Lesson 17

Parameters and Graphs

These materials, when encountered before Algebra 1, Unit 6, Lesson 17 support success in that lesson.

17.1: Which One Doesn’t Belong: Triangles (10 minutes)

Warm-up

This warm-up prompts students to carefully analyze and compare figures that are transformed in the plane. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students know and how they talk about transformations. Terms like translation and reflection are great, but for the purposes of this lesson and the associated Algebra 1 lesson, less formal terms like shift or flip are also acceptable.

Launch

Arrange students in groups of 2–4. Display the figures for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

Student Facing

Each figure shows triangle \(ABC\), and its image after a transformation, \(A’B’C’\). Which one doesn’t belong? 

A

Two triangles in an x y plane.

B

Two triangles in an x y plane.

C

Two triangles in an x y plane.

D

Two triangles in an x y plane.

 

Student Response

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

Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as slide, shift, flip, translate, rotate, or dilate. Also, press students on unsubstantiated claims.

17.2: Describe the Change (20 minutes)

Launch

Provide access to graphing technology.

Student Facing

  1. Use graphing technology to graph each equation. Describe how each graph changes from the previous graph and draw a sketch of the change. 
    equation description of change sketch of graph
    \(y=x^2\) original graph
    Parabola in the x y plane.
    \(y = (x-5)^2\)  
    Parabola in the x y plane.
    \(y=(x-5)^2+4\)  
    Parabola in the x y plane.
  2. Describe the change in the given sketch and write an equation that you think would generate that change.
    equation description of change sketch of graph
    \(y=x^2\) original graph
    Parabola in the x y plane.
       
    Two parabolas in the x y plane.
       
    Three parabolas in the x y plane.
  3. How would the graph of \(y=\text-2x^2-3\) compare to the graph of \(y=2x^2-3\)?

Student Response

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

Activity Synthesis

Ask selected students to share how they decided to create sketches of new functions based on the original function, and how they decided to describe changes to the graph. Relate the changes in these graphs to the visual display used to synthesize the previous lesson. Instead of the equation used in the display, now we are using

  • \(y=(x-5)^2 + 4\). This is a graph of \(y=x^2\), shifted 5 to the right and 4 up
  • \(y=2x^2-3\). This is a graph of \(y=x^2\), made narrower and shifted 3 down

17.3: Select a Function (10 minutes)

Activity

The purpose of this activity is to apply what students noticed in order to select a function whose graph will have an intended difference from the graph of \(y=x^2\).

Launch

Encourage students to predict which function meets each description without using graphing technology. Then, use technology to check.

Student Facing

Let’s call the graph of \(y=x^2\) “the original graph.”

Select the function that will affect the original graph in the way described.

  1. Shift the vertex of the graph left 1 unit.
  2. Shift the vertex of the graph up 1 unit.
  3. Shift the vertex of the graph right 1 unit and up 1 unit.
  4. Make the original graph narrower.
  5. Make the original graph narrower, and shift the vertex 1 unit to the right.
  • \(y=x^2+1\)
  • \(y=(x+1)^2\)
  • \(y=3x^2\)
  • \(y=(x-1)^2+1\)
  • \(y=3(x-1)^2\)

Student Response

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

Activity Synthesis

Invite students to check their responses with graphing technology, or demonstrate this. Display an equation like this, and ask students to summarize how values of \(a\) (other than 1), and \(h\) and \(k\) (other than 0) affect the graph of \(y=x^2\).

\(y=a(x-h)^2+k\)

  • When \(a\) is positive and greater than 1, the graph is narrower. When \(a\) is positive and less than 1, the graph is wider.
  • When \(h\) is positive (so the equation says \((x-h)\)), the graph shifts to the right. Negative, it shifts left.
  • When \(k\) is positive, the graph shifts up. Negative, it shifts down.