Linear Models
40: chapter 3, 41: chapter 8, 115: 1.2, 130: 1.3, 141: 4.1, 150A- review
Objective 1: Average Rate of Change and Linear Functions
Example: The following table represents a comparison of the cost to manufacture items, to their retail price. Values are in dollars.
Cost to Manufacture
Retail Price
10 25
15 35
25 42
30 50
Let f be a function with these ordered pairs as points. f is the retail price as a function of the cost to manufacture. Find the average rate of change of f from:
1. 10 to 15
2. 15 to 25
3. 25 to 30
The average rate of change for a function
( )
2 1( )
1 22 1
f from x to x is
f x f x
x x
−
−
Example: The following table represents a comparison of the age of a car to the value of the car.
Age of car (in years) Value of car (in dollars)
2 40000
6 35000
8 32500
12 27500
Let f be a function with these ordered pairs as points. f is the value of the car as a function of the age of the car. Find the average rate of change of f from:
1. 2 to 6
2. 6 to 8
3. 8 to 12
Example: Let f be defined by the following set of ordered pairs.
x y 5 8 7 5 11 -1 13 -4
Pause the video to try this one on your own, then restart when you are ready to check your answer.
For each problem, let f be defined by the set of ordered pairs in the table.
1.
a) Determine the average rate of change between each two consecutive points.
b) Determine if f is a linear function.
1. Find the average rate of change from a. 5 to 7
b. 7 to 11
c. 11 to 13
2. Is f a linear function?
x y -2 2 4 4 10 6 13 8
2.
a) Determine the average rate of change between each two consecutive points.
b) Determine if f a linear function is.
Restart when you are ready to check your answers.
Objective 2: Find a Linear Model given Two Pairs of Data
The slope of a line is defined as:
2 1
2 1
y y
slope m
x x
= = −
−
Let (x,y) be any point on the line.
( )
2 1
2 1
1 1
1 1
y y
m x x
y y
m x x
y y m x x
= −
−
= −
−
− = −
x y 1 10 5 4 7 1 11 -5
Point-slope form for the equation of a line:
( )
( )
1 1
1 1
y - y = m x - x
where m is the slope and
x , y is a point on the line
Example: Find the equation of the line through the points (3,-5) and (7,2). Write your answer in slope- intercept form.
Example: A manufacturing firm had a profit of $30,000 in the first year of selling a new product. They had a profit of $66,000 in the fourth year. The ratio of change in profit to change in time is constant.
Find an equation that relates the company’s profit and time. Find and interpret the slope.
Slope-intercept form for the equation of a line:
( )
y = mx + b
where m is the slope
and 0,b is the y - intercept.
( )
1 1
y − y = m x − x
Let (0,b) be the y-intercept of the line.
( 0 )
y b m x
y b mx
y mx b
− = −
− =
= +
Example: Jamel’s candle making shop uses 600 pounds of wax to create 500 candles. They use 800 pounds to make 750 candles. Assuming the rate of candles to pounds of wax is constant, develop a linear model that indicates candles as a function of pounds of wax. Find and interpret the slope.
Example: The average price of a new home sold in the U.S. was $342,800 in 2014. In 2009 the average price of a new home sold in the U.S. was $268,200. Let y be the average price of a new home. Let x be number of years, x = 0 represents 2009. Write a linear model that indicates average price in terms of years since 2009. Find and interpret the slope and y-intercept.
Pause the video to try this one on your own, then restart when you are ready to check your answer.
1. Alonzo’s toy manufacturing has determined that if they sell a bike for $60, they will sell 2000 per day. If they raise the price to $80, they will only sell 1500 per day. Assume that the ratio of change in price to change in daily sales is constant. Let x be the price and y be the number of bikes sold per day.
Find a linear equation that models the number of bikes to the sales price.
2. The speed that sound travels through sea water is related to the temperature of the water. This is a linear relationship. Sound travels through sea water at a speed of 331 m/sec when the temperature
0
C. The speed is 343 m/sec when the temperature is
20
C. Create a linear model for the speed of sound in sea water as a function of the temperature of the water. Find and interpret the slope and y-intercept.3. The number of physical therapists in the state of Washington was 26 thousand in 2015. That number grew to 32 thousand by 2021. Assume that the number of therapists is linearly related to the year.
Create a linear model that relates the number of physical therapists in Washington to the number of years since 2015. Let y be the number of thousands of physical therapists, and x be the number of years since 2015. Let x = 0 represent 2015. Find and interpret the slope and y-intercept.
Restart when you are ready to check your answers.
Objective 3: Economic Applications
Example: If Scooter’s Market sells apples at $9 per box, then the demand will be 200,000 boxes and the supply will be 320,000 boxes. If they sell the apples at $8.50 per box, then the demand will be 300,000 boxes and the supply will be 270,000 boxes. Assume that the relationship between supply and price is linear, and the relationship between demand and price is linear.
a) Find a price-supply equation for this situation. Let y = the price and x = the supply in thousands of boxes.
The price that a company charges for their product depends on supply and demand. If supply is much greater than demand, the price will need to drop in order to sell the product.
An Equilibrium Point is when supply equals demand.
b) Find a price-demand equation for this situation. Let y = the price and x = the demand in thousands of boxes.
c) Find the equilibrium point for your model.
Example: The supply and demand equations for selling t-shirts are:
( )
( )
600 50
1200 25
S p p
D p p
= − +
= −
where p is the price of a t-shirt. Find the equilibrium price and quantity.
(Note that the variables are switched here from our previous example. The independent variable p (typically x) is the price and the dependent variable (typically y) is the supply or demand.)
Example: Suppose that the revenue equation for a company selling water bottles is R(x) = 5x, and the cost equation is C(x)= 1.5x + 11,500. Find the break-even point for the company.
Pause the video to try this one on your own, then restart when you are ready to check your answer.
1. If Ezekiel’s Market sells a t-shirt for $8.40, then the demand will be 3800 t-shirts and the supply will be 5000 t-shirts. If they sell the t-shirts for $5.40, then the demand will be 5300 t-shirts and the supply will be 2000 t-shirts. Assume that the relationship between supply and price is linear, and the
relationship between demand and price is linear.
a) Find a price-supply equation for this situation. Let y = the price and x = the supply.
b) Find a price-demand equation for this situation. Let y = the price and x = the demand.
c) Find the equilibrium point for your model.
2. The supply and demand equations for selling sandwiches are:
( )
( )
2000 3000
10000 1000
S p p
D p p
= − +
= −
where p is the price of a sandwich. Find the equilibrium price and quantity.
3. Suppose that the revenue equation for a company selling toy trains is R(x) = 12x, and the cost equation is C(x)= 10x + 15000. Find the break-even point for the company.
Restart when you are ready to check your answers.
Objective 4: Domain, Slope and y-intercept
Example: Hertz rents cars for $45 plus $8 per mile. Create a linear model for the cost of renting a car.
Find and interpret the slope and y-intercept. What is the implied domain?
Example: Pacific Phone company charges 40¢ for the first minute and 11¢ for each additional minute or portion of a minute after the first minute. Create a linear model for the cost of a phone call. Find and interpret the slope and y-intercept.
Example: Antoinette’s Bike shop has daily fixed costs of $2300 and each bicycle will cost $110 to manufacture. Write a linear equation that expresses the cost C of manufacturing x bicycles in a day. Find and interpret the slope and y-intercept. What is the implied domain?
Example: Kiona works at Newley’s department store. She works on commission. She earns $400 per week plus 15% of her sales. Create a linear model for her weekly income.
Example: A bucket initially contains one gallon of water. Water is leaking out of the bucket at a rate of 0.06 gallons per minute. Write a linear equation that indicates the amount of water y in the bucket after x minutes. Find and interpret the slope and y-intercept.
Example: A farmer purchased a tractor for $170,000. She assumes it will have a trade-in value of
$75,000 in ten years. She uses straight-line depreciation to determine the value of her tractor. Write a linear model for the value of the tractor x years after it is purchased. Find and interpret the slope and y- intercept. What is the implied domain?
Example: A company purchased a new machine for $100,000. The company will depreciate the machine using straight-line depreciation over 8 years. Write a linear model for the value of the machine x years after it is purchased. Find and interpret the slope and y-intercept. What is the implied domain?
Pause the video to try this one on your own, then restart when you are ready to check your answer.
1. Hertz rents cars for $62 plus $9 per mile. Create a linear model for the cost of renting a car. Find and interpret the slope and y-intercept. What is the implied domain?
2. Hazel’s Bike shop has daily fixed costs of $1800 and each bicycle will cost $90 to manufacture. Write a linear equation that expresses the cost C of manufacturing x bicycles in a day. Find and interpret the slope and y-intercept.
3. Keime sells cars for Nissan. She works on commission. She earns $300 per week plus 10% of her sales. Create a linear model for her weekly income.
4. A company purchased a new machine for $55,000. The company will depreciate the machine using straight-line depreciation over 10 years. Write a linear model for the value of the machine x years after it is purchased. Find and interpret the slope and y-intercept. What is the implied domain?
Restart when you are ready to check your answers.
