3x 2 + 7x + 2. A 8-6 Factor. Step 1. Step 3 Step 4. Step 2. Step 1 Step 2 Step 3 Step 4
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1 A 8-6 Factor. Step 1 3x 2 + 7x + 2 Step 2 Step 3 Step 4 3x 2 + 7x + 2 3x 2 + 7x + 2 Step 1 Step 2 Step 3 Step 4 Factor. 1. 3x 2 + 4x +1 = 2. 3x 2 +10x + 3 = 3. 3x 2 +13x + 4 =
2
3 A 8-6 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 +11x + 5 = 6. 2x 2 + 4x + 2 = 7. 2x 2 + 5x + 3 = 8. 2x 2 + 6x + 4 = 9. 2x 2 + 7x + 5 = 10. 2x 2 +13x + 6 = 11. 2x 2 11x 6 = 12. 2x 2 +11x 6 = 13. 2x 2 13x + 6 = 14. 2x 2 +19x + 24 = 15. 2x 2 13x 24 = 16. 2x 2 +13x 24 = 17. 2x 2 19x + 24 = 18. 4x 2 9 =
4 19. 3x 2 2x 5= 20. 2x 2 + 3x 9= 21. 3x 2 8x + 4 = 22. 3x 2 8x + 4 = 23. 6x 2 + 5x 6= 24. 4x 2 15x 25= 28. Investigate 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 Factored form -6 Standard Form -5-4 x-intercept -3-2 y-intercepts -1 0 Vertex 29. Investigate 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-intercept x y y-intercepts Vertex
5 A 8-7 Standard form of a quadratic y = ax 2 + bx + c Consider y = x 2 + 6x + 5. Find the coefficients a, b, and c and graph the parabola. a= b= c= Formula for finding the vertex of a parabola x = b 2a Use the formula to find the vertex of the parabola. Vertex form of a second degree polynomial y = a(x h) 2 + k with vertex (h, k). 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 = x 2 +10x +16. Find the Use the formula to find the vertex of Find the parameters a, h and k. the parabola. a= h= k= coefficients a, b, and c and graph the parabola. a= b= c= Rewrite the quadratic equation in vertex form. Consider y = x 2 10x 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.
6 Consider y = x 2 22x Calculate the vertex and rewrite in vertex form. Consider y = x 2 26x Calculate the vertex and rewrite in vertex form. Consider y = x x 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) y = (x 10) 2 11 y = (x +12) y = (x 14) 2 y = x 2 +15
7 A 8-8 Standard Form y = ax 2 + bx + c 1. Graph y = x 2 + 6x + 5 Discriminant-Tells us how many x- intercepts d = b 2 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 2. Calculate the discriminant of y = x 2 + 6x + 5. Tell how many x-intercepts there are. a= b= c= A 8-8 Name BDFM? Why? 1. Graph y = x 2 +8x Calculate the discriminant of y = x 2 +8x +16. Tell how many x- intercepts there are. a= b= c= 3. Graph y = x 2 + 6x Calculate the discriminant of y = x 2 + 6x +13. Tell how many x- intercepts there are. a= b= c= Factor. 5. 2x 2 + 5x + 3 = 6. 2x 2 + 6x + 4 = 7. 2x 2 + 7x + 5 =
8 8. Graph y = x 2 7x How many x-intercepts does y = x 2 7x + 6 have? Prove using algebra. 10. Show the calculations for finding the vertex of y = x 2 7x Graph y = x 2 + 2x How many x-intercepts does y = x 2 + 2x 24 have? Prove it. 13. Show the calculations for finding the vertex of y = x 2 + 2x 24. Factor 14. 2x 2 +13x + 6 = 15. 2x 2 11x 6 = 17. 2x 2 +11x 6 =
9 A 8-9 Graph y = (x +1)(x 3) x-intercepts: Zero Product Property: if ab=0 then a=0, b=0 or a=b=0 Solve for x using the Zero Product Property (x +1)(x 3) = 0 Solve for x by getting the x alone. x 2 + 6x 16 = 0 x 2 16 = 0 5x 2 +8 = 53 Check your answer: 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. x 2 +8x +12 = 0 9. x x + 24 = x 2 x 6 = x 2 9x 10 = x 2 7x + 6 = 0
10 Solve for x by getting x alone. 13. x 2 = x 2 36 = x 2 81= x 2 50 = x = x = x 2 11= x 2 56 = x 2 14 = x 2 22 = 76 Factor x 2 +13x + 6 = 24. 2x 2 + 5x 3= 25. 2x 2 9x + 4 = Consider y = 2x 2 20x Calculate the discriminant. How many x-intercepts are there? 27. Calculate the vertex. 28. Write the equation in vertex form.
11 A 8-10 The Quadratic Formula If 0 = ax 2 + bx + c x = Then b ± b 2 4ac 2a Solve for x. x 2 +11x + 24 = 0 By factoring By using the quadratic formula. a= b= c=
12
13 A 8-10 Classwork Name BDFM? Why? 1. Solve for x. x 2 +12x + 20 = 0 By factoring By using the quadratic formula. a= b= c= 2 2. Solve for x. x 7x + 6 = 0 By factoring By using the quadratic formula. a= b= c= Solve for x by getting x alone. 3. x 2 = x 2 64 = 0 5. x 2 65 = 0 6. x 2 66 = 0 7. x = 0
14 2 8. Solve for x. x + 4x 12 = 0 By factoring By using the quadratic formula. a= b= c= 9. Solve for x. 2x 2 + 7x + 3 = 0 By factoring By using the quadratic formula. a= b= c= Solve for x by getting x alone x 2 12 = x = x = x 2 15 = x 2 20 = 76
15 A 8-11 Find the POIs of By Graphing " y = x + 4 # $ y = x 2 + 2x 2 Using Algebra POIs: Solve for x. Solve for y. A 8-11 Name BDFM? Why? Find the POIs of By Graphing " y = 2x +8 # $ y = x 2 4x + 5 Using Algebra POIs: Solve for x. Solve for y. Find the POIs of By Graphing " y = x 5 # $ y = x 2 10x +19 Using Algebra POIs: Solve for x. Solve for y.
16 Find the POIs of By Graphing " y = 2x + 9 # $ y = x 2 4x + 4 Using Algebra POIs: Solve for x. Solve for y. A rocket is launched from a platform. Its path is modeled by the equation y = x 2 + 6x +1. Graph the rocket s flight. How high is the platform that the rocket is launched from? How high is the rocket when it hits the ground? When does it hit the ground? When does the rocket reach its maximum height? How high is it? y-intercept vertex x-intercepts
17 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 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 By Graphing # y = x 5 $ % y = x 2 + 7x + 3 Using Algebra POIs: Solve for x. Solve for y.
18 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) e. y = (x +12) f. y = (x +14) 2 15 Solve for x. Solve for x by factoring. Solve for x by factoring. (x 1000)(2x + 3) = 0 2 x + 9x + 20 = x 2 +15x +18 = 0 Solve for x by getting x alone. Solve for x using the quadratic formula. 13. x 2 60 = x 2 11= x 2 +17x + 6 = A rocket is launched from a platform. Its path is modeled by the equation y = x 2 + 4x + 6. Graph the rocket s flight. How many feet above the ground is the platform? How high is the rocket when it hits the ground? When does it hit the ground? When does the rocket reach its maximum height? How high is it? y-intercept vertex x-intercepts
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