23.1: Values of a Function (10 minutes)
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.
Here are graphs that represent two functions, \(f\) and \(g\), defined by:
\(f(1)\) can be expressed in words as “the value of \(f\) when \(x\) is 1.” Find or compute:
- the value of \(f\) when \(x\) is 1
Can you find an \(x\) value that would make \(f(x)\):
- Less than 1?
- Greater than 10,000?
\(g(9)\) can be expressed in words as “the value of \(g\) when \(x\) is 9.” Find or compute:
- the value of \(g\) when \(x\) is 9
Can you find an \(x\) value that would make \(g(x)\):
- Greater than 7?
- Less than -10,000?
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.
23.2: Maximums and Minimums (15 minutes)
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.
Design Principle(s): Support sense-making; Cultivate conversation
Supports accessibility for: Memory; Organization
The graph that represents \(p(x)=(x-8)^2+1\) has its vertex at \((8,1)\). Here is one way to show, without graphing, that \((8,1)\) corresponds to the minimum value of \(p\).
- When \(x=8\), the value of \((x-8)^2\) is 0, because \((8-8)^2 = 0^2 = 0\).
- Squaring any number always results in a positive number, so when \(x\) is any value other than 8, \((x-8)\) will be a number other than 0, and when squared, \((x-8)^2\) will be positive.
- Any positive number is greater than 0, so when \(x \neq 8\), the value of \((x-8)^2\) will be greater than when \(x= 8\). In other words, \(p\) has the least value when \(x=8\).
Use similar reasoning to explain why the point \((4,1)\) corresponds to the maximum value of \(q\), defined by \(q(x) = \text-2(x-4)^2+1\).
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.
|maximum or minimum?|
Are you ready for more?
Here is a portion of the graph of function \(q\), defined by \(q(x)=\text-x^2+14x-40\).
\(ABCD\) is a rectangle. Points \(A\) and \(B\) coincide with the \(x\)-intercepts of the graph, and segment \(CD\) just touches the vertex of the graph.
Find the area of \(ABCD\). Show your reasoning.
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.
23.3: All the World’s a Stage (10 minutes)
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.
Design Principle: Support sense-making
Supports accessibility for: Memory; Organization
A function \(A\), defined by \(p(600-75p)\), describes the revenue collected from the sales of tickets for Performance A, a musical.
The graph represents a function \(B\) that models the revenue collected from the sales of tickets for Performance B, a Shakespearean comedy.
In both functions, \(p\) represents the price of one ticket, and both revenues and prices are measured in dollars.
Without creating a graph of \(A\), determine which performance gives the greater maximum revenue when tickets are \(p\) dollars each. Explain or show your reasoning.
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.
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: \(\text-x^2-6x-2.\)
Ask students to consider how they would go about deciding whether the function had a maximum or minimum value, and how they would determine what the maximum or minimum value is. It is not necessaty to actually determine this value for the example. After a few minutes of quiet think time, invite them to share their approach with a partner. Time permitting, select a few students to share with the class. Some possible approaches are:
- Since the coefficient of \(x^2\) 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 \(y\)-coordinate, so that the function has a maximum value, or a smallest \(y\)-coordinate, so the function has a minimum value.
- I could use the quadratic equation to find the zeros, and since I know the vertex is exactly in between the zeros, I could detemine 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 was larger or smaller.
- I could rewrite the expression in vertex form to see the coordinates of the vertex. Then, I could think about \(x\) values on either side, and whether their corresponding \(y\) values were greater than or less than the \(y\)-coordinate of the vertex.
23.4: Cool-down - Looking for The Greatest or the Least (5 minutes)
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 \(f\) and \(g\), defined by \(f(x)=\text-(x+5)^2+4\) and \(g(x)=x^2+6x-1\).
- The vertex of the graph of \(f\) is \((\text-5,4)\) and the graph is a U shape that opens downward.
- No other points on the graph of \(f\) (no matter how much we zoom out) are higher than \((\text-5,4)\), so we can say that \(f\) has a maximum of 4, and that this occurs when \(x=\text-5\).
- The vertex of the graph of \(g\) is at \((\text-3, \text-10)\) and the graph is a U shape that opens upward.
- No other points on the graph (no matter how much we zoom out) are lower than \((\text-3,\text-10)\), so we can say that \(g\) has a minimum of -10, and that this occurs when \(x = \text-3\).
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 \((\text-3, \text-10)\) is a minimum or maximum of \(g\), we can rewrite \(x^2+6x-1\) in vertex form, which is \((x+3)^2-10\). Let’s look at the squared term in \((x+3)^2-10\).
- When \(x=\text-3\), \((x+3)\) is 0, so \((x+3)^2\) is also 0.
- When \(x\) is not -3, the expression \((x+3)\) will be a non-zero number, and \((x+3)^2\) will be positive (squaring any number gives a positive result).
- Because a squared number cannot have a value less than 0, \((x+3)^2\) has the least value when \(x=3\).
To see if \((\text-5,4)\) is a minimum or maximum of \(f\), let’s look at the squared term in \(\text-(x+5)^2+4\).
- When \(x=\text-5\), \((x+5)\) is 0, so \((x+5)^2\) is also 0.
- When \(x\) is not -5, the expression \((x+5)\) will be non-zero, so \((x+5)^2\) will be positive. The expression \(\text-(x+5)^2\) has a negative coefficient of -1, however. Multiplying \((x+5)^2\) (which is positive when \(x \neq \text-5\)) by a negative number results in a negative number.
- Because a negative number is always less than 0, the value of \(\text-(x+5)^2+4\) will always be less when \(x \neq \text-5\) than when \(x=\text-5\). This means \(x =\text-5\) gives the greatest value of \(f\).