Piecewise Functions
10 min
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.
Student Task
A 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.
A
B
C
D
Sample Response
Graph B. Sample explanation: There are two different pricing rules for the price of yogurt—one rule for buying between 0 and 8 ounces and one for buying amounts greater than 8 ounces and up to 12 ounces.
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 p represents the price of yogurt for a serving of w ounces, then the rules would be:
p(w)=\begin {cases} \begin {align} &0.50w, &\quad& 0<w></w></span></p> <p>Display the notation for all to see, and demonstrate how to read the notation: "Function <span class="math">\(p has a value of 0.50w if the input w is greater than 0 and is no more than 8. Function p 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:
\(p(w)=\begin {cases} \begin {align} &0.50w &\text{if} \quad & 0
Standards
- F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
- F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
- F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
- F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
- F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
- F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
- F-IF.C·Analyze functions using different representations
- HSF-IF.B.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <span>For example, if the function <span class="math">\(h(n)\)</span> gives the number of person-hours it takes to assemble <span class="math">\(n\)</span> engines in a factory, then the positive integers would be an appropriate domain for the function.</span>
- HSF-IF.C·Analyze functions using different representations.
- F-IF.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
- F-IF.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
- F-IF.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
- F-IF.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
- F-IF.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
- HSF-IF.C.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.