Lesson 12

Connections between Graphs and Equations

These materials, when encountered before Algebra 1, Unit 5, Lesson 12 support success in that lesson.

12.1: Math Talk: Evaluating a Function (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for evaluating functions at a given value of the input. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to construct a table of values for a given function.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Here is a function: \(g(x)=100-5x\)

Evaluate mentally:

\(g(0)\)

\(g(1)\)

\(g(4)\)

\(g(20)\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?” Sketch a blank graph for all to see.
  • “What shape do we expect this graph to be?” (A line, because the output is 100 when the input is 0, and then the output decreases by 5 every time the input increases by 1.)
  • “What are some points on the graph?” (\((0,100),(1,95),(4,80),(20,0)\))
  • “Where can you see the 100 and the -5 on the graph?” (100 is the vertical intercept and -5 is the slope.)

12.2: Bank Accounts (20 minutes)

Activity

If students also use technology to create the tables, they are using appropriate tools strategically (MP5).

Launch

Provide access to graphing technology. Students may need some assistance adjusting their graphing window to see the relevant features of each graph. Either help them set the window before they start to a domain of \(\text-2 \le x \le 8\) and range of \(\text-60 \le y \le 600\), or wait until they have attempted to graph at least one function and remind them of what they have learned about changing the graphing window.

Student Facing

Each function represents the amount in a bank account after \(t\) weeks.

\(A(t) = 500\)

\(B(t) = 500 + 40t\)

\(C(t) = 500 - 40t\)

\(D(t) = 500 \boldcdot (1.5)^t\)

\(E(t) = 500 \boldcdot (0.75)^t\)

 

  1. Make a table for each bank account showing the money in the account at 0, 1, 2, and 3 weeks.
  2. Describe in words how the money in the account is changing week by week.
  3. Use technology to create a graph of each function. How can you see your description in each graph?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Focus the discussion on the connection between the equations, descriptions and graphs. Discuss questions such as:

  • “What do all functions have in common?” (The vertical intercept is 500, each bank account starts with $500)
  • “How does the money change each week? How do we see this change in the graph?"
  • “How are \(B\) and \(C\) the same and different?” (They are the same because they are each linear, and each start with \$500 and change by \$40 per week. They are different because B increases and C decreases (or the rate of change is positive versus negative.)
  • “How are \(D\) and \(E\) the same and different?” (They are the same because they are each exponential, and each start with \$500. They are different because \(D\) increases and \(E\) decreases (or the growth factor is greater than 1 versus less than 1.))

12.3: Build a New Function (20 minutes)

Activity

Students continue to work with the functions from the previous activity, but strategically modify them so that they represent new situations. Then, they select a graph that represents the new situation, and use technology to see if their new equation is correct (by seeing whether its graph matches the one they chose).

Launch

Continue to provide access to graphing technology. Since this work is a continuation of the previous activity, students should be able to get started without much intervention.

Student Facing

Consider the same five functions:

\(A(t) = 500\)

\(B(t) = 500 + 40t\)

\(C(t) = 500 - 40t\)

\(D(t) = 500 \boldcdot (1.5)^t\)

\(E(t) = 500 \boldcdot (0.75)^t\)

 

  1. Starting with one of the functions, change it so that it represents an account that . . .
    1. Starts with a balance of \$300, and loses \$40 each week.
    2. Starts with a balance of \$500, and gains \$15 each week.
    3. Starts with a balance of \$500, and loses \(\frac{1}{10}\) of its value each week.
    4. Starts with a balance of \$700, and gains \(\frac{3}{10}\) of its value each week.
  2. Here are four graphs. Which graph matches each of your new equations?

    graph 1

    graph, Starts with a balance of $700, each y coordinate is the fraction 3 over 10 greater than the previous y coordinate.

    graph 2

    Y intercept = 300, each point’s y coordinate is 30 less than previous point 

    graph 3

    Y intercept = 500, each point’s y coordinate is one tenth of previous point 

    graph 4

    Y intercept = 500, each point’s y coordinate is 15 more than previous point 

  3. To check, use technology to graph your equations. Make sure to use the same graphing window. Check that the graph of your equation matches the graph you chose.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Discuss questions such as:

  • “What were some strategies you used for building new functions to represent new situations?”
  • “In exponential functions, what are the differences in an equation representing a situation that grows, and one that represents a situation that shrinks?”
  • “Where can you see each rate of change and growth factor on the graph?”