Lesson 5

This warm-up refreshes the idea of evaluating and solving equations, preparing students for the main work of the lesson. It reminds students that to solve a variable equation is to find one or more values for the variable that would make the equation true. In subsequent activities, students will work with equations in function notation to find unknown input or output values.

Invite students to share their response and strategy for answering the questions.

To find the values of \(p\) in the second set of questions, some students may have guessed and checked, but many students should have recognized that they could solve the equations for \(p\) (either before or after substituting the value of \(q\)). If no students mentioned solving the equations, bring it to their attention.

Remind students that to solve \(12=4+0.8p\) is to find the value of \(p\) that makes the equation true.

In this activity, students work with rules of functions that arise from situations and continue to make connections between different representations of functions. They use rules to evaluate functions and interpret the input and output values in context. For instance, they see that \(B(1)\) represents the cost of using 1 gigabyte of data beyond the monthly allowance of Data Plan B, and find its value by computing \(10(1)+25\).

Students also use rules of functions to solve for an unknown input value. For example, given the rule \(B(x) = 10x + 25\) and the statement \(B(x)=50\), they work to find a value of \(x\) that makes \(B(x) = 50\) true. They do so by:

• guessing and checking
• analyzing the graph of \(y=10x+25\) and identifying the \(x\)-value when \(y\) is 50
• reasoning that a budget of \$50 means that only \$25 is available for extra data, and at \$10 per gigabyte, it means 2.5 gigabytes of data
• solving \(10x+25=50\) algebraically

Identify students who use different strategies and ask them to share their thinking later.

Arrange students in groups of 2.

As a class, read the opening paragraphs and the first question in the activity statement. Ask students, “What do \(A(1)\) and \(B(1)\) represent in this situation? What does the 1 in each expression mean?” Give students a minute of quiet think time and then time to discuss their thinking with their partner.

Invite students to share their interpretations, in particular the meaning of “allowance.” Most data plans include a basic amount of data at a fixed cost but start charging an additional fee if usage goes beyond this allowance. Before students begin the activity, make sure they see that an input value of 1 represents usage of 1 gigabyte of data beyond the monthly allowance.

Students may question if \(A\) is a function at all, because unlike \(B\) or other function rules they have seen so far, \(A(x)\) is defined with a constant instead of an expression containing the dependent variable. Or they may wonder why \(A(x)\) has the same value no matter what the input value is. Ask students to recall the definition of function and to consider whether each input value gives only one output value. Because it does, even though it is always the same output value, \(A\) is still a function.

Select students to share their interpretations of the two data plans. Make sure students see that:

• The equation \(A(x)=60\) tells us that, regardless of the extra gigabytes of data used, \(x\), the cost, \(A(x)\) is always 60.
• The \(10x\) in the rule of \(B(x)\) tells us that each extra gigabyte of data used costs \$10, and that there is a \$25 fixed fee.

Explain to students that the two functions here are linear functions because the output of each function changes at a constant rate relative to the input. Option B involves a rate of change of \$10 per gigabyte of data over the monthly allowance, while Option A has a rate of change of \$0 per gigabyte over the allowance.

Then, ask students how they went about graphing the functions. Students are likely to have plotted some input-output pairs of each function. If no students mention identifying the slope and vertical intercept of each graph, ask them about it.

Next, focus the discussion on students’ response to the last question and how they found out the gigabytes of data that could be bought with \$50 under Option B. Select previously identified students to share their strategies, in the order listed in the Activity Narrative. If no one mentions solving using a graph or solving \(10x+25=50\), bring these up.

Explain the following points to help students connect some key ideas:

• We can graph functions like \(A(x) = 60\) and \(B(x)=10x+25\) without plotting individual coordinate pairs.

  • \(A(x)\) is the output of function \(A\) and is represented by vertical values on a coordinate plane. The vertical values are typically labeled with the variable \(y\), so we can write \(y=A(x)\) and graph \(y=60\) to represent function \(A\).

  • Likewise, \(B(x)\) is the output of function \(B\) and is represented by vertical values on a plane. We can write \(y=B(x)\) and graph \(y=10x+25\) to represent function \(B\).

• To solve equations like \(B(x) = 50\) means to find one or more values of \(x\) that make the equation true. We can do this, among other ways, by using the graph of \(B\) and by solving an equation algebraically.

  • On the graph of \(B\), we can look for one or more values of \(x\) that correspond to the vertical value of 50. This might involve some estimating.

  • Because \(B(x)\) is equal to \(10x + 25\) and \(B(x)\) is also equal to 50, we can write \(10x+25=50\) and solve the equation.

If time permits, invite students to share which option they believe the college student should choose and why.

In this optional activity, students learn to use graphing technology to graph an equation in function notation, evaluate the function at a specific input value, create a table of inputs and outputs, and identify the coordinates of the points along a graph. They also revisit how to set an appropriate graphing window.

Invite students to share any insights they had while using the graphing tool and techniques to evaluate expressions and solve equations. In what ways might the tool and techniques be handy? When might they be limited?

Also discuss any issues that students encountered while completing the task—technical or otherwise.

If desired, consider showing another way to obtain input-output pairs of a function in Desmos.

Let’s assign a new input variable, say \(n\), to function \(B\). If we enter \((n, B(n))\) in the expression list, activate a slider for \(n\), and enable the option to label points, the graph will show the coordinate pair for any value of \(n\).