MAT 117 College Algebra Module 2 Linear Functions

1
MAT 117 College AlgebraModule 2Linear Functions
Thinking about constant change!
2
Stacking Books

• Imagine that youre creating a stack of books
  (All the same size)
• What quantities can be measured in this
  situation?
• What quantities are changing in this situation?
  How can you represent the things that are
  changing?

3
Stacking Books

• 3. Using the books or another method, determine
  the height of a stack of books and record your
  data in the table below.

4
Stacking Books

• 4. As the number of books changes from 1 to
  2, how does the height of the stack change? As
  the number of books changes from 2 to 3, how does
  the height of the stack change? As the number of
  books changes from 3 to 4, how does the height of
  the stack change? As the number of books changes
  from 1 to 3, how does the height of the stack
  change? As the number of books changes from 3 to
  5, how does the height change?
• 5. For every increase of one book, by what
  amount does the height change? For every increase
  in two books, by what amount does the height
  change?
• 6. What is the ratio of height to number of
  books? What is the ratio of the change in height
  to the change in the number of books? Does the
  ratio hold for any amount of change in height and
  any amount of change in books in this situation?
  Explain.
• 7. What are the units for the ratios that you
  determined?
• 8. What does the ratio of the changes represent?

5
Stacking Books

• 9. If the bookshelves in our classroom are 84cm
  across, how many books can we put on each shelf?
  Explain your thinking.
• 10. If the bookshelves in our classroom are 100
  cm across, how many books can we put on each
  shelf? Explain your thinking.

6

• In your groups, create a formula that will allow
  you to produce the height of the stack (in
  cm.) for any number of books. What do your
  letters (variables) represent?
• Describe what each letter in your formula
  represents in the context of this situation.
• 13. Using your formula, determine how the height
  changes as the number of books increases from 25
  to 28.
• 14. In your groups, create a graph that
  represents the height of the stack in terms of
  (as a function of) the number of books.
• 15. Use your formula to determine the number of
  books in a stack that is 129.5 centimeters high.
  Explain why your procedure works.

7
Connecting to Proportionality

• Can you say that the number of books is
  proportional to the height?
• -In your groups build an argument either for or
  against this claim.

8
Multiple Representations - Tables
9
Multiple Representations - Graph
10

• Now consider the situation of measuring the
  height of the book stack from the floor instead
  of the table (leaving the stack on the table).

• What new quantity do you need to measure?
• How does this new height change the algebraic
  function? The graph? The table?
• Is the relationship between height of books and
  number of books still proportional?

11
Water Flowing Out of a Tub

• You just finished taking a bath and began to
  drain the water from the tub. The water is
  draining at a rate of 1.5 gallons per minute.
  After draining for 4 minutes, there is still 38
  gallons in the tub.
• In your groups, answer the following.
• How many gallons of water were in the tub before
  you began draining?
• Define a function that relates the number of
  gallons in the tub and the number of minutes
  since you began draining.???
• Sketch a graph that shows how many gallons were
  in the tub at each moment in time for the first
  10 minutes when the water was draining.

12
Water Flowing Out of a Tub

• You just finished taking a bath and began to
  drain the water from the tub. The water is
  draining at a rate of 1.5 gallons per minute.
  After draining for 4 minutes, there is still 38
  gallons in the tub.
• What is the meaning of the y-intercept?
• What is the rate of change of the function and
  what does it tell you about this situation?
• How does the amount of water in the tub change
  when the draining time increases from 2 minutes
  to 6 minutes.
• How does the amount of water in the tub change
  when the draining time increases from 10 minutes
  to 15 minutes.

13
Calculating Constant Rate of Change
14
Group Activity Hotel

• Hotel management finds that if they spend no
  money on renovations, they will be able to rent
  100 rooms per night. They find that for every
  5000 spent on renovation, they will be able to
  rent an additional 25 rooms.
• What two quantities are co-varying (changing
  together) in this situation?
• If n is the number of rooms the hotel can rent
  each night and a is the amount of money spent on
  renovations, create a function that describes the
  number of rooms that can be rented in terms of
  (as a function of) the amount of money spent on
  renovation. Represent this function using a
  table, formula and graph and discuss how the
  three representations are related.
• If hotel management wants to rent 200 rooms per
  night, how much do they need to spend on
  renovations?
• If hotel management spends 22,000 on
  renovations, how many rooms will they be able to
  rent per night?

15
Telephone Companies

• Company A charges .37 per minute and company B
  charges 13.95 per month plus .22 per minute.
• Determine a function that describes the cost per
  minute for one month of service for company A.
  Determine a function that describes the cost per
  minute for one month of service of company B.
• For what number of minutes is company B
• cheaper than company A?

16
College Meal Plans

• In a college meal plan you pay a membership fee
  then all your meals are at a fixed price per
  meal.
• If 30 meals cost 152.50 and 60 meals cost 250,
  find the membership fee and the price per meal.
• Write the function for the cost of a meal plan,
  C, in terms of the number of meals, n.
• Sketch a graph of the function.
• Similar to finding the equation of a line given 2
  points
• Example (2,5) and (-1,7)

17
Creating Linear Functions

• Write a linear function with slope of 5 and
  y-intercept of (0,-2).
• Write a linear function with a slope of 4/3 and a
  point (6,4)
• Write a linear function that goes through the
  points (-2,9) and (5,3).
• Alternative Notation f(-2)9 f(5)3

18
What is a function???

• For every input, theres exactly one output
• Relation/Rule between two things
• Inputs Outputs
• Domain Range (Possible inputs/outputs)
• Vertical Line Test
• Notation f(2)4
• f(input)output f(independent)dependent

19
Functions - Formulas

• f(x)2x3
• Is this a function? Why/why not?
• Is it possible to input the same value multiple
  times and get different outputs?
• So, for this function
• f(1)5
• f(2)7
• f(3)9

20
Functions - Tables

• The following table shows Average Monthly
  Rainfall at Chicago OHare Airport

Is the relation of Month to Rainfall a function?
That is, given the month, can you tell how much
rainfall there has been in that month? Given
the rainfall, can you tell what month it is?
21
Functions - Graphs

• Is the following a function? Why/why not?

22
Functions - Words

• Are the following relationships functions?
  Why/why not?
• Phone numbers to Houses
• Houses to Phone numbers
• License Plates to Cars
• Names to People
• People to Names
• Social Security Numbers to People

23
Answer the following

• Let f(t) be the number of people, in millions,
  who own cell phones t years after 1990. Explain
  the meaning of
• f(10)100.3 f(15) 126.2
• A bug starts out 10 ft from a light, flies closer
  to the light, then farther away, then closer than
  before, then farther away. Finally the bug hits
  the bulb and flies off.
• Sketch the distance of the bug from the light as
  a function of time on a graph.

24
Population

• Suppose we know the population of a city is
  23,000 in 1982 and 21,000 in 1986.
• Assuming the population has been declining at a
  constant rate since 1970, find a formula for the
  population as a function of time, t
• Let t be the number of years since 1970 (i.e. t
  0 corresponds to the year 1970)
• Use your model (function) to determine the
  population in the year 2000.
• When will the population reach 0 people?