Lesson 20: Graphing Quadratic...
Transcript of Lesson 20: Graphing Quadratic...
ALGEBRA I
Lesson 20: Graphing Quadratic Functions
Opening Exercise
1. The science class created a ball launcher that could accommodate a heavy ball. They moved the launcher to the roof of a 23-story building and launched an 8.8-pound shot put straight up into the air. (Note: Olympic and high school women use the 8.8-pound shot put in track and field competitions.) The motion is described by the function β(π‘π‘) = β16π‘π‘2 + 32π‘π‘ + 240, where β(π‘π‘) represents the height, in feet, of the shot put above the ground with respect to time π‘π‘ in seconds. (Important: No one was harmed during this experiment!)
A. Graph the function, and identify the key features of the graph.
B. After how many seconds does the shot put hit the
ground?
C. What is the maximum height of the shot put?
D. What is the value of β(0), and what does it mean for this problem?
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Ball Launcher Data
Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
2. Matching Match each graph with the correct equation. Be prepared to share how you know the equation and graph are a match.
A.
i. 1 ( 3)( 1)2
y x x= β β +
B.
ii. 5 ( 3)( 1)4
y x x= β +
C.
iii. 1 ( 3)( 1)4
y x x= β +
D.
iv. 5 ( 3)( 1)4
y x x= β β +
E.
v. 1 ( 3)( 1)4
y x x= β β +
Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
3. Discussion What information did you use to match the graph to its equation?
4. Solve the following equation.
π₯π₯2 + 6π₯π₯ β 40 = 0
5. Consider the equation π¦π¦ = π₯π₯2 + 6π₯π₯ β 40.
A. Given this quadratic equation, find the point(s) where the graph crosses the π₯π₯-axis.
B. Earlier in this unit, we learned about the symmetrical nature of the graph of a quadratic function. How can we use that information to find the vertex for the graph?
C. How could we find the π¦π¦-intercept (where the graph crosses the π¦π¦-axis and where π₯π₯ = 0)?
D. What else can we say about the graph based on our knowledge of the symmetrical nature of the graph of a quadratic function? Can we determine the coordinates of any other points?
E. Plot the points you know for this equation on the grid above, and connect them to show the graph of the equation.
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Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
Practice Exercises
Graph the following functions, and identify key features of the graph. 6. ππ(π₯π₯) = β (π₯π₯ + 2)(π₯π₯ β 5) 7. ππ(π₯π₯) = π₯π₯2 β 5π₯π₯ β 24
Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
8. ππ(π₯π₯) = 5(π₯π₯ β 2)(π₯π₯ β 3) 9. ππ(π₯π₯) = β6π₯π₯2 + 42π₯π₯ β 60
Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
10. Consider the graph of the quadratic function at the right with π₯π₯-intercepts β4 and 2.
A. Write a formula for a possible quadratic function, in factored form, that the graph represents using ππ as a constant factor.
B. The π¦π¦ βintercept of the graph is β16. Use the π¦π¦-intercept to adjust your function by finding the constant factor ππ.
11. Given the π₯π₯-intercepts for the graph of a quadratic function, write a possible equation for the quadratic function, in factored form.
A. π₯π₯-intercepts: 0 and 3 B. π₯π₯-intercepts: β1 and 1
C. π₯π₯-intercepts: β5 and 10 D. π₯π₯-intercepts: 12 and 4
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
12. Consider the graph of the quadratic function shown at the right with π₯π₯-intercept β2.
A. Write a formula for a possible quadratic function, in factored form, that the graph represents using ππ as a constant factor.
B. The π¦π¦-intercept of the graph is 4. Use the π¦π¦-intercept to adjust your function by finding the constant factor ππ.
Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
Lesson Summary
οΏ½ When we have a quadratic function in factored form, we can find its π₯π₯-intercepts, π¦π¦-intercept, axis of symmetry, and vertex.
οΏ½ For any quadratic equation, the roots are the solution(s) where π¦π¦ = 0, and these solutions correspond to the points where the graph of the equation crosses the π₯π₯-axis.
οΏ½ A quadratic equation can be written in the form π¦π¦ = ππ(π₯π₯ βππ)(π₯π₯ β ππ), where ππ and ππ are the roots of the function. Since the π₯π₯-value of the vertex is the average of the π₯π₯-values of the two roots, we can substitute that value back into the equation to find the π¦π¦-value of the vertex. If we set π₯π₯ = 0, we can find the π¦π¦-intercept.
Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
Homework Problem Set Graph the following and identify the key features of the graph.
1. ππ(π₯π₯) = (π₯π₯ β 2)(π₯π₯ + 7)
2. β(π₯π₯) = β3(π₯π₯ β 2)(π₯π₯ + 2)
Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
3. ππ(π₯π₯) = β2(π₯π₯ β 2)(π₯π₯ + 7)
4. β(π₯π₯) = π₯π₯2 β 16
Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
5. ππ(π₯π₯) = π₯π₯2 β 2π₯π₯ + 1
6. ππ(π₯π₯) = 4π₯π₯2 + 20π₯π₯ + 24
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
7. A rocket is launched from a cliff. The relationship between the
height of the rocket, β, in feet, and the time since its launch, π‘π‘, in seconds, can be represented by the following function:
β(π‘π‘) = β16π‘π‘2 + 80π‘π‘ + 384. A. Sketch the graph of the motion of the rocket.
B. When does the rocket hit the ground?
C. When does the rocket reach its maximum height?
D. What is the maximum height the rocket reaches?
E. At what height was the rocket launched?
8. Given the π₯π₯-intercepts for the graph of a quadratic function, write a possible formula for the quadratic function, in factored form.
A. π₯π₯-intercepts: β1 and β6 B. π₯π₯-intercepts: β2 and 23
C. π₯π₯-intercepts: β3 and 0 D. π₯π₯-intercept: 7
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
9. Suppose a quadratic function is such that its graph has π₯π₯-intercepts of β3 and 2 and a π¦π¦-intercept of 6. A. Write a formula for the quadratic function.
B. Sketch the graph of the function.
Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20
ALGEBRA I
Lesson 20: Graphing Quadratic Functions Unit 11: More with Quadratics β Factored Form
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Hart Interactive β Algebra 1 M4 Lesson 20