hart interactive – algebra 1 lesson 17...

ALGEBRA I

Lesson 17: Graphing Quadratic Functions from Factored Form, 𝒇𝒇(𝒙𝒙) = 𝒂𝒂(𝒙𝒙 −𝒎𝒎)(𝒙𝒙 − 𝒏𝒏)

Opening Exercise

1. Solve the following equation.

𝑥𝑥2 + 6𝑥𝑥 − 40 = 0

2. Consider the equation 𝑦𝑦 = 𝑥𝑥2 + 6𝑥𝑥 − 40.

A. Given this quadratic equation, can you find the point(s) where the graph crosses the 𝑥𝑥-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 𝑦𝑦-intercept (where the graph crosses the 𝑦𝑦-axis and where 𝑥𝑥 = 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 17 Graphing Quadratic Functions from Factored Form, 𝑓𝑓(𝑥𝑥) = 𝑎𝑎(𝑥𝑥 −𝑚𝑚)(𝑥𝑥 − 𝑛𝑛) Unit 11: More with Quadratics – Factored Form

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Hart Interactive – Algebra 1 M4 Lesson 17

ALGEBRA I

Practice Exercises

Graph the following functions, and identify key features of the graph. 3. 𝑓𝑓(𝑥𝑥) = − (𝑥𝑥 + 2)(𝑥𝑥 − 5) 4. 𝑔𝑔(𝑥𝑥) = 𝑥𝑥2 − 5𝑥𝑥 − 24

5. 𝑓𝑓(𝑥𝑥) = 5(𝑥𝑥 − 2)(𝑥𝑥 − 3) 6. 𝑝𝑝(𝑥𝑥) = −6𝑥𝑥2 + 42𝑥𝑥 − 60


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ALGEBRA I

7. Consider the graph of the quadratic function at the right with 𝑥𝑥-intercepts −4 and 2.

A. Write a formula for a possible quadratic function, in factored form, that the graph represents using 𝑎𝑎 as a constant factor.

B. The 𝑦𝑦 −intercept of the graph is −16. Use the 𝑦𝑦-intercept to adjust your function by finding the constant factor 𝑎𝑎.

8. Given the 𝑥𝑥-intercepts for the graph of a quadratic function, write a possible formula for the quadratic function, in factored form.

a. 𝑥𝑥-intercepts: 0 and 3 b. 𝑥𝑥-intercepts: −1 and 1

c. 𝑥𝑥-intercepts: −5 and 10 d. 𝑥𝑥-intercepts: 12 and 4


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ALGEBRA I

9. Consider the graph of the quadratic function shown at the right with 𝑥𝑥-intercept −2.

A. Write a formula for a possible quadratic function, in factored form, that the graph represents using 𝑎𝑎 as a constant factor.

B. The 𝑦𝑦-intercept of the graph is 4. Use the 𝑦𝑦-intercept to adjust your function by finding the constant factor 𝑎𝑎.


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ALGEBRA I

10. 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 ℎ(𝑡𝑡) = −16𝑡𝑡2 + 32𝑡𝑡 + 240, where ℎ(𝑡𝑡) represents the height, in feet, of the shot put above the ground with respect to time 𝑡𝑡 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 ℎ(0), and what does it mean for this problem?


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ALGEBRA I

Lesson Summary

� When we have a quadratic function in factored form, we can find its 𝑥𝑥-intercepts, 𝑦𝑦-intercept, axis of symmetry, and vertex.

� For any quadratic equation, the roots are the solution(s) where 𝑦𝑦 = 0, and these solutions correspond to the points where the graph of the equation crosses the 𝑥𝑥-axis.

� A quadratic equation can be written in the form 𝑦𝑦 = 𝑎𝑎(𝑥𝑥 −𝑚𝑚)(𝑥𝑥 − 𝑛𝑛), where 𝑚𝑚 and 𝑛𝑛 are the roots of the function. Since the 𝑥𝑥-value of the vertex is the average of the 𝑥𝑥-values of the two roots, we can substitute that value back into the equation to find the 𝑦𝑦-value of the vertex. If we set 𝑥𝑥 = 0, we can find the 𝑦𝑦-intercept.


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ALGEBRA I

Homework Problem Set Graph the following and identify the key features of the graph.

1. 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 − 2)(𝑥𝑥 + 7)

2. ℎ(𝑥𝑥) = −3(𝑥𝑥 − 2)(𝑥𝑥 + 2)

3. 𝑔𝑔(𝑥𝑥) = −2(𝑥𝑥 − 2)(𝑥𝑥 + 7)

4. ℎ(𝑥𝑥) = 𝑥𝑥2 − 16


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ALGEBRA I

5. 𝑝𝑝(𝑥𝑥) = 𝑥𝑥2 − 2𝑥𝑥 + 1

6. 𝑞𝑞(𝑥𝑥) = 4𝑥𝑥2 + 20𝑥𝑥 + 24

7. A rocket is launched from a cliff. The relationship between the

height of the rocket, ℎ, in feet, and the time since its launch, 𝑡𝑡, in seconds, can be represented by the following function:

ℎ(𝑡𝑡) = −16𝑡𝑡2 + 80𝑡𝑡 + 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? 0

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ALGEBRA I

8. Given the 𝑥𝑥-intercepts for the graph of a quadratic function, write a possible formula for the quadratic function, in factored form.

A. 𝑥𝑥-intercepts: −1 and −6 B. 𝑥𝑥-intercepts: −2 and 23

C. 𝑥𝑥-intercepts: −3 and 0 D. 𝑥𝑥-intercept: 7

9. Suppose a quadratic function is such that its graph has 𝑥𝑥-intercepts of −3 and 2 and a 𝑦𝑦-intercept of 6. A. Write a formula for the quadratic function.

B. Sketch the graph of the function.


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ALGEBRA I


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hart interactive – algebra 1 lesson 17...

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