Unit 5, Lesson 12
Piecewise Functions
Algebra 1
12.1
Warm-up
Instructional Routines
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NoneActivity Narrative
This Warm-up presents a context for making sense of piecewise functions. Students are given a situation in which one quantity (price) is a function of another (ounces of yogurt), but different rules apply to different values of input. Then, they identify a graph that represents the function.
Students learn that a function can be defined by a set of rules, and that the graph of such a function has different features for different parts of the domain.
Launch
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NoneA self-serve frozen yogurt store sells servings up to 12 ounces. It charges \$0.50 per ounce for a serving between 0 and 8 ounces and \$4 for any serving greater than 8 ounces and up to 12 ounces.
Choose the graph that represents the price as a function of the weight of a serving of yogurt. Be prepared to explain how you know.
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Student Response
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Building on Student Thinking
Activity Synthesis
Invite students to share their response and reasoning. Discuss questions such as:
- “How much would it cost to get a 4-ounce serving?” (\$2)
- “How much for a 10-ounce serving?” (\$4)
- “You just applied two different rules. How did you know which one to use in each case?” (It depended on the weight, on whether it was greater or less than 8 ounces.)
- “How much would it cost to get an 8-ounce serving?” (\$4)
Explain that the relationship between the serving weight of yogurt and the price is an example of a piecewise function, in which different rules are applied to different input values to find the output values. Point out that the graph is made up of two (in this case linear) pieces that correspond to the two rules.
Ask students if they can think of other situations that could be represented by piecewise functions. Students may bring up examples, such as parking rates, postage or shipping rates, or pricing by age.
Explain that one way to write the rules for this type of function is by using the “cases” notation. The rule for each interval of input is treated as a separate case, with the output listed first, followed by the interval of the input.
If the function represents the price of yogurt for a serving of ounces, then the rules would be:
Display the notation for all to see, and demonstrate how to read the notation: "Function has a value of if the input is greater than 0 and is no more than 8. Function has a value of 4 if the input is greater than 8 and is no more than 12."
Sometimes, instead of a comma, the conditional word "if" is used in the notation:
Lesson Synthesis
Tell students that function gives the cost, in dollars, of parking for hours in the parking lot of the stadium. It can be represented with this equation in cases notation:
Discuss questions such as:
- "What do the numbers 3, 6, 10, and 15 mean?" (he different values of output, in this case, cost in dollars)
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"How would you explain the parking rates in words?" (One possibility:
- Up to 30 minutes: \$3.00
- More than 30 minutes but no more than 1 hour: \$6.00
- More than 1 hour but no more than 2 hours: \$10.00
- More than 2 hours but no more than 8 hours: \$15.00)
- "Why would we call a piecewise function?" (Different rules apply to different intervals of input.)
- "What is ? What about ?" (\$3.00; \$10.00)
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"Here is an incomplete graph of function . As is, why is this not a graph of a function?" (When the input is 0.5, 1, and 2, the graph shows two possible outputs.)
- "How can we complete the graph so that it does represent ?" (Add open or closed circles to indicate which output values correspond to 0.5, 1, and 2 hours of parking.)
- "How should we mark the endpoints of the segments?" (See completed graph.)
- "What are the domain and range of this function? (Assuming the parking lot is not accessible for longer than 8 hours, the domain is hours greater than 0 and no more than 8. The range includes only 3, 6, 10, and 15, in dollars.)
Student Lesson Summary
A piecewise function has different descriptions, or rules, for different parts of its domain.
Function gives the train fare, in dollars, for a child who is years old based on these rules:
- Free for children under 5
- \$5 for children who are at least 5 but younger than 11
- \$7 for children who are at least 11 but younger than 16
The different prices for different ages tell us that function is a piecewise function.
The graph of a piecewise function is often composed of pieces or segments. The pieces could be connected or disconnected. When disconnected, the graph appears to have breaks or steps.
Here is a graph that represents .
A graph, with origin O. The horizontal axis, age, years, scale from 0 to 16 by 1s. The vertical axis, train fare, dollars, scale from 9 to 8 by 1s. A horizontal line segment, with open circle endpoints, begins at 0 comma 0 and ends at 5 comma 0. A horizontal line segment begins with a closed circle at 5 comma 5 and ends with an open circle at 11 comma 5. A line segment begins with a closed circle at 11 comma 7 and ends with an open circle at 16 comma 7.
It is important to consider the value of the function at the points where the rule changes, or where the graph “breaks.” For instance, when a child is exactly 5 years old, is the ride free, or does it cost \$5?
On the graph, one segment ends at and another segment starts at , but the function cannot have both 0 and 5 as outputs when the input is 5!
Based on the fare rules, the ride is free only if the child is under 5, which means:
- is false. On the graph, the point is marked with an open circle to indicate that it is not included in the first segment (which represents ages that qualify for a free ride).
- is true. The point has a closed circle to indicate that it is included in the middle segment (which represents ages that qualify for the \$5 fare).
The same reasoning applies when deciding how and should be shown on the graph.
- is true because 11-year-olds ride for \$7. The point is a closed circle.
- is false because a 16-year-old no longer qualifies for a child’s fare. The point is an open circle.
The fare rules can be expressed with function notation: