Unit 7, Lesson 17
Parameters and Graphs
Extra Support Materials for Algebra 1
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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.
Arrange students in groups of 2–4. Display the figures for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three images that go together and can explain why. Next, tell students to share their response with their group and then to work together to find as many sets of three as they can.
Each figure shows triangle , and its image after a transformation, . Which three go together? Why do they go together?
ATwo triangles in an x y plane. Horizontal axis scale negative 5 to 5 by 1s. Vertical axis scale negative 5 to 5 by 1s. Triangle P Q R. Point P is at 0 comma 0. Point Q is at 0 comma 2. Point R is at 3 comma 0. Triangle P prime Q prime R prime. Point P prime is at negative 1.5 comma 0. Point Q prime is at negative 1.5 comma 4. Point R prime is at 4.5 comma 0.
BTwo triangles in an x y plane. Horizontal axis scale negative 5 to 5 by 1s. Vertical axis scale negative 5 to 5 by 1s. Triangle P Q R. Point P is at 0 comma 0. Point Q is at 0 comma 2. Point R is at 3 comma 0. Triangle P prime Q prime R prime. Point P prime is at 0 comma 0. Point Q prime is at negative 1.5 comma 1.414 comma negative 1.414. Point R prime is at negative 2.12 comma negative 2.121.
CTwo triangles in an x y plane. Horizontal axis scale negative 5 to 5 by 1s. Vertical axis scale negative 5 to 5 by 1s. Triangle P Q R. Point P is at 0 comma 0. Point Q is at 0 comma 2. Point R is at 3 comma 0. Triangle P prime Q prime R prime. Point P prime is at negative 2 comma negative 3. Point Q prime is at negative 2 comma negative 1. Point R prime is at 1 comma negative 3.
DTwo triangles in an x y plane. Horizontal axis scale negative 5 to 5 by 1s. Vertical axis scale negative 5 to 5 by 1s. Triangle P Q R. Point P is at 0 comma 0. Point Q is at 0 comma 2. Point R is at 3 comma 0. Triangle P prime Q prime R prime. Point P prime is at 0 comma 0. Point Q prime is at 0 comma negative 2. Point R prime is at 3 comma 0.
Invite each group to share one reason why a particular set of three go together. 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 three go together, attend to students’ explanations and ensure that 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,” and to clarify their reasoning as needed. Consider asking:
Graphing technology is needed for every student.
| equation | description of change | sketch of graph |
|---|---|---|
| original graph | Parabola in the x y plane. Horizontal axis scale negative 4 to 10 by 1s. Vertical axis scale negative 4 to 8. Parabola has a vertex minimum at 0 comma 0, opens up. It passes through the points negative 2 comma 2 and 2 comma 2.
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Parabola in the x y plane. Horizontal axis scale negative 4 to 10 by 1s. Vertical axis scale negative 4 to 8. Parabola has a vertex minimum at 0 comma 0, opens up. It passes through the points negative 2 comma 2 and 2 comma 2.
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Parabola in the x y plane. Horizontal axis scale negative 4 to 10 by 1s. Vertical axis scale negative 4 to 8. Parabola has a vertex minimum at 0 comma 0, opens up. It passes through the points negative 2 comma 2 and 2 comma 2.
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| equation | description of change | sketch of graph |
|---|---|---|
| original graph | Parabola in the x y plane. Horizontal axis scale negative 7 to 7 by 1s. Vertical axis scale negative 7 to 12. Parabola has a vertex minimum at 0 comma 0, opens up. It passes through the points negative 2 comma 2 and 2 comma 2.
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Two parabolas in the x y plane. Horizontal axis scale negative 7 to 7 by 1s. Vertical axis scale negative 7 to 12. A black parabola has a vertex minimum at 0 comma 0, opens up. It passes through the points negative 2 comma 2 and 2 comma 2. A blue parabola has a vertex minimum at 0 comma 0, opens up. It passes through the points 2 comma 8 and negative 2 comma 8.
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Three parabolas in the x y plane. Horizontal axis scale negative 7 to 7 by 1s. Vertical axis scale negative 7 to 12. A black parabola has a vertex minimum at 0 comma 0, opens up. It passes through the points negative 2 comma 2 and 2 comma 2. A blue parabola has a vertex minimum at 0 comma 0, opens up. It passes through the points 2 comma 8 and negative 2 comma 8. A red parabola has a vertex minimum at 0 comma negative 3, opens up. It passes through the points negative 2 comma 5 and 2 comma 5.
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The goal of this discussion is to see how parts of equations affect the resulting graph. Here are some questions for discussion:
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