lesson 10: getting enough informationmrpunpanichgulmath.weebly.com/.../a1_m4_l10_se.pdf · hart...

ALGEBRA I

Hart Interactive – Algebra 1 M4 Lesson 10

Lesson 10: Getting Enough Information

Opening Activity

1. Sketch a graph of a parabola that has its vertex at (-2, -9).

2. A. You are now told that the y-intercept for this parabola is at (0, -5). Redraw your sketch to include this new information.

B. What changed on your parabola from Exercise 1?

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Lesson 10: Getting Enough Information Unit 11: More with Quadratics – Factored Form

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ALGEBRA I


3. A. One more piece of data has come in about your parabola sketch. You now have x-intercepts at (-5, 0) and (1, 0). Remember it still has to have a vertex at (-2, -9) and a y-intercept at (0, -5).

B. What changed on your parabola from Exercise 2?

4. Write the equation of the parabola in vertex form. ___________________________________________

5. Write the equation of the parabola in standard form. _________________________________________

6. A. Substitute each x-intercept into your standard form equation. What value do you get for y?

B. Why does this make sense?


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ALGEBRA I


7. Another form of the equation for this parabola is y = (x + 5)(x – 1). This is called factored form and gives us information about the x-intercepts. How does this form of the equation help us find the x-intercepts of -5 and 1?

For each exercise below, tell what information you can get from the form of the equation and then sketch a graph of the parabola.

8.

Form of the Equation What can you easily find? Find it! Sketch of the Parabola

Vertex: y = (x + 1)2 – 4

Standard: y = x2 + 2x – 3

Factored: y = (x – 1)(x + 3)

9.


Vertex: y = – (x – 1)2 + 9

Standard: y = – x2 + 2x + 8

Factored: y = – (x – 4)(x + 2)


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ALGEBRA I


10. Besides being useful when graphing parabolas, the x-intercepts also help us solve real-life problems. In the ball toss activity from Lesson 1, what information did the x-intercept provide?

11. A science class designed a ball launcher and tested it by shooting a tennis ball straight up from the top of a 𝟏𝟏𝟏𝟏-story building. They determined that the motion of the ball could be described by the function:

ℎ(𝑡𝑡) = −16𝑡𝑡2 + 144𝑡𝑡 + 160, where 𝒕𝒕 represents the time the ball is in the air in seconds and 𝒉𝒉(𝒕𝒕) represents the height, in feet, of the ball above the ground at time 𝒕𝒕. A. What information can be found easily with this form of the equation? What does it represent in real life?

B. Their equation in vertex form is 29( ) 16 484

2h t t= − − + . What do we now know about the flight of this

tennis ball?

C. With a graph, we can see the number of seconds it takes for the ball to reach its peak and how long it takes to hit the ground. How might the factored form of the equation help us find this information?


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ALGEBRA I


12. Below is a graph of the motion of the tennis ball. A. From the graph, determine one of the x-intercepts. B. Estimate the other x-intercept. Why is that x-intercept not shown on the graph?

13. The factored form of the equation is y = -16(t + 1)(t – 10). Does this confirm your estimates in Exercise 12?


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ALGEBRA I


Looking for Patterns

For each parabola graph below, determine the x-intercepts, vertex and axis of symmetry.

14. 15. 16.

x-intercepts: _____ and _____ x-intercepts: _____ and _____ x-intercepts: _____ and _____

Vertex: ( _____, _____) Vertex: ( _____, _____) Vertex: ( _____, _____)

Axis of Symmetry: x = _____ Axis of Symmetry: x = _____ Axis of Symmetry: x = _____

Discussion

17. From Lesson 6, we found that the standard form, y = ax2 + bx + c, gives you the axis of symmetry using

the equation 2bxa

= − . How could you find the axis of symmetry if you knew the x-intercepts? Use your

work from Exercises 14 – 16 to help you find the pattern.

18. If you have the axis of symmetry, how can you find the vertex?


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ALGEBRA I


Homework Problem Set

For each exercise below, tell what information you can get from the form of the equation and then sketch a graph of the parabola.

1.

Form of the Equation What can you easily find? Find it!

Sketch of the Parabola

Vertex: y = x2 – 16

Standard: y = x2 – 16

Factored: y = (x – 4)(x + 4)

2.


Vertex: y = – (x – 2)2 + 9

Standard: y = – x2 + 4x + 5

Factored: y = – (x – 5)(x + 1)


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ALGEBRA I


SPIRAL REVIEW

Multiply the following binomials; note that every binomial given in the problems below is a polynomial in one variable, 𝒙𝒙, with a degree of one. Write the answers in standard form, which in this case takes the form 𝒂𝒂𝒙𝒙𝟐𝟐 + 𝒃𝒃𝒙𝒙 + 𝒄𝒄, where 𝒂𝒂, 𝒃𝒃, and 𝒄𝒄 are constants.

3. (𝑥𝑥 + 1)(𝑥𝑥 − 7) 4. (𝑥𝑥 + 9)(𝑥𝑥 + 2)

5. (𝑥𝑥 − 5)(𝑥𝑥 − 3) 6. �𝑥𝑥 + 152 � (𝑥𝑥 − 1)

7. Describe any patterns you noticed in Problems 3 – 6.

8. The square parking lot at Gene Simon’s Donut Palace is going to be enlarged so that there will be an additional 30 ft. of parking space in the front of the lot and an additional 30 ft. of parking space on the side of the lot, as shown in the figure below. Write an expression in terms of 𝑥𝑥 that can be used to represent the area of the new parking lot. Explain how your solution is demonstrated in the area model.

𝑥𝑥

30

30

𝑥𝑥


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lesson 10: getting enough informationmrpunpanichgulmath.weebly.com/.../a1_m4_l10_se.pdf · hart...

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