Unit 4, Lesson 14
Graphs That Represent Situations (Optional)
Integrated Math 2
14.1
Warm-up
Instructional Routines
NoneMaterials
To Gather
Graphing technologyActivity 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 (-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.
Activity
NoneThe height in inches of a frog's jump is modeled by the equation , where the time, , after it jumped is measured in seconds.
- Find and . 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.
Activity Synthesis
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 and , because and both have a value of 0.
- is the maximum height of the frog because is halfway between the horizontal intercepts. If we were to graph it, would be the vertex.
- The graph opens downward because the coefficient of the squared term is a negative number (-75).
Lesson Synthesis
The examples that students saw in this lesson were related to the motion of projectiles. Emphasize that the graphs that represent quadratic functions that model other kinds of situations can also be interpreted in context. For example, in an earlier lesson we saw a quadratic function defined by . It described the revenue in thousands of dollars collected from ticket sales, as a function of the price of one ticket.
- “What can you tell about the ticket sales and revenue from the equation?” (If the ticket is sold for 0 dollars, or if , the revenue will be 0.)
- “In what form is the quadratic expression defining written?” (factored form) “If we rewrite it in standard form, what new information would we gain?” (In standard form, the equation is . It gives us the vertical intercept, but that is not new information. We saw that one of the horizontal intercepts is at , so the vertical intercept is known.)
- “Here is the graph representing the same function. What information can we determine from the horizontal intercepts, the vertical intercept, and the vertex of the graph?” (The horizontal intercepts tell us that no revenue will be collected if the tickets are sold at \$0 and \$8. The vertical intercept gives the same information: that if the ticket is free, there will be no revenue. The vertex tells us that the greatest revenue will be collected when the tickets are sold at \$4 each, and that amount is \$1,200.)
Student Lesson Summary
Let’s say that a tennis ball is hit straight up into the air, and its height, in feet, above the ground is modeled by the equation , where represents the time, in seconds, after the ball is hit. Here is a graph that represents the function, from the time the tennis ball was hit until the time it reached the ground.
In the graph, we can see some information we already know, and some new information:
- The 4 in the equation means the graph of the function intersects the vertical axis at 4. It shows that the tennis ball was 4 feet off the ground at , when it was hit.
- The horizontal intercept is . It tells us that the tennis ball hit the ground 1 second after it was hit.
- The vertex of the graph is at approximately . This means that about 0.4 second after the ball was hit, it reached the maximum height of about 6.3 feet.
The equation can be written in factored form as . From this form, we can see that the zeros of the function are and . The negative zero, , is not meaningful in this situation, because the time before the ball was hit is irrelevant.