Name Algebra 1B notes and class problems
October 1, 2008 “Application problems using mx + b” page 6

Application problems using mx + b

To get the slope m: look for the rate of change

In most real-world application problems involving linear functions, the slope number represents a rateofchange. Here are some examples of rates of change:

□  a rate at which some quantity is growing

□  a speed at which an object is moving

□  a dollar amount that is charged for each item or for each person

Here are two words often seen in word problems that give a hint about which number is a rate and gives the slope.

□  per (for example: if a problem says “55 miles per hour” the 55 is probably the slope)

□  each (for example: if a problem says “10 dollars each” the 10 is probably the slope)

To get the intercept b, look for the starting amount

Many real-world situations involve a calculation that has a rate and a starting amount.
If you know both of these numbers, you can use a slope-intercept (mx + b) formula to model the problem. The starting amount gives you the y-intercept and the rate gives you the slope.

m = the slope = the rate
b = the y-intercept = the starting amount

In application problems it’s often best to write slope-intercept equations using function notation. So the equation typically looks like f(x) = mx + b, or sometimes you may wish to pick a letter different than f to convey the meaning of the output variable.

Example: Abby rakes leaves for her neighbors. She charges $10, plus $5 per hour.

Name the variables… Let x = time worked in hours, P(x) = amount paid in dollars.

Identify the slope and y-intercept values…
slope: m = 5 (from $5 per hour)
y-intercept: b = 10 (because the starting charge is $10)

Now write the equation… P(x) = 5x + 10.

Problems

1. Here are the problem situations from several word problems you have seen before.
Identify the number in each problem that is the slope of the linear function.
Then write a function formula in the form f(x) = mx + b.

a. Concert tickets cost $30 each, plus a service charge of $10 for the whole order.

b. Users at a gym must pay $50 for a membership, plus $3 for each time they use the gym.

c. A car travels on a highway at the speed limit of 50 miles per hour. (The input is the amount of travel time; the output is the distance traveled.)

d. Each can of regular soda contains 40 grams of sugar. (The input is a number of cans ofsoda; the output is the amount of sugar contained in the cans of soda.)

e. A mountain climber begins a hike at an elevation of 6000 feet, then climbs such that herelevation increases by 1000 feet per hour.

f. At the start of a carnival, you have 50 ride tickets. Each time you ride the roller coaster, youhave to pay 6 tickets.

g. A refreshment stand is selling drinks at a rate of 100 drinks per hour.

h. A pizzeria is running a special where the first pizza ordered costs $12, and eachadditional pizza costs $8. [Hint: On this problem, you will need to write a function formula that correctly expresses the “each additional” idea, then take a couple of algebra steps to rewrite the formula into mx + b form.]


2. At the beginning of a rainstorm, the water level of a river is 6 feet.
As it rains, the water rises at a steady rate of 2 feet per hour.

a. What are the two variables in this situation? Describe them in words.

x =

f(x) =

b. What numbers are the slope and the intercept? m = ______, b = ______

c. Write the function formula: f(x) = ______

d. What is the river level after 4 hours of rainfall? (Show how you get your answer.)

e. The river will flood if the level reaches 20 feet. After how many hours would this happen? (Show how you get your answer.)

f. On graph paper: Make a graph of function f(x) from the rainstorm problem (see part c for the formula). Follow the usual expectations about careful drawing and complete labeling.

3. A pet owner has 4 bags of cat food. Each week, his cats eat of a bag of food.

Let x = the number of weeks that pass; f(x) = how many bags of cat food remain.

a. What numbers are the slope and the intercept? m = ______, b = ______

b. Write the function formula: f(x) = ______

c. After 6 weeks, how much cat food will be left? (Show work.)

d. After how many weeks will the food run out? (Show work.)

e. On graph paper: Make a graph of function f(x) from the cat food problem (see part b for the formula). Follow the usual expectations about careful drawing and complete labeling.


4. Suppose a library currently has 8,000 books, and is buying 500 more books per year.

Let x = the number of years from now.

a. Describe the output variable in words. f(x) = ______

b. Write the function formula.

c. How many years will it take for the library collection to reach 12,000 books?

d. On graph paper: Make a graph of function f(x) from the library problem (see part c for the formula). Follow the usual expectations about careful drawing and complete labeling.

5. A car’s gas mileage is 30 miles per gallon.

Let x = the number of gallons of gas that the car has used.

a. Give a word description of an output variable. f(x) = ______

b. What is the starting value in this problem?
Hint: If x = 0, how far does the car go?

c. Write the function formula.

d. Chicago is 750 miles away. How many gallons of gas are needed to drive to Chicago?

e. On graph paper: Make a graph of function f(x) from the car problem (see part c for the formula). Follow the usual expectations about careful drawing and complete labeling.


6. A donut shop charges 40 cents for each donut, plus a one-time charge of 20 cents (topay for the box, the service, etc.).

Let x stand for the number of donuts bought; let f(x) stand for the cost of the order in dollars.

a. Write the function formula equation.

b. What is the price of a box of 12 donuts? Remember to show work.

c. A donut order costs $8.60. How many donuts were bought?

d. Another donut order costs $15.00. How many donuts were bought?

e. Only one of these three dollar amounts could be the cost of a donut order:

$10.00, $11.00, $12.00.

Tell which one, and explain why the other prices aren’t possible.


7. These questions ask you to make up your own word problems. Be creative. Try to make your problems about different situations from what you’ve seen in previous problems.

a. Make up your own word problem for which the function formula would be f(x) = 3x + 20.

b. Make up your own word problem for which the function formula would be f(x) = 10 x.

c. Make up your own word problem for which the function formula would be f(x) = 80 – 5x.