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
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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
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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
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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.
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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 =
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4
Check
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