Chapter 9

1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = -2x2 – 4x + 2

domain=

range=

x y

-4 -3 -2 -1 0 1 2 3 4

y = x2 + 6x -3

domain=

range=

x y

-4 -3 -2 -1 0 1 2 3 4

2. Compare these graphs to the linear equation graphs that we have studies previously. How are they different?

3. Define the words and sketch a pictureparabola: vertex:

Function of a (the coefficient of the x2 in the quadratic equation y = ax2 +bx +c)

Now you will use the graphing calculator to explore the different roles of a, b, and c in the quadratic equation y = ax2 +bx +c

4. Set the window to x-min=-5, x-max=5, x-scl=1 and the same for y.

5. Put these 4 equations into the y= and then press graph. Observe the differences as the graphs are drawn.:

y= x2 y= 2x2 y= 4x2 y= 8x2

6. What happens as the value of a gets larger?

7. What do you think will happen as the value of a gets smaller?

Graph y= x2 y= .5x2 y= .25x2 y= .1x2

8. What did happen as the value of a got smaller?

9. What do you think will happen when the value of a is negative?

Graph: y= -x2 y= -2x2 y= -4x2 y= -8x2

10. What did happen when you made a negative?11. What will these graphs look like:

Graph: y= -x2 y= -.5x2 y= -.25x2 y= -.1x2

12. Explain how you can tell what the parabola will look like based on the value of a in the quadratic equation y = ax2 +bx +c?

Function of c (the constant in the quadratic equation y = ax2 +bx +c)

13. What is the constant in the linear equation y = mx + b. What did it tell you about the graph?

What effect do you think the value of c has on the graph in the quadratic equation y = ax2 +bx +c?

14. Put each set of 4 equations into the y= and then press graph. Observe the differences as the graphs are drawn.

y= x2 y= x2 + 1 y= x2 + 2 y= x2 + 3

y= x2 y= x2 - 1 y= x2 - 2 y= x2 – 3

y= -x2 y= -x2 + 1 y= -x2 + 2 y= -x2 + 3

y= -x2 y= -x2 - 1 y= -x2 - 2 y= -x2 - 3

15. Explain what you can tell by the value of the c in the quadratic equation y = ax2 +bx +c?

***Homework: pg. 538 #7-28

(Correct using the teacher’s edition. Help each other understand any missed problems. Ask teacher for assistance for any parts that are still causing you trouble prior to taking the quiz.)

Quiz 1 Score: ___________________________

16. Define and sketch a picture:vertex Minimum value

Maximum value axis of symmetry

Example: Graph the function . Make a table of values. What are the

domain and range?

****Notice that each point is reflected over the axis of symmetry, not the y-axis.

The domain is all real numbers. The range is y > 0

17. On graph paper: graph the function y = -3x2. What are the domain and range?

18. My cat played with my homework paper and put a big hole in it. Am I still able to answer the questions and if so, what is the answer?

What is the vertex:

What are the x-intercepts:

Function of b (the coefficient of the x in the quadratic equation y = ax2 +bx +c)

19. What effect do you think changing the value of b will have on the graph?

20. Using the graphing calculator, put each of the 4 equations into the y= and then press graph:

y= x2 y= x2 + x y= x2 + 2x y= x2 + 3x y= x2 y= x2 - x y= x2 - 2x y= x2 - 3x

21. What happened to the parabola as the value of b changed?

22. What is the y-intercept in all the graphs above? Why?

Axis of Symmetry/Vertex

23. To find the axis of symmetry, use the equation (Remember, the axis of

symmetry is just the coordinate value of x at the vertex)

24. What is the axis of symmetry for the following quadratic equation:y= 2x2 + 2x

25. What is the vertex of the equation? (Find the vertex by plugging the axis of symmetry in for x in the equation. Solve for y.)

26. Graph it on the calculator to check your answer.

27. Find the vertex of the following equations:

y= 2x2 + 4x y= 2x2 + 6x y=4x2 – 7x + 3

28. Internet: Watch montgomerycollege.edu/algebra quadratic equations “graphing quadratic equations”

Another way to Graph y= ax 2 + bx + c

You know you can graph a quadratic equation by making a table of values and drawing the parabola. Here is another way to graph it:29. (on graph paper) Follow the directions below to graph y= –3x2+ 6x + 5

o Find the axis of symmetry

o Find the y-coordinate of the vertex by substituting the axis of symmetry in for x

o Plot the vertex

o Plot the y-intercept (remember which letter, a, b, or c was the y-intercept?)

o Reflect the y-intercept coordinate over the axis of symmetry.

o Choose another value for x and plug it into the equation to find the y. Plot it. Reflect it across the axis of symmetry.

30. (on graph paper) Follow the directions above to graph y= 2x2 + 12x + 10

***Homework: pg. 544 #7-17 (even), 16-19, 20-25 (even), 26-27, 28-33 (even) (Correct using the teacher’s edition. Help each other understand any missed problems. Ask teacher for assistance for any parts that are still causing you trouble prior to taking the quiz.)

Quiz 2 Score: ___________________________

31. Internet: Watch montgomerycollege.edu/algebra quadratic equations “zero property”

Roots of the Equation/Zeros of the Function

One way to solve a quadratic equation ax2 +bx + c = 0 is to graph the related quadratic function y = ax2 +bx + c. The solutions to the equation are the x-intercepts of the related function. The solutions are called roots of the equation or zeros of the function.

The solutions to the equation are the x-intercepts!

32. Why are the x-intercepts the solution to the equation ax2 +bx + c = 0?

33. Find the solutions of each equation by graphing (on graph paper):

x2 – 16 = 0 3x2 + 6 = 0 x2 – 25 = -25

Solving Using Square Roots

You don’t have to graph to find the solutions to the quadratic equation. You can solve by using square roots.

34. Find the solutions to each equation:

m2 – 36 = 0 3x2 + 15 = 0 4d2 + 16 = 16

***Homework: pg. 551 #20-30 even

Solve by factoring

35. Factor each polynomial. (Yes, this is from chapter 8)(If it’s not in standard form, put it in standard form……y= ax2+ bx + c before factoring.)

x2+ 7x + 12 2x2 + 5x – 3 6x2 – 5x – 6

x2 – 8x – 48 2x2 – 5x = 88

You can use factoring and the Zero-Product Property to solve quadratic equations.

Therefore, to solve 0 = x2 + 5x + 6 first factor it to (x + 3)(x + 2) and then set each one to zero to find the solutions.

x + 3 = 0 x + 2 = 0 x = -3 x = -2

The solutions are -3 or -2

36. What are the solutions to each equation?

m2 – 5m – 14 = 0 4x2 – 21x = 18 2a2 – 15a + 18 = 0

***Homework pg. 558 #14-24 even

Quiz 3 Score: ___________________________

Graphing Inequalities

37. On graph paper, graph the quadratic equation y= x2 – 3x – 4 Remember with linear equations that > or < meant a dashed line and ≥ or ≤ meant a solid line.Also, ≥ and > meant shade above the line and ≤ and < meant shade below the line.

This is no different.

38. On graph paper, graph the quadratic inequality y < x2 – 3x – 439. On graph paper, graph the quadratic inequality y > x2 – 3x – 4

Quadratic Formula

40. Internet: Watch montgomerycollege.edu/algebra quadratic equations “quadratic formula”

41. Write the quadratic formula.

Solve using the quadratic formula. Round answers to the nearest hundredth if necessary.42. 0= x2 – 5x + 6 43. 0= x2 – 2x – 8 44. 117= x2 – 4x

45. 7c2 + 8c + 1 = 0 46. 2w2 – 28w = –98 47. 2n2 – 6n = 8

Quadratic equations can have two, one, or no real-number solutions. Before you solve a quadratic equation, you can determine how many real-number solutions it has by using the discriminant. The discriminant is the expression under the radical sign in the quadratic formula.

The discriminant of a quadratic equation ( b2 – 4ac) can be positive, zero, or negative.

Find the number of real-number solutions to each equation.51. x2 – 8x + 7 = 0 52. 2x2 – 5x + 16 = 0 53. 7x2 + 12x – 21 = 0 54. 2x2 + 4x + 2 = 0

***Homework pg. 571-572 #7-22 even, 29-34

Quiz 4 Score: ___________________________

42. Internet: www.teachertube.com Search for quadratic formula and watch some of the videos.

***Homework p. 587Problem Solving

1. Suppose that a pizza must fit into a box with a base that is 12 in. long and 12 in. wide. You can use the quadratic function A = лr2 to find the area of a pizza in terms of its radius.

A. What values of r make sense for this situation?B. Make a table for those values. Round values of A to the nearest tenth. C. Graph the function.

r A

2. Suppose a person is riding in a hot-air balloon, 144 feet above the ground. He drops and apple. The height of the apple above the ground is given by the formula h = -16t2 + 144, where h is height in feet and t is time in seconds.

A. Graph the function using the graphing calculator.B. How far has the apple fallen from time t = 0 to t = 1?C. Does the apple fall as far from time t = 1 to t = 2 as it does from time t = 0 to t

= 1? Why or why not?

3. In professional fireworks displays, aerial fireworks carry “stars” upward, ignite them, and project them into the air. Suppose a particular star is projected from an aerial firework at a starting height of 520 ft with an initial upward velocity of 72 ft/s. How long will it take for the star to reach its maximum height? How far above the ground will it be? (The equation h = -16t2 + 72t + 520 gives the star’s height h in feet at time t in seconds.)

Hint: What point on the parabola will be the maximum height? How do you find the coordinates of that point?

4. Suppose you have 80 ft. of fence to enclose a rectangular garden. The function A = 40x – x2 gives you the area of the garden in square feet where x is the width in feet. Graph the function on the graphing calculator.

A. What width gives you the maximum gardening area?B. What is the maximum area?

5. A city is planning a circular duck pond for a new park. The depth of the pond will be 4 ft. Because of water resources, the maximum volume will be 20,000 ft3. Find the radius of the pond. Use the equation V = лr2h where V is the volume, r is the radius, and h is the depth.

6. Suppose you want to make a rectangle like the one shown.A. Estimate each dimension of the rectangle to the nearest integer.B. Write a quadratic equation and use the quadratic formula to find each

dimension to the nearest hundredth.

60 ft2 x

x+1