Finding Unknown Inputs
5 min
Narrative
This task reminds students that they can use a graph of a function to gain some information about the situation that the function models.
Every question can be answered, or at least estimated, by analyzing the graph. As students work, look for those who use the given equation to solve or verify the answers. For example, the first question can be answered by evaluating h(1) and the second question by evaluating h(8). Invite them to share their strategy during Activity Synthesis.
Launch
Given their earlier work on quadratic functions, students should be familiar with projectiles. If needed, give a brief orientation on the context. Tell students that there are devices that use compressed air or other means to generate a great amount of force and launch a potato or similar-sized object.
If desired and time permits, find and show a short video clip of someone using an air-powered or a catapult-type device. Warn students that some of these devices can be dangerous and they shouldn’t try to build one without help from an adult.
Student Task
A mechanical device is used to launch a potato vertically into the air. The potato is launched from a platform 20 feet above the ground, with an initial vertical velocity of 92 feet per second.
The function h(t)=-16t2+92t+20 models the height of the potato over the ground, in feet, t seconds after launch.
Here is the graph representing the function.
For each question, be prepared to explain your reasoning.
- What is the height of the potato 1 second after launch?
- 8 seconds after launch, will the potato still be in the air?
- Will the potato reach 120 feet? If so, when will it happen?
- When will the potato hit the ground?
Sample Response
- 96 feet (or another height that approximates 96 feet)
- No.
- Yes, the potato’s height will be 120 feet at two times: approximately 1.5 seconds and 4.3 seconds after launch.
- after approximately 6 seconds
Activity Synthesis (Teacher Notes)
Focus the discussion on how students used the graph to help them answer the questions. Encourage students to use precise mathematical vocabulary in their explanation. Invite students, especially those who do not rely solely on the graph, to share their responses and reasoning.
Point out that we can gather quite a bit of information about the function from the graph, but the information may not be precise.
- “Is there a way to get more exact answers rather than estimates?” (Some questions can be answered by evaluating the function. For questions that are not easy to calculate at this point, we can only estimate from the graph.)
- “Which questions in the activity could be answered by calculating?” (the first two questions, about where the potato is after some specified number of seconds) “Which questions were not as easy to figure out by calculation?” (those about when the potato reaches certain heights)
- “How can we verify the time when the potato is 120 feet above the ground? The exact time when it hits the ground?” (We can find the t-values that make h(t) equal 120 by looking at the graph and then evaluating the function at those values of t. To find the exact time the potato hits the ground, we would need to find the zeros of h by solving the equation h(t)=0.)
Tell students that in this unit, they will investigate how answers to these questions could be calculated rather than estimated from a graph or approximated by guessing and checking.
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.
- F-LE.6·Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity.
- F-LE.6·Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity.
- 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>
20 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Finding Unknown Inputs
5 min
Narrative
This task reminds students that they can use a graph of a function to gain some information about the situation that the function models.
Every question can be answered, or at least estimated, by analyzing the graph. As students work, look for those who use the given equation to solve or verify the answers. For example, the first question can be answered by evaluating h(1) and the second question by evaluating h(8). Invite them to share their strategy during Activity Synthesis.
Launch
Given their earlier work on quadratic functions, students should be familiar with projectiles. If needed, give a brief orientation on the context. Tell students that there are devices that use compressed air or other means to generate a great amount of force and launch a potato or similar-sized object.
If desired and time permits, find and show a short video clip of someone using an air-powered or a catapult-type device. Warn students that some of these devices can be dangerous and they shouldn’t try to build one without help from an adult.
Student Task
A mechanical device is used to launch a potato vertically into the air. The potato is launched from a platform 20 feet above the ground, with an initial vertical velocity of 92 feet per second.
The function h(t)=-16t2+92t+20 models the height of the potato over the ground, in feet, t seconds after launch.
Here is the graph representing the function.
For each question, be prepared to explain your reasoning.
- What is the height of the potato 1 second after launch?
- 8 seconds after launch, will the potato still be in the air?
- Will the potato reach 120 feet? If so, when will it happen?
- When will the potato hit the ground?
Sample Response
- 96 feet (or another height that approximates 96 feet)
- No.
- Yes, the potato’s height will be 120 feet at two times: approximately 1.5 seconds and 4.3 seconds after launch.
- after approximately 6 seconds
Activity Synthesis (Teacher Notes)
Focus the discussion on how students used the graph to help them answer the questions. Encourage students to use precise mathematical vocabulary in their explanation. Invite students, especially those who do not rely solely on the graph, to share their responses and reasoning.
Point out that we can gather quite a bit of information about the function from the graph, but the information may not be precise.
- “Is there a way to get more exact answers rather than estimates?” (Some questions can be answered by evaluating the function. For questions that are not easy to calculate at this point, we can only estimate from the graph.)
- “Which questions in the activity could be answered by calculating?” (the first two questions, about where the potato is after some specified number of seconds) “Which questions were not as easy to figure out by calculation?” (those about when the potato reaches certain heights)
- “How can we verify the time when the potato is 120 feet above the ground? The exact time when it hits the ground?” (We can find the t-values that make h(t) equal 120 by looking at the graph and then evaluating the function at those values of t. To find the exact time the potato hits the ground, we would need to find the zeros of h by solving the equation h(t)=0.)
Tell students that in this unit, they will investigate how answers to these questions could be calculated rather than estimated from a graph or approximated by guessing and checking.
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.
- F-LE.6·Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity.
- F-LE.6·Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity.
- 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>