**2.6 Piecewise linear functions**

Commonly, a function isn’t just one line, but rather consists of two or more pieces, each piece being a line with a different slope. Such a function is known as a piecewise linear function. Such a function is graphed by graphing each piece.

**2.6.1: Piecewise linear function: Day’s earnings with overtime pay.**

Start

2x speed

Hours worked

Wages ($)

20

1

2

3

4

5

6

7

8

9

10

11

12

13

14

40

60

80

100

120

140

160

y = 10x

$10/hr

$15/hr

10

10

1

15

1

Piece

Piece

$155

keyboard_arrow_downCaptions

**2.6.2: Piecewise linear function: Day’s earnings considering overtime pay.**

Consider the example above.

1)

Looking at the graph above, how much does a person earn for a day if the person worked 9 hours?

$80

$95

$100

2)

No one line represents the function. To graph, the first piece’s line is:

0-8 hours: y = 10x

The second piece’s line starts at the first piece’s end, meaning x is 8 and y is 80. What is that line’s slope?

10

15

3)

Drawing a line with slope 15, starting at x of 8 and y of 80, can be done by drawing a second point, obtained by going 1 unit to the right and up by how many units?

1

15

95

4)

No one equation can represent the piecewise linear function. Instead, the function is described like this:

0-8 hours: y = 10x

8+ hours: y = 80 + 15(x – ?)

(The 8+ means “8 or more”).

1

8

5)

For more than 8 hours worked, the total earnings are y = 80 + 15(x – 8). A person works a long 20 hour day. How many dollars does the person earn for that long day?

$200

$260

$380

6)

Suppose the company above pays double-time (20 dollars/hour) for above 12 hours. The equations are then:

0-8 hours: y = 10x

8-12 hours: y = 80 + 15(x – 8)

12+ hours: y = ___ + 20(x – 12)

80

140

155

**2.6.3: Water companies penalize wasteful water use with higher rates for more gallons: A piecewise linear function.**

Start

2x speed

120

100

80

60

40

20

Daily cost (cents)

Gallons used per day

25

50

75

100

125

150

175

200

0.2 cents/gal

0.5 cents/gal

1 cent/gal

225

250

36

30

24

18

12

6

Monthly cost ($)

45

Continues

28

keyboard_arrow_downCaptions

**2.6.4: Piecewise linear function: Water use.**

Consider the example above.

1)

What is the rate per gallon for an apartment that uses under 100 gallons per day?

0.2 cents/gal

0.5 cents/gal

1 cent/gal

2)

What is the rate per gallon for an apartment that uses over 150 gallons per day?

0.2 cents/gal

0.5 cents/gal

1 cent/gal

3)

Knowing that using 150 gallons in a day costs 45 cents, and every gallon above 150 gal costs 1 cent, what would 400 gallons cost? Hint: Add the cost for those additional 400 – 150 = 250 gal to the 45 cents.

295 cents

400 cents

4)

Which equation calculates total water cost for water use between 100 and 150 gal?

y = 0.2x

y = 0.5x

y = 20 + 0.5x

y = 20 + 0.5(x – 100)

**2.6.5: Companies take advantage of people’s lack of math skills: Health insurance example.**

Start

2x speed

6000

5000

4000

3000

2000

1000

6000

5000

4000

3000

2000

1000

User pays ($)

Estimated medical costs ($)

$500 ded plan

$2000 ded plan

$1000 ded plan

1400

1900

Better if estimated medical costs are low

Better if estimated medical costs are high

Never better

2750

500

keyboard_arrow_downCaptions

**2.6.6: Figuring out insurance plans.**

Refer to the graph for insurance plans in the animation above.

1)

Use the graph to predict the amount the user pays for medical costs of $800 (the cost of a sprained ankle) if the user has the $1,000 deductible plan (orange graph).

Round to the nearest $500.

$

CheckShow answer

2)

Which deductible plan is cheapest for someone who has medical costs of $800 (the cost of a sprained ankle)?

Type 500, 1000, or 2000.

$ deductible plan

CheckShow answer

3)

Use the graph to predict the user pay for medical costs of $5,000 (the cost of breaking a bone) if the user has the $500 deductible plan (red graph).

Round to the nearest 1000.

$

CheckShow answer

4)

Which deductible plan is cheapest for someone who has medical costs of $5000 (the cost of breaking a bone)?

Type 500, 1000, or 2000.

$ deductible plan

CheckShow answer

5)

Which plan is never the cheapest for any level of medical costs?

Type 500, 1000, or 2000.

$ deductible plan

CheckShow answer

6)

For the $2,000 deductible plan, the user pays $1,400 in yearly fees, the $2,000 deductible, and 20% of all medical costs over $2,000. What would the user pay for total medical costs of $15,000 (the cost of having a baby)?

$

CheckShow answer

7)

For the $500 deductible plan, the user pays $2,750 in yearly fees, the $500 deductible, and 20% of all medical costs over $500. What would the user pay for total medical costs of $15,000 (the cost of having a baby)?

$

CheckShow answer

8)

How much does a user save with the $2000 deductible plan over the $500 deductible plan for medical costs of $15,000 (the cost of having a baby)?

$

CheckShow answer

**2.6.1: Piecewise linear functions.**

404272.1880072.qx3zqy7

Start

A company pays $10 per hour for up to 8 hours of work. What is the equation for total pay (y) for hours worked (x), for 0-8 hours of work (x)? y-intercept: Slope: Equation: y =

1

2

3

4

Check

Next

1

2

3

4

How much does a user save with the $2000 deductible plan over the $500 deductible plan for medical costs of $15,000 (the cost of having a baby)?

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