% annually, how many stores does the restaurant operate in 2007?

Example:

The value of a mortgage brokerage company has decreased from 7.5 million dollars in 2010 to 6 million dollars in 2012.

a) Assuming continuous exponential decay, find a formula of the form P

b) Find when the worth of the company will be 4 million dollars.

Depreciation, Appreciation, Compounded, Compounded Continuously.

Examples:

1. My "Clown at your Party" purchases a van to drive to parties for $20,000. The value of the van depreciates at a rate of 9% each year. Write an exponential decay model for the value of the van and find the value of the van after 8 years.

2. A house was purchased in 2012 for $150,000 If the value of the home increases by 3% each year, what will the house be worth in the year 2015?

3. Your deposit $100.000 in an account earning 7.5% annual interest, compounded monthly. What will your balance be in 10 years?

4. George Washington's wooden teeth are decaying rapidly. P(in grams) is the weight of the tooth after t years and can be modeled by P = 3.2(0.98)

5. You deposit $43,128 in an account that earns 2.5% annual interest compounded continuously. What will be the account balance after 12 years?

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