A 8-6 Factor. Step 1 3x2 + 7x + 2

Step 2 3x2 + 7x + 2

Step 3

Step 4 3x2 + 7x + 2

Step 1

Step 2 Step 3 Step 4

Factor. 1. 3x2 + 4x +1=

2. 3x2 +10x +3 =

3. 3x2 +13x + 4 =

A 8-6 Name_____________________________BDFM?_________Why?____________________________________ Factor. 1. 2x2 +3x +1=

2. 2x2 + 5x + 2 =

3. 2x2 + 7x +3 =

4. 2x2 + 9x + 4 =

5. 2x2 +11x + 5=

6. 2x2 + 4x + 2 =

7. 2x2 + 5x +3=

8. 2x2 + 6x + 4 =

9. 2x2 + 7x + 5=

10. 2x2 +13x + 6 =

11. 2x2 −11x − 6 =

12. 2x2 +11x − 6 =

13. 2x2 −13x + 6 =

14. 2x2 +19x + 24=

15. 2x2 −13x − 24 =

16. 2x2 +13x − 24 =

17. 2x2 −19x + 24=

18. 4x2 − 9 =

19.

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3x 2 − 2x − 5=

20.

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2x 2 + 3x − 9=

21.

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3x 2 − 8x + 4=

22.

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3x 2 − 8x + 4=

23.

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6x 2 + 5x − 6=

24.

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4x 2 −15x − 25=

28. Investigate

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y = x 2 + 4x + 3 Circle the x-intercepts, put a star next to the y-intercept, and put a ‘v’ next to the vertex. Table

x y

-6 -5 -4

-3 -2 -1 0

Factored form Standard Form x-intercept y-intercepts Vertex

29. Investigate

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y = (x − 3)2 −1 Circle the x-intercepts, put a star next to the y-intercept, and put a ‘v’ next to the vertex. This problem is in vertex form. Can you see why after you graph it? Table

x y

-6

-5

-4

-3

-2

-1

0

x-intercept y-intercepts Vertex

A 8-7 Standard form of a quadratic

y = ax2 + bx + c Formula for f inding the vertex

of a parabola

x = −b2a

Vertex form of a second degree polynomial

y = a(x − h)2 + k with vertex (h, k).

Consider y = x2 + 6x + 5 . Find the coefficients a, b, and c and graph the parabola.a=____ b=_____ c=_____

Use the formula to find the vertex of the parabola.

Find the parameters a, h and k. a=_____ h=_____ k=_____ Rewrite the quadratic equation in vertex form.

A 8-7 Name_____________________________________BDFM?____________Why?______________________________ Consider y = x2 +10x +16 . Find the coefficients a, b, and c and graph the parabola.a=____ b=_____ c=_____

Use the formula to find the vertex of the parabola.

Find the parameters a, h and k. a=_____ h=_____ k=_____ Rewrite the quadratic equation in vertex form.

Consider y = x2 −10x + 21 . Find the coefficients a, b, and c and graph the parabola.a=____ b=_____ c=_____

Use the formula to find the vertex of the parabola.

Find the parameters a, h and k. a=_____ h=_____ k=_____ Rewrite the quadratic equation in vertex form.

Consider y = x2 − 22x +133 . Calculate the vertex and rewrite in vertex form.

Consider y = x2 − 26x +155 . Calculate the vertex and rewrite in vertex form.

Consider y = x2 +30x + 241. Calculate the vertex and rewrite in vertex form.

Given the coefficients, write the quadratic equation in standard form. a=1 b=2 c=3

a=-7 b=-8 c=-9 a=-1 b=10 c=-60 a=3 b=0 c=1 a=10 b=4 c=0

Given the parameters, write the vertex and the quadratic equation in vertex form. a=1 h=4 k=10

a=1 h=5 k=-9 a=1 h=-3 k=-2 a=1 h=0 k=5 a=1 h=6 k=0

Given the vertex form, write the vertex. y = (x −8)2 + 9 y = (x −10)2 −11 y = (x +12)2 +13 y = (x −14)2 y = x2 +15

A 8-8 Standard Form y = ax2 + bx + c

Discriminant-Tells us how many x-intercepts d = b2 − 4ac

If d is positive, there are two x-ints

If d is zero,

there is one x-int

If d is negative, there are zero x-ints

1. Graph y = x2 + 6x + 5 a=________b=________ c=________

2. Calculate the discriminant of y = x2 + 6x + 5 . Tell how many x-intercepts there are.

A 8-8 Name____________________________________BDFM?_________Why?_____________________________ 1. Graph y = x2 +8x +16 a=________b=________c=________

2. Calculate the discriminant of y = x2 +8x +16 . Tell how many x-intercepts there are.

3. Graph y = x2 + 6x +13 a=________ b=________ c=________

4. Calculate the discriminant of y = x2 + 6x +13 . Tell how many x-intercepts there are.

Factor. 5. 2x2 + 5x +3= 6. 2x2 + 6x + 4 =

7. 2x2 + 7x + 5=

8. Graph y = x2 − 7x + 6

9. How many x-intercepts does y = x2 − 7x + 6 have? Prove using algebra.

10. Show the calculations for finding the vertex of y = x2 − 7x + 6 .

11. Graph y = x2 + 2x − 24

12. How many x-intercepts doesy = x2 + 2x − 24 have? Prove it.

13. Show the calculations for finding the vertex of y = x2 + 2x − 24 .

Factor 14. 2x2 +13x + 6 =

15. 2x2 −11x − 6 =

17. 2x2 +11x − 6 =

A 8-9

Solve for x using the ‘Zero Product Property’ Solve for x by getting the x alone. Graph y = (x +1)(x −3) x-intercepts:

(x +1)(x −3) = 0 Check your answer:

x2 + 6x −16 = 0 x2 −16 = 0 5x2 +8 = 53

A 8-9 Name_______________________________BDFM?_______Why?____________________________________ 1. Graph y = (x + 7)(x +1) x-intercepts

Solve for x using the zero product property. (x +1)(x + 7) = 0

2. Graph y = (x − 2)(x −8) x-intercepts

Solve for x using the zero product property. (x − 2)(x −8) = 0

Check your answer:

Check your answer:

Solve for x using the zero product property. 3. (x + 5)(x +11) = 0

4. (x − 4)(x −12) = 0

5. (x +19)(x −1) = 0

6. x(x − 7) = 0 7. x(x + 2)(x −3) = 0

8. x2 +8x +12 = 0

9. x2 + 25x + 24 = 0 10. x2 − x − 6 = 0 11. x2 − 9x −10 = 0

12. x2 − 7x + 6 = 0

Zero Product Property: if ab=0 then a=0, b=0 or a=b=0

Solve for x by getting x alone. 13. x2 = 25

14. x2 −36 = 0 15. x2 −81= 0 16. x2 − 50 = 0 17. x2 +16 = 0

18. 4x2 +10 = 46

19. 5x2 −11= 69 20. 8x2 − 56 =144 21. 3x2 −14 = 94 22. 2x2 − 22 = 76

Factor. 23. 2x2 +13x + 6 =

24. 2x2 + 5x −3=

25. 2x2 − 9x + 4 =

Consider y = 2x2 − 20x + 53 26. Calculate the discriminant. How many x-intercepts are there?

27. Calculate the vertex. 28. Write the equation in vertex form.

A 8-10 The Quadratic Formula If

0 = ax2 + bx + c

Then

aacbbx

242 −±−

=

Solve for x. x2 +11x + 24 = 0 By factoring

By using the quadratic formula. a=_____b=_____c=_____

A 8-10 Classwork Name________________________________BDFM?___________Why?______________________ 1. Solve for x. x2 +12x + 20 = 0 By factoring

By using the quadratic formula. a=_____b=_____c=_____

2. Solve for x. 0672 =+− xx By factoring

By using the quadratic formula. a=_____b=_____c=_____

Solve for x by getting x alone. 3. x2 =100

4. x2 − 64 = 0 5. x2 − 65= 0 6. x2 − 66 = 0 7. x2 + 25= 0

8. Solve for x. 01242 =−+ xx By factoring

By using the quadratic formula. a=_____b=_____c=_____

9. Solve for x. 2x2 + 7x +3= 0 By factoring

By using the quadratic formula. a=_____b=_____c=_____

Solve for x by getting x alone. 10. 3x2 −12 = 63

11. 5x2 + 7 = 27 12. 6x2 +10 =106 13. 2x2 −15=101 14. 3x2 − 20 = 76

A 8-11

Find the POIs of y = x + 4y = x2 + 2x − 2

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By Graphing Using Algebra POIs:

Solve for x. Solve for y.

A 8-11 Name______________________________BDFM?________Why?___________________________________

Find the POIs of y = −2x +8y = x2 − 4x + 5

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By Graphing Using Algebra POIs:

Solve for x. Solve for y.

Find the POIs of y = x − 5y = x2 −10x +19

"#$

By Graphing Using Algebra POIs:

Solve for x. Solve for y.

Find the POIs of y = 2x + 9y = −x2 − 4x + 4

"#$

By Graphing Using Algebra POIs:

Solve for x. Solve for y.

A rocket is launched from a platform. Its path is modeled by the equation y = −x2 + 6x +1 . Graph the rocket’s flight. y-intercept_________ vertex__________ x-intercepts________________________

How high is the platform that the rocket is launched from? When does the rocket reach its maximum height? How high is it?

How high is the rocket when it hits the ground? When does it hit the ground?

A 8-Review Part 2 Name_________________________________BDFM?_______Why?____________________________ Factor. 1.

€

2x 2 + 3x +1 =

2.

€

2x 2 + 5x + 2 =

3.

€

2x 2 + 7x + 3 =

4.

€

2x 2 − 9x + 4 =

5.

€

2x 2 − 5x − 3 =

6.

€

2x 2 − 7x −15 =

7. a) Consider

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y = x 2 − 6x + 5 . Find the coefficients a, b, and c and graph the parabola.a=_____ b=_____ c=_____

b) Use the formula to find the vertex of the parabola.

c) Use the discriminant to determine the number of x-intercepts.

8. Find the POIs of

€

y = x − 5y = x 2 + 7x + 3

# $ %

By Graphing Using Algebra POIs:

Solve for x.

Solve for y.

9. Rewrite in vertex form or determine the vertex. a. Vertex (4, 5)

b. Vertex (5, -6) c. Vertex (-7, -8)

d. y = (x −10)2 +11

e. y = (x +12)2 +13

f. y = (x +14)2 −15

Solve for x. Solve for x by factoring. Solve for x by factoring. 10. 12. 12. 3x2 +15x +18 = 0 Solve for x by getting x alone. Solve for x using the quadratic formula. 13. x2 − 60 = 0 14. 5x2 −11= 69 15. 5x2 +17x + 6 = 0 16. A rocket is launched from a platform. Its path is modeled by the equation

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y = −x 2 + 4x + 6. Graph the rocket’s flight. y-intercept_________ vertex__________ x-intercepts________________________

How many feet above the ground is the platform? When does the rocket reach its maximum height? How high is it?

How high is the rocket when it hits the ground? When does it hit the ground?

(x −1000)(2x +3) = 0 02092 =++ xx