Your teacher will give you instructions for completing the table.
A customer at a bagel shop is buying 13 bagels. The shopkeeper says, “That would be $16.25.”
Jada, Priya, and Han, who are in the shop, all think it is a mistake.
Jada says to her friends, “Shouldn’t the total be $13.25?”
Priya says, “I think it should be $13.00.”
Han says, “No, I think it should be $11.25.”
Explain how the shopkeeper, Jada, Priya, and Han could all be right.
number of bagels
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1.2
Activity
Standards Alignment
Building On
8.F.5
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
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.
Three days in a row, a dog owner tied his dog’s 5-foot-long leash to a post outside a store while he ran into the store to get a drink. Each time, the owner returned within minutes.
The dog’s movement each day is described here.
Day 1: The dog walked around the entire time while waiting for its owner.
Day 2: The dog walked around for the first minute and then laid down until its owner returned.
Day 3: The dog tried to follow its owner into the store but was stopped by the leash. Then, it started walking around the post in one direction. It kept walking until its leash was completely wound up around the post. The dog stayed there until its owner returned.
Each day, the dog was 1.5 feet away from the post when the owner left.
Each day, 60 seconds after the owner left, the dog was 4 feet from the post.
Your teacher will assign one of the days for you to analyze.
Sketch a graph that could represent the relationship between the dog’s distance from the post, in feet, and time, in seconds, since the owner left.
Day
1.3
Activity
Standards Alignment
Building On
8.F.5
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
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).
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.
Here are two pairs of quantities from a situation you’ve seen in this lesson. Each pair has a relationship that can be defined as a function.
time, in seconds, since the dog owner left and the total number of times the dog has barked
time, in seconds, since the owner left and the total distance, in feet, that the dog has walked while waiting
Choose one pair of quantities, and express their relationship as a function.
In that function, which variable is independent? Which one is dependent?
Write a sentence of the form “ is a function of .”
Sketch a possible graph of the relationship on the coordinate plane. Be sure to label and indicate a scale on each axis, and be prepared to explain your reasoning.
Student Lesson Summary
A relationship between two quantities is a function if there is exactly one output for each input. We call the input the independent variable and the output the dependent variable.
Let’s look at the relationship between the amount of time since a plane takes off, in seconds, and the plane’s height above the ground, in feet.
These two quantities form a function if time is the independent variable (the input) and height is the dependent variable (the output). This is because at any amount of time since takeoff, the plane could be at only one height above the ground.
For example, 50 seconds after takeoff, the plane might have a height of 180 feet. At that moment, it cannot be simultaneously 180 feet and 95 feet above the ground.
For any input, there is only one possible output, so the height of the plane is a function of the time since takeoff.
The two quantities do not form a function, however, if we consider height as the input and time as the output. This is because the plane can be at the same height for multiple lengths of times since takeoff.
For instance, the plane will likely be 100 feet above the ground shortly after takeoff as well as shortly before landing.
For an input, there are multiple possible outputs, so the time since takeoff is not a function of the height of the plane.
Functions can be represented in many ways—with a verbal description, a table of values, a graph, an expression or an equation, or a set of ordered pairs.
When a function is represented with a graph, each point on the graph is a specific pair of an input and output.
Here is a graph that shows the height of a plane as a function of time since takeoff.
Graph of a function, origin O. Horizontal axis from 0 to 300, by 50s, time in seconds. Vertical axis from 0 to 1500, by 250s, height in feet. Function begins at zero and goes through 50 comma 180, 250 comma 1500, and 250 comma 1500. Point 125 comma 400 is noted.
It is a function because there is one output for each input. The point on the function's graph tells us that 125 seconds after takeoff, the height of the plane is 400 feet.
Here is a graph that shows the time since takeoff as the output and the height of the plane as the input.
This is not a function because an input of 100 feet has two possible outputs.
A dependent variable is a variable that represents the output of a function.
For example, the equation defines as a function of .
The variable is called the independent variable because its value is chosen.
The variable is called the dependent variable because it depends on . Once a value is chosen for , the value of is determined.
A function is a rule that takes inputs from one set and assigns them to outputs from another set. Each input is assigned exactly one output.
An independent variable is a variable that represents the input of a function.
For example, the equation defines as a function of .
The variable is called the independent variable because its value is chosen.
The variable is called the dependent variable because it depends on . Once a value is chosen for , the value of is determined.
Standards Alignment
Building On
8.F.1
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
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).
