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9-12 Algebra 1 Unit 7 Lesson 17

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K-5 Math 6-8 Math •9-12 Math

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•Algebra 1 Geometry Algebra 2 Extra Support Materials for Algebra 1

Unit

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 •Unit 7 Unit 8

Modeling Prompts

Lesson

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 Lesson 16 •Lesson 17

K-5
6-8
9-12

Algebra 1
Geometry
Algebra 2
Extra Support Materials for Algebra 1

Integrated Math 1
Integrated Math 2
Integrated Math 3

Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8

Modeling Prompts

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Lesson 2
Lesson 3
Lesson 4
Lesson 5
Lesson 6
Lesson 7
Lesson 8
Lesson 9
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Lesson 11
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Lesson 13
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Lesson 17

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Lesson 17

17.1 Warm-up 17.2 Activity 17.3 Activity 17.4 Activity Student Lesson Summary

Unit 7, Lesson 17

Changing the Vertex

Algebra 1

17.1

Warm-up

Graphs of Two Functions

Here are graphs representing two functions, and , given by and .

<p>Two graphs representing functions in x y plane, origin O.</p>

Which graph represents each function? Explain how you know.
Where does the graph of meet the -axis? Explain how you know.

Are You Ready for More?

17.2

Activity

Shifting the Graph

How would you change the equation so that the vertex of the graph of the new equation is located at the following coordinates and so that the graph opens as described?
1. , opens upward
2. , opens upward
3. , opens downward
Use graphing technology to verify your predictions. Adjust your equations if necessary.
Kiran graphed the equation and noticed that the vertex is at . He changed the equation to and saw that the graph shifted 3 units to the right and the vertex is now at .

Next, he graphed the equation and observed that the vertex is at . Kiran thought, “If I change the squared term to , the graph of will be 5 units to the right and the vertex will be at .”

Do you agree with Kiran? Explain or show your reasoning.

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17.3

Activity

A Peanut Jumping over a Wall

Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to model the path of the main character, an animated peanut, jumping over a wall.

<p>two peanuts with wire arms and legs on purple background</p>

Mai uses the equation to represent the path of the jump. represents the height of the peanut as a function of the horizontal distance, , that it travels.

On the screen, the base of the wall is located at , with the top of the wall at . The dashed curve in the picture shows the graph of 1 equation that Mai tried, where the peanut fails to make it over the wall.

<p>Graph of the path of a peanut’s jump.</p>

What are the values of and in this equation?
Starting with Mai’s equation, choose values for and that will guarantee that the peanut stays on the screen but also makes it over the wall. Be prepared to explain your reasoning.

17.4

Activity

Smiley Face

Do you see 2 “eyes” and a smiling “mouth” on the graph? The 3 arcs on the graph all represent quadratic functions that were initially defined by , but whose equations were later modified.

Write equations to represent each curve in the smiley face.
What domain is used for each function to create this graph?

<p>Left eye, Opens down, Vertex at -2 comma 50. Right eye, opens down, vertex = 2 comma 50. Mouth, opens up, vertex = 0 comma 10.</p>

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Student Lesson Summary

The graphs of , and all have the same shape but their locations are different. The graph that represents has its vertex at .

<p>Three parabolas in x y plane, origin O.</p>

Notice that adding 12 to raises the graph by 12 units, so the vertex of that graph is at . Replacing with shifts the graph 3 units to the left, so the vertex is now at .

We can also shift a graph both horizontally and vertically.

The graph that represents will have the same shape as but it will be shifted 12 units up and 3 units to the left. Its vertex is at .

<p>Two parabolas in x y plane, origin O.</p>

The graph representing the equation has the same vertex at , but because the squared term is multiplied by a negative number, the graph is flipped over horizontally, so that it opens downward.

<p>Two parabolas in x y plane, origin O.</p>

Glossary

None

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