Graphs That Represent Situations
5 min
Teacher Prep
Required Preparation
Narrative
In this Warm-up, students begin to apply their new understandings about graphs to reason contextually about quadratic functions. They evaluate a simple quadratic function, find its maximum, and interpret these values in context. The input values to be evaluated produce an output of 0, reminding students of the meaning of the zeros of a function and their connection to the horizontal intercepts (x-intercepts) of the graph.
To find the maximum value of the function, students could graph the function, but it would be more efficient to apply what they learned about the connection between the horizontal intercepts and the vertex (without graphing). Identify students who make this connection, and ask them to share their thinking in the discussion.
Launch
Give students access to a calculator.
Student Task
The height in inches of a frog's jump is modeled by the equation h(t)=60t−75t2, where the time, t, after it jumped is measured in seconds.
- Find h(0) and h(0.8). What do these values mean in terms of the frog’s jump?
- How much time after it jumped did the frog reach the maximum height? Explain how you know.
Sample Response
- h(0)=h(0.8)=0. This means that the frog is on the ground when t=0 (before jumping) and when t=0.8 (when the frog lands on the ground.)
- The frog reached its maximum height at t=0.4 as that is the halfway point of the jump. If we graph the function, the vertex of the graph will have a horizontal coordinate (t-coordinate) of 0.4.
Activity Synthesis (Teacher Notes)
Select students to share their responses and explanations. Even though students are not asked to graph the function, make sure that they begin to connect the quadratic expression that defines the function to the features of the graph representing that function.
Ask students: “If we graph the equation, what would the graph look like? Where would the intercepts be? Would the graph open upward or downward? Where would the vertex be?”
Highlight the following points:
- The graph intersects the horizontal axis at t=0 and t=0.8, because h(0) and h(0.8) both have a value of 0.
- h(0.4) is the maximum height of the frog because t=0.4 is halfway between the horizontal intercepts. If we were to graph it, (0.4,h(0.4)) would be the vertex.
- The graph opens downward because the coefficient of the squared term is a negative number (-75).
Standards
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- 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.A.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
15 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Graphs That Represent Situations
5 min
Teacher Prep
Required Preparation
Narrative
In this Warm-up, students begin to apply their new understandings about graphs to reason contextually about quadratic functions. They evaluate a simple quadratic function, find its maximum, and interpret these values in context. The input values to be evaluated produce an output of 0, reminding students of the meaning of the zeros of a function and their connection to the horizontal intercepts (x-intercepts) of the graph.
To find the maximum value of the function, students could graph the function, but it would be more efficient to apply what they learned about the connection between the horizontal intercepts and the vertex (without graphing). Identify students who make this connection, and ask them to share their thinking in the discussion.
Launch
Give students access to a calculator.
Student Task
The height in inches of a frog's jump is modeled by the equation h(t)=60t−75t2, where the time, t, after it jumped is measured in seconds.
- Find h(0) and h(0.8). What do these values mean in terms of the frog’s jump?
- How much time after it jumped did the frog reach the maximum height? Explain how you know.
Sample Response
- h(0)=h(0.8)=0. This means that the frog is on the ground when t=0 (before jumping) and when t=0.8 (when the frog lands on the ground.)
- The frog reached its maximum height at t=0.4 as that is the halfway point of the jump. If we graph the function, the vertex of the graph will have a horizontal coordinate (t-coordinate) of 0.4.
Activity Synthesis (Teacher Notes)
Select students to share their responses and explanations. Even though students are not asked to graph the function, make sure that they begin to connect the quadratic expression that defines the function to the features of the graph representing that function.
Ask students: “If we graph the equation, what would the graph look like? Where would the intercepts be? Would the graph open upward or downward? Where would the vertex be?”
Highlight the following points:
- The graph intersects the horizontal axis at t=0 and t=0.8, because h(0) and h(0.8) both have a value of 0.
- h(0.4) is the maximum height of the frog because t=0.4 is halfway between the horizontal intercepts. If we were to graph it, (0.4,h(0.4)) would be the vertex.
- The graph opens downward because the coefficient of the squared term is a negative number (-75).
Standards
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- 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.A.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.