Applications of linear functions (Tillämpning av linjära funktioner)

Ämne / Topic

Using functions to describe real-world situations.

Inlärningsmål / Learning Goals

Be able to determine values from graphs of real-world situations.

Be able to write the function to describe real-world situations.

Google Slides & Videos

3. Aktiviteter / Key Activities

In this lesson the teacher will guide learners on how to apply their knowledge of linear functions to real world situations.

To do this, the teacher will go through two examples with the learners, before giving them an example to go through with a partner.

Example 1

Joe’s Warehouse has banquet facilities to accommodate a maximum of 250 people. When the manager quotes a price for a banquet she is including the cost of renting the room plus the cost of the meal. A banquet for 70 people costs $1300. For 120 people, the price is $2200.

1. Plot a graph of cost versus the number of people.

2. From the graph, estimate the cost of a banquet for 150 people.

3. Determine the slope of the line. What quantity does the slope of the line represent?

4. Write an equation to model this real-life situation.

a. On the x-axis is the number of people and on the y-axis is the cost of the banquet.

The equation to model the real-life situation is y=18x+40. The variables should be changed to match the labels on the axes. The equation that best models the situation is c=18n+40 where ‘c’ represents the cost and ‘n’ represents the number of people.

Example 2

Some college students who plan on becoming math teachers decide to set up a tutoring service for high school math students. One student was charged $25 for 3 hours of tutoring. Another student was charged $55 for 7 hours of tutoring. The relationship between the cost and time is linear.

1. What is the independent variable?

2. What is the dependent variable?

3. What are two data values for this relationship?

4. Draw a graph of cost versus time.

5. Determine an equation to model the situation.

6. What is the significance of the slope?

7. What is the cost-intercept?

8. What does the cost-intercept represent?

1. The cost for tutoring depends upon the amount of time. The independent variable is the time.

2. The dependent variable is the cost.

3. Two data values for this relationship are (3, 25) and (7, 55).

4. On the x-axis is the time in hours and on the y-axis is the cost in dollars.

5. Use the two data values (3, 25) and (7, 55) to calculate the slope of the line. m=7.5.Determine the y-intercept of the graph.

y = mx + b

25 = 7.5 (3) + b

25 = 22.5 + b

25 - 22.5 = b

2.5 = b

The relationship is cost in dollars versus time in hours. The equation could also be written as

c = 7.5x + 2.5

6. The slope of 7.5 means that it costs $15.00 for 2 hours of tutoring. If the slope is expressed as a decimal, it means that it costs $7.50 for 1 hour of tutoring.

7. The cost-intercept is the y-intercept. The y-intercept is (0, 2.50). This value could represent the cost of having a scheduled time or the cost that must be paid for cancelling the appointment. In a problem like this, the y-intercept must represent a meaningful quantity for the problem.

Example 3

Juan drove from his mother’s home to his sister’s home. After driving for 20 minutes he was 62 miles away from his sister’s home and after driving for 32 minutes he was only 38 miles away. The time driving and the distance away from his sister’s home form a linear relationship.

1. What is the independent variable? What is the dependent variable?

2. What are the two data values?

3. Draw a graph to represent this problem. Label the axis appropriately.

4. Write an equation expressing distance in terms of time driving.

5. What is the slope and what is its meaning in this problem?

6. What is the time-intercept and what does it represent?

7. What is the distance-intercept and what does it represent?

8. How far is Juan from his sister’s home after he had been driving for 35 minutes?

a. The independent variable is the time driving. The dependent variable is the distance.

b. The two data values are (20, 62) and (32, 38).

c. On the x-axis is the time in minutes and on the y-axis is the distance in miles.

d. (20, 62) and (32, 38) are the coordinates that will be used to calculate the slope of the line.

y = y2 - y1

x2 - x1

y = 38 - 62

32 - 20

y = -24

12

y = -2

y = mx + b

62 = -2 (20)+ b

62 = - 40 + b

62 + 40 = b

102 = b

y = mx + b

y = -2x + 102

e. The slope is

−2=−2/1 = -2 (miles)/1 . The slope means that for each minute of driving, the distance that Juan has to drive to his sister’s home is reduced by 2 miles.

f. The time-intercept is actually the x-intercept. This value is:

d = -2t + 102

0 = -2t + 102

2t = 102

2t = 102

2 = 2

t = 51 minutes

The time-intercept is 51 minutes and this represents the time it took Juan to drive from his mother’s home to his sister’s home.

g. The distance-intercept is the y-intercept. This value has been calculated as (0, 102). The distance-intercept represents the distance between his mother’s home and his sister’s home. The distance is 102 miles.

d = -2t + 102

d = -2 (35) + 102

d = -70 + 102

d = 32 miles

After driving for 35 minutes, Juan is 32 miles from his sister’s home.

4. Kolla vad eleverna kan / Check for understanding / Exit ticket

Review

Players on the school soccer team are selling candles to raise money for an upcoming trip. Each player has 24 candles to sell. If a player sells 4 candles a profit of $30 is made. If he sells 12 candles a profit of $70 is made. The profit and the number of candles sold form a linear relation.

1. State the dependent and the independent variables.

2. What are the two data values for this relation?

3. Draw a graph and label the axis.

4. Determine an equation to model this situation.

5. What is the slope and what does it mean in this problem?

6. Find the profit-intercept and explain what it represents.

7. Calculate the maximum profit that a player can make.

8. Write a suitable domain and range.

9. If a player makes a profit of $90, how many candles did he sell?

10. Is this data continuous, discrete, or neither? Justify your answer.

Jacob leaves his summer cottage and drives home. After driving for 5 hours, he is 112 km from home, and after 7 hours, he is 15 km from home. Assume that the distance from home and the number of hours driving form a linear relationship.

11. State the dependent and the independent variables.

12. What are the two data values for this relationship?

13. Represent this linear relationship graphically.

14. Determine the equation to model this situation.

15. What is the slope and what does it represent?

16. Find the distance-intercept and its real-life meaning in this problem.

17. How long did it take Jacob to drive from his summer cottage to home?

18. Write a suitable domain and range.

19. How far was Jacob from home after driving 4 hours?

20. How long had Jacob been driving when he was 209 km from home?

5. Stängning / Closure / Summary of lesson

Review real-world situations. Learners will come up with their own examples and swap them with a partner, who will complete the exercise. Partners will mark each others work and provide feedback.