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Unit 8, Lesson 23

Using Quadratic Expressions in Vertex Form to Solve Problems

Algebra 1

23.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students recall the meaning of maximum or minimum value of a function, which they learned in a previous unit. They also practice interpreting the language related to maximum and minimum values of functions.

Launch

Ask students to describe some situations in which people use the words “minimum” and “maximum.” For example, we might say there is a minimum age for voting or for getting a driver’s license, or that roads and highways have maximum speed limits.

Then, ask students what the words “minimum” and “maximum” mean more generally. We might think of a minimum as the least, the least possible, or the least allowable value, and a maximum as the greatest, the greatest possible, or the greatest allowable value.

Activity

None

Here are graphs that represent two functions, and , defined by these equations:

can be expressed in words as “the value of when is 1.” Find or compute:
1. the value of when is 1
Does have a maximum, minimum, or neither? If it has a maximum or minimum, what is the greatest or least value can have?
can be expressed in words as “the value of when is 9.” Find or compute:
1. the value of when is 9
Does have a maximum, minimum, or neither? If it has a maximum or minimum, what is the greatest or least value can have?

Student Response

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Building on Student Thinking

Some students may struggle to relate the -coordinates of points on a graph with the outputs of a function. Earlier in the course, students learned that the graph of a function is the graph of the equation . Consider having students label the coordinates of points on each graph and then complete the statements such as “The point on the graph means .” Another approach would be to have students organize the points on the graphs into tables with headers and and and .

Activity Synthesis

Discuss with students:

“Why does not have a maximum value?” (We can always use larger values of in both the positive and negative directions to get greater and greater values of .)
“Why does not have a minimum value?” (We can always find an input that makes the value of less and less.)

Emphasize that we can find an input that makes the value of as great as we want and that makes as small as we want.

Remind students that:

A maximum value of a function is a value of a function that is greater than or equal to all the other values. It corresponds to the highest -value on the graph of the function.
A minimum value of a function is a value of a function that is less than or equal to all the other values. It corresponds to the lowest -value on the graph of the function.
For quadratic functions, there is only one maximum or minimum value.

23.2

Activity

Instructional Routines

MLR8: Discussion Supports

Materials

None

Activity Narrative

The goal of this activity is to use the vertex form to find out if a vertex represents the minimum or the maximum value of the function. To do this, students rely on the behavior of a quadratic function, the structure of the expression, and some properties of operations (MP7).

Because using structure is central to the work here, graphing technology is not an appropriate tool.

Launch

Arrange students in groups of 2, and ask students to keep their materials closed.

Display the equation for all to see. Ask students how they would determine, without graphing, if the vertex of the graph, , is a maximum or a minimum. Give partners a moment to discuss their thinking. Solicit a few ideas from the class before asking students to proceed to the activity.

Ask students to take a couple of minutes of quiet think time to make sense of the line of reasoning in the first question and then discuss their understanding with their partner.

MLR8 Discussion Supports. Display sentence frames to help partners explain their reasoning for the first question. Invite Partner A to begin with this sentence frame: “The point is a maximum value because . . . .” “First, I _____ because . . . .” or “I noticed _____, so I . . . .” Invite the listener, Partner B, to press for additional details by asking, “What happens when ?” “What happens when you square ?” or “Can you say that a different way?” This will help students justify if a vertex represents the minimum or maximum value of a function..
Advances: Conversing, Representing

23.3

Activity

Instructional Routines

MLR6: Three Reads

Materials

None

Activity Narrative

Students may intuitively think of graphing the function for Performance A, comparing the two graphs, and seeing which one has a greater -value at its vertex. While this strategy is both effective and efficient, the task asks students to decide which function has a greater maximum without graphing, in order to encourage them to reason algebraically.

Monitor for students taking different approaches. Some possible strategies:

Use the factored form of the expression for Performance A to find the -intercepts. Then, find the midpoint of the two intercepts, identify its -value, evaluate the expression at that value, and compare the output with the -value of the vertex of the other graph.
Convert the expression representing Performance A to vertex form, identify the vertex, and compare the -value to that of the vertex of the graph for Performance B.

This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.

Lesson Synthesis

The purpose of this lesson is for students to understand what is meant by a minimum or maximum value of a function and ways to approach finding a maximum or minimum value given a function expressed in any form. Display this expression for all to see:

Ask students to consider how they would go about deciding whether the function has a maximum or minimum value, and how they would determine what the maximum or minimum value is. It is not necessary to actually determine this value for the example. After a few minutes of quiet think time, invite students to share their approaches with a partner. If time allows, select a few students to share with the class. Some possible approaches are:

Since the coefficient of is negative, I know the graph of the function opens downward, so the function has a maximum value.
I could use technology to graph the function and see if the graph has a largest -coordinate. If so, the function has a maximum value. If it has a smallest -coordinate, the function has a minimum value.
I could use the quadratic equation to find the zeros, and since I know the vertex is halfway between the zeros, I could determine the value of the midpoint of the two zeros and then substitute that value into the function to determine its output. Then I could compare this output to any other output value and see if it is larger or smaller.
I could rewrite the expression in vertex form to see the coordinates of the vertex. Then, I could think about -values on either side and whether their corresponding -values were greater than or less than the -coordinate of the vertex.

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Student Lesson Summary

Any quadratic function has either a maximum or a minimum value. We can tell whether a quadratic function has a maximum or a minimum by observing the vertex of its graph.

Here are graphs representing functions and , defined by and .

<p>Graph of function f on a grid. X axis from negative 8 to 1, by 1’s. Y axis from negative 6 to 4, by 2’s. Origin, O. Parabola opens downward with vertex at negative 35 comma 4.</p>

The vertex of the graph of is , and the graph is a parabola that opens downward.
No other points on the graph of (no matter how much we zoom out) are higher than , so we can say that has a maximum of 4, and that this occurs when .

<p>Graph of function g on a grid. X axis from negative 8 to 1, by 1’s. Y axis from negative 16 to 4, by 4’s. Origin, O. Graph opens upward with vertex at negative 3 comma negative 10.</p>

The vertex of the graph of is at , and the graph is a parabola that opens upward.
No other points on the graph (no matter how much we zoom out) are lower than , so we can say that has a minimum of -10, and that this occurs when .

We know that a quadratic expression in vertex form can reveal the vertex of the graph, so we don’t actually have to graph the expression. But how do we know, without graphing, if the vertex corresponds to a maximum or a minimum value of a function?

The vertex form can give us that information as well!

To see if is a minimum or maximum of , we can rewrite in vertex form, which is . Let’s look at the squared term in .

When , is 0, so is also 0.
When is not -3, the expression is a nonzero number, and is positive.
Because a squared number cannot have a value less than 0, has the least value when .

To see if is a minimum or maximum of , let’s look at the squared term in .

When , is 0, so is also 0.
When is not -5, the expression is nonzero, so is positive. The expression has a coefficient of -1, however. Multiplying (which is positive when ) by a negative number results in a negative number.
Because a negative number is always less than 0, the value of will always be less when than when . This means gives the greatest value of .

The graph that represents has its vertex at . Here is one way to show, without graphing, that corresponds to the minimum value of .
- When , the value of is 0, because .
- Squaring any nonzero number always results in a positive number, so when is any value other than 8, will be a number other than 0, and when the expression is squared, , it will be positive.
- Any positive number is greater than 0, so when , the value of will be greater than when . In other words, has the least value when .
Use similar reasoning to explain why the point corresponds to the maximum value of , defined by .
Here are some quadratic functions and the coordinates of the vertex of the graph of each. Determine if the vertex corresponds to the maximum or the minimum value of the function. Be prepared to explain how you know.

equation	coordinates of the vertex	maximum or minimum?

Launch

Arrange students in groups of 2.

Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the graph, without revealing the questions.

For the first read, read the problem aloud, then ask, “What is this situation about?” (revenue collected from two performances) Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured. (the vertex of the graph, the intercepts of the graph)
After the third read, reveal the question: “Without creating a graph of , determine which performance gives the greater maximum revenue when tickets are dollars each. Explain or show your reasoning,” and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read (find the vertex of by starting with the -intercepts and using symmetry).

Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, check in with students within the first 2–3 minutes of work time. Look for students who are working to identify the zeros or convert the equation for Performance A into vertex form. If students are stuck, remind them of previous experiences converting expressions into vertex form, and encourage them to write the vertex of Performance B next to the graph to compare. If a display was created in the previous lesson, use the display as a reference.
Supports accessibility for: Organization, Attention

Activity

None

Function , defined by , describes the revenue collected from the sales of tickets for Performance A, a musical.

The graph represents a function, , that models the revenue collected from the sales of tickets for Performance B, a Shakespearean comedy.

In both functions, represents the price of one ticket, and both revenues and prices are measured in dollars.

Without creating a graph of , determine which performance gives the greater maximum revenue when tickets are dollars each. Explain or show your reasoning.

Student Response

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Lesson 23

Using Quadratic Expressions in Vertex Form to Solve Problems

Warm-up

Standards Alignment

Building On

HSF-IF.A.2

Addressing

HSF-IF.C

Building Toward

HSA-SSE.B.3.b

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

HSF-IF.A.2

Addressing

HSA-SSE.B.3.b

Building Toward

Standards Alignment

Building On

Addressing

HSF-IF.C.9

Building Toward