Lesson 23

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.

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.

Some students may struggle to relate the \(y\)-coordinates of the points on a graph with the outputs of a function. Earlier in the course, students learned that the graph of a function \(f\) is the graph of the equation \(y=f(x)\). Consider having students label the coordinates of points on each graph and then complete the statements such as “The point \((3,2)\) on the graph means \(2=f(3)\).” Another approach would be to have students organize the points on the graphs into tables with headers \(x\) and \(f(x)\) and \(x\) and \(g(x)\).

Discuss with students:

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

Emphasize that we can find an input that makes the value of \(f\) as great as we want and that makes \(g\) 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 \(y\) 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 point on the graph of the function.
• For quadratic functions, there is only one maximum or minimum value.

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.

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

Display the equation \(p(x)=(x-8)^2+1\) for all to see. Ask students how they would determine, without graphing, if the vertex of the graph, \((8,1)\), 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. 

Focus the discussion on students’ responses to the first question. Invite students to share their explanations and highlight reasoning that makes use of structure (as shown in the sample response).

• Squaring an expression results in a positive value or a zero value.
• Zero is the least possible value of a squared expression.
• Multiplying a squared expression (which is either positive or zero) by a negative number gives either a negative or zero value.
• The graph of a quadratic function is symmetrical across the vertex. If an input value on one side of the vertex \((h,k)\) produces an output that is less than \(k\), then \((h,k)\) represents the maximum value of the function. Similarly, if an input value on one side of the vertex \((h,k)\) produces an output that is more than \(k\), then \((h,k)\) represents the minimum value of the function.

Students may intuitively think of graphing the function for performance A, comparing the two graphs, and seeing which one has a greater \(y\) 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 \(x\)-intercepts. Then, find the midpoint of the two intercepts, identify its \(x\) value, evaluate the expression at that value, and compare the output with the \(y\) value of the vertex of the other graph.
• Convert the expression representing performance A to vertex form, identify the vertex, and compare the \(y\) value to that of the vertex on the graph for performance B.

Arrange students in groups of 2. Give students time to read the task statement and think quietly about how they would go about comparing the two functions. Then, ask them to share their thoughts with their partner before beginning to work on the problem.

Invite previously identified students to share their solutions. Ask students to explain why they decided to take the steps that they did. Highlight any connections made between the structure of an expression defining a function, points on its graph, and the meaning of any values in this situation.