1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = -2x 2 4x + 2 x y.
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1 1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = -2x 2 4x + 2 x y y = x 2 + 6x -3 x y domain= range= domain= range= Compare these graphs to the linear equation graphs that we have studies previously. How are they different? 3. Define the words and sketch a picture parabola: vertex: Function of a (the coefficient of the x 2 in the quadratic equation y = ax 2 +bx +c) Now you will use the graphing calculator to explore the different roles of a, b, and c in the quadratic equation y = ax 2 +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= x 2 y= 2x 2 y= 4x 2 y= 8x 2 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= x 2 y=.5x 2 y=.25x 2 y=.1x 2 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= -x 2 y= -2x 2 y= -4x 2 y= -8x 2
2 10. What did happen when you made a negative? 11. What will these graphs look like: Graph: y= -x 2 y= -.5x 2 y= -.25x 2 y= -.1x Explain how you can tell what the parabola will look like based on the value of a in the quadratic equation y = ax 2 +bx +c? Function of c (the constant in the quadratic equation y = ax 2 +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 = ax 2 +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= x 2 y= x y= x y= x y= x 2 y= x 2-1 y= x 2-2 y= x 2 3 y= -x 2 y= -x y= -x y= -x y= -x 2 y= -x 2-1 y= -x 2-2 y= -x Explain what you can tell by the value of the c in the quadratic equation y = ax 2 +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:
3 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 > On graph paper: graph the function y = -3x 2. 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 = ax 2 +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= x 2 y= x 2 + x y= x 2 + 2x y= x 2 + 3x y= x 2 y= x 2 - x y= x 2-2x y= x 2-3x 21. What happened to the parabola as the value of b changed? 22. What is the y-intercept in all the graphs above? Why?
4 Axis of Symmetry/Vertex 23. To find the axis of symmetry, use the equation coordinate value of x at the vertex) b x (Remember, the axis of symmetry is just the 2a 24. What is the axis of symmetry for the following quadratic equation: y= 2x 2 + 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= 2x 2 + 4x y= 2x 2 + 6x y=4x 2 7x 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= 3x 2 + 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= 2x x + 10 ***Homework: pg. 544 #7-17 (even), 16-19, (even), 26-27, (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:
5 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 ax 2 +bx + c = 0 is to graph the related quadratic function y = ax 2 +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 ax 2 +bx + c = 0? 33. Find the solutions of each equation by graphing (on graph paper): x 2 16 = 0 3x = 0 x 2 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: m 2 36 = 0 3x = 0 4d = 16 ***Homework: pg. 551 #20-30 even
6 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= ax 2 + bx + c before factoring.) x 2 + 7x x 2 + 5x 3 6x 2 5x 6 x 2 8x 48 2x 2 5x = 88 You can use factoring and the Zero-Product Property to solve quadratic equations. Therefore, to solve 0 = x 2 + 5x + 6 first factor it to (x + 3)(x + 2) and then set each one to zero to find the solutions. 36. What are the solutions to each equation? x + 3 = 0 x + 2 = 0 x = -3 x = -2 The solutions are -3 or -2 m 2 5m 14 = 0 4x 2 21x = 18 2a 2 15a + 18 = 0 ***Homework pg. 558 #14-24 even Quiz 3 Score:
7 Graphing Inequalities 37. On graph paper, graph the quadratic equation y= x 2 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 < x 2 3x On graph paper, graph the quadratic inequality y > x 2 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 = x 2 5x = x 2 2x = x 2 4x 45. 7c 2 + 8c + 1 = w 2 28w = n 2 6n = h 2 2h 2 = x 2 + 3x = s 2 10s + 14 = 0
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 ( b 2 4ac) can be positive, zero, or negative. Find the number of real-number solutions to each equation. 51. x 2 8x + 7 = x 2 5x + 16 = x x 21 = x 2 + 4x + 2 = 0 ***Homework pg #7-22 even, Quiz 4 Score: 42. Internet: Search for quadratic formula and watch some of the videos. ***Homework p. 587
9 Problem 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 = лr 2 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 = -16t , 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 = -16t t 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 x 2 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 ft 3. Find the radius of the pond. Use the equation V = лr 2 h 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. x 60 ft 2 x+1
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