Lesson 16

Graphing from the Vertex Form

These materials, when encountered before Algebra 1, Unit 6, Lesson 16 support success in that lesson.

16.1: Math Talk: When $x$ Is -7 (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for adding and squaring signed numbers, and to help them see evaluating an expression like \(\text-(x+4)^2+5\) for an input as a sequence of operations in a conventional order. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to evaluate a quadratic expression for a negative value of the variable.

Students notice and make use of structure if they notice that they can substitute their previous answer for part of the next expression (MP7). For example, if they notice that \(\text-(x+4)^2\) is the negation of the previous expression.

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

Evaluate each expression when \(x\) is -7:

\(x+4\)

\((x+4)^2\)

\(\text-(x+4)^2\)

\(\text-(x+4)^2+5\)

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?”

16.2: Four Functions (20 minutes)

Activity

In this activity, students create a table consisting of the coordinates of the points around a graph’s vertex. This prepares them to reason using the structure of expressions in the associated Algebra 1 lesson, where they perform a similar analysis without being prompted to create a table first.

Launch

Display the functions \(f\) and \(g\) for all to see.

\(f(x) = (x-4)^2\)

\(g(x) = \text-(x-4)^2\)

Ask: “How are they alike? How are they different?”

This activity is meant to be done without a calculator, as performing the calculations will help students notice the structure of the outputs of the function for different inputs. However, if the calculations will prevent any students from accessing the task, provide access to a scientific calculator.

Student Facing

  1. Complete the table of values for each function.

    \(f(x)=(x-4)^2\)

    \(x\) 0 1 2 3 4 5 6 7
    \(f(x)\)                

    \(g(x)=\text-(x-4)^2\)

    \(x\) 0 1 2 3 4 5 6 7
    \(g(x)\)                
  2. Use the completed tables to answer these questions:
    1. What are the coordinates of the vertex of each graph? How can you tell?
    2. Does the graph of function \(f\) open up or down? How can you tell?
    3. Does the graph of function \(g\) open up or down? How can you tell?
  3. Suppose function \(h\) is defined by \(h(x) = (x-4)^2 + 5\) and function \(j\) is defined by \(j(x) = \text-(x-4)^2 + 5\). Make predictions about the graph of each function using the questions here. If you get stuck, try creating a tables of values.
    1. What are the coordinates of the vertex of the graph of \(h\) and \(j\)?
    2. Which way—up or down—does the graph of each function open? How do you know?

Student Response

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

Activity Synthesis

Display a blank coordinate plane in order to plot points from the table to illustrate the discussion.

  • What patterns do you see in the tables?
  • How can a table help us identify the vertex of a graph? (There are two conditions: it is the greatest or least value of the function, but also the values around it have symmetry.)
  • How can it help us visualize the rest of the graph? (Comparing the output values on either side of the vertex gives us information about the opening of the parabola.)

16.3: Four More Functions (20 minutes)

Activity

In order to match each definition with a function, students could use the structure of the vertex form noticed in this lesson and in earlier Algebra 1 lessons, or they could start by substituting different values into each equation and compare those to the tables.

Launch

Point out that the first question is just asking for a rough sketch, rather than carefully scaled axes and plotted points. We are essentially just looking for the vertex to be located in the correct quadrant and labeled correctly, and for the graph to open up or down as appropriate.

Student Facing

Here are some tables of values that represent quadratic functions.

\(x\) 2 3 4 5 6 7 8
\(t(x)\) -11 -2 1 -2 -11 -26 -47
\(x\) -2 -1 0 1 2 3 4
\(u(x)\) 13 4 1 4 13 28 49
\(x\) -1 0 1 2 3 4 5
\(v(x)\) 76 49 28 13 4 1 4
\(x\) -4 -3 -2 -1 0 1 2
\(w(x)\) -47 -26 -11 -2 1 -2 -11
  1. Make a rough sketch of a graph of each function. Label the vertex of each graph with its coordinates.
     
    Coordinate plane. Horizontal axis -5 to 5, by 1’s. Vertical axis, -5 to 5, by 1’s.
    Coordinate plane. Horizontal axis -5 to 5, by 1’s. Vertical axis, -5 to 5, by 1’s.
    Coordinate plane. Horizontal axis -5 to 5, by 1’s. Vertical axis, -5 to 5, by 1’s.
    Coordinate plane. Horizontal axis -5 to 5, by 1’s. Vertical axis, -5 to 5, by 1’s.

     
  2. Here are some expressions that define quadratic functions. Match each function \(t\), \(u\), \(v\), and \(w\) with an expression that defines it. 
    1. \(3x^2 + 1\)
    2. \(\text-3(x-4)^2+1\)
    3. \(3(x-4)^2 + 1\)
    4. \(\text-3x^2 + 1\)

Student Response

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

Activity Synthesis

Ask students to share how they decided which table went with each expression. The key is to relate the parameters in the expression to the vertex of the graph. Point out that the expression defining \(u\), for example, could be written \(3(x-0)^2+1\), to make it more obvious that the vertex of the graph is \((0,1)\). Consider creating a display like this with the parameters labeled, and leaving it displayed in the class for reference.

Quadratic functions, annotated.