Function Notation
10 min
Narrative
The goal of this Warm-up is to motivate the need for a notation that can be used to communicate about functions.
Students analyze three graphs from an earlier lesson, interpret various points on the graphs, and use their analyses to answer questions about the situations. This work requires students to make careful connections between points on the graphs, pairs of input and output values, and verbal descriptions of the functions. Students find that unless each feature and the function being referred to is clearly articulated, which could be tedious to do, what they wish to communicate about the function may be ambiguous or unclear.
When answering the last two questions, students are likely to find the prompts lacking in specificity and to probe: “for which day?” Suggest that they answer based on their interpretation of the questions.
Then, look for students who assume that the questions refer to one particular function and those who assume they refer to all three functions (and consequently answer them for each function). Ask them to share their interpretations during the whole-class discussion.
Student Task
Here are the graphs of some situations you saw before. Each graph represents the distance of a dog from a post as a function of time since the dog owner left to purchase something from a store. Distance is measured in feet, and time is measured in seconds.
-
Use the given graphs to answer these questions about each of the three days:
-
How far away was the dog from the post 60 seconds after the owner left?
Day 1:
Day 2:
Day 3:
-
How far away was the dog from the post when the owner left?
Day 1:
Day 2:
Day 3:
-
The owner returned 160 seconds after he left. How far away was the dog from the post at that time?
Day 1:
Day 2:
Day 3:
-
How many seconds passed before the dog reached the farthest point from the post it could reach?
Day 1:
Day 2:
Day 3:
-
- Consider the statement, “The dog was 2 feet away from the post after 80 seconds.” Do you agree with the statement?
- What was the distance of the dog from the post 100 seconds after the owner left?
Sample Response
-
Day 1:
- 4 feet
- 1.5 feet
- about 3.7 feet
- about 50 seconds
Day 2:
- 4 feet
- 1.5 feet
- 4 feet
- about 34 seconds
Day 3:
- 4 feet
- 1.5 feet
- 0.5 foot
- about 3 seconds
- Sample response: This is true for Day 3, but not the other days. More information about the statement is needed.
- Sample response: On Day 1, the dog was about 2.1 feet from the post. On Day 2, the dog was 4 feet from the post. On Day 3, the dog was about 2.1 feet from the post again.
Activity Synthesis (Teacher Notes)
Invite students to share their response to the first set of questions.
To help illustrate that it could be tedious to refer to a specific part of a function fully and precisely, ask each question completely for each of the three days. (For instance, “How far away was the dog from the post 60 seconds after the owner left on Day 1? How far away was the dog from the post 60 seconds after the owner left on Day 2?”) If students offer a numerical value (for instance, “1.5 feet”) without stating what question it answers or to what quantity it corresponds to, ask them to clarify.
Next, select previously identified students to share their responses to the last two questions. Regardless of whether students chose to answer them for a particular day or for all three days, point out that the answers depend on the day. When the day (or the function) is not specified, it is unclear what information is sought.
Explain that sometimes we need to be pretty specific when talking about functions. But being specific could require many words and become burdensome. Tell students that they will learn about a way to describe functions clearly and succinctly.
Standards
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- HSF-IF.B.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. <span>Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.</span>
- F-IF.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
- F-IF.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
- F-IF.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
- HSF-IF.A.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If <span class="math">\(f\)</span> is a function and <span class="math">\(x\)</span> is an element of its domain, then <span class="math">\(f(x)\)</span> denotes the output of <span class="math">\(f\)</span> corresponding to the input <span class="math">\(x\)</span>. The graph of <span class="math">\(f\)</span> is the graph of the equation <span class="math">\(y = f(x)\)</span>.
15 min
10 min
Knowledge Components
All skills for this lesson
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Function Notation
10 min
Narrative
The goal of this Warm-up is to motivate the need for a notation that can be used to communicate about functions.
Students analyze three graphs from an earlier lesson, interpret various points on the graphs, and use their analyses to answer questions about the situations. This work requires students to make careful connections between points on the graphs, pairs of input and output values, and verbal descriptions of the functions. Students find that unless each feature and the function being referred to is clearly articulated, which could be tedious to do, what they wish to communicate about the function may be ambiguous or unclear.
When answering the last two questions, students are likely to find the prompts lacking in specificity and to probe: “for which day?” Suggest that they answer based on their interpretation of the questions.
Then, look for students who assume that the questions refer to one particular function and those who assume they refer to all three functions (and consequently answer them for each function). Ask them to share their interpretations during the whole-class discussion.
Student Task
Here are the graphs of some situations you saw before. Each graph represents the distance of a dog from a post as a function of time since the dog owner left to purchase something from a store. Distance is measured in feet, and time is measured in seconds.
-
Use the given graphs to answer these questions about each of the three days:
-
How far away was the dog from the post 60 seconds after the owner left?
Day 1:
Day 2:
Day 3:
-
How far away was the dog from the post when the owner left?
Day 1:
Day 2:
Day 3:
-
The owner returned 160 seconds after he left. How far away was the dog from the post at that time?
Day 1:
Day 2:
Day 3:
-
How many seconds passed before the dog reached the farthest point from the post it could reach?
Day 1:
Day 2:
Day 3:
-
- Consider the statement, “The dog was 2 feet away from the post after 80 seconds.” Do you agree with the statement?
- What was the distance of the dog from the post 100 seconds after the owner left?
Sample Response
-
Day 1:
- 4 feet
- 1.5 feet
- about 3.7 feet
- about 50 seconds
Day 2:
- 4 feet
- 1.5 feet
- 4 feet
- about 34 seconds
Day 3:
- 4 feet
- 1.5 feet
- 0.5 foot
- about 3 seconds
- Sample response: This is true for Day 3, but not the other days. More information about the statement is needed.
- Sample response: On Day 1, the dog was about 2.1 feet from the post. On Day 2, the dog was 4 feet from the post. On Day 3, the dog was about 2.1 feet from the post again.
Activity Synthesis (Teacher Notes)
Invite students to share their response to the first set of questions.
To help illustrate that it could be tedious to refer to a specific part of a function fully and precisely, ask each question completely for each of the three days. (For instance, “How far away was the dog from the post 60 seconds after the owner left on Day 1? How far away was the dog from the post 60 seconds after the owner left on Day 2?”) If students offer a numerical value (for instance, “1.5 feet”) without stating what question it answers or to what quantity it corresponds to, ask them to clarify.
Next, select previously identified students to share their responses to the last two questions. Regardless of whether students chose to answer them for a particular day or for all three days, point out that the answers depend on the day. When the day (or the function) is not specified, it is unclear what information is sought.
Explain that sometimes we need to be pretty specific when talking about functions. But being specific could require many words and become burdensome. Tell students that they will learn about a way to describe functions clearly and succinctly.
Standards
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- HSF-IF.B.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. <span>Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.</span>
- F-IF.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
- F-IF.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
- F-IF.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
- HSF-IF.A.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If <span class="math">\(f\)</span> is a function and <span class="math">\(x\)</span> is an element of its domain, then <span class="math">\(f(x)\)</span> denotes the output of <span class="math">\(f\)</span> corresponding to the input <span class="math">\(x\)</span>. The graph of <span class="math">\(f\)</span> is the graph of the equation <span class="math">\(y = f(x)\)</span>.