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Do Now!

Do Now!. Evaluate each expression for a = -1, b = 3, and c = -2. 1) 2a – b 2 + c 2) b 2 – 4ac 2a Graph each function. 3) y = x 2 Factor each expression. 4) 4x 2 + 4x + 1 5) m 2 – 7m – 18. Chapter 9: Quadratic Functions and Equations.

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Do Now!

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  1. Do Now! Evaluate each expression for a = -1, b = 3, and c = -2. 1) 2a – b2 + c 2) b2 – 4ac 2a Graph each function. 3) y = x2 Factor each expression. 4) 4x2 + 4x + 1 5) m2 – 7m – 18

  2. Chapter 9: Quadratic Functions and Equations SWBAT solve, graph, compare, and identify Quadratic Functions and Equations.

  3. 9-1 Quadratic Graphs and Their Properties Vocabulary: Quadratic Function Standard Form of a Quadratic Function Quadratic Parent Function Parabola Axis of Symmetry Vertex Minimum Maximum

  4. 9-1 Quadratic Graphs and Their Properties Vocabulary: Quadratic Function: a function that can be written in the form of y = ax2 + bx + c Standard Form of a Quadratic Function: y = ax2 + bx + c Quadratic Parent Function: The simplest quadratic function f(x) = x2 or y = x2 Parabola: The U-Shaped curve of a quadratic function Axis of Symmetry: The fold or line that divides the parabola into 2 matching halves Vertex: the highest or lowest point of a parabola Minimum: Lowest point of the parabola Maximum: Highest point of the parabola

  5. Standard Form of a Quadratic Function Examples: y = 3x2 y = x2 + 9 y = x2 – x - 2

  6. Identifying a Vertex • What are the coordinates of the vertex of each graph? Is it a minimum of maximum? Which direction does the parabola open? DOWNWARD What is the highest point of the parabola? (0, 3) Is this vertex a maximum or minimum? Maximum

  7. You Do! • What is the vertex of the graph? Is it a minimum or maximum? • (-2, -3); minimum

  8. Graphing y = ax2 Graph the function y = 1/3 x2 . Make a table of values. What are the domain and range? The domain is all real numbers. The range is y ≥ 0.

  9. You Do! Graph the function y = -3x2. What are the domain and range? Domain: All real numbers, Range: y ≤ 0.

  10. Comparing Widths of Parabolas What is the order, from widest to narrowest, of the graphs of the quadratic functions: f(x) = -4x2 f(x) = ¼ x2 f(x) = x2

  11. Comparing Widths of Parabolas What is the order, from widest to narrowest, of the graphs of the quadratic functions: f(x) = -4x2 f(x) = ¼ x2 f(x) = ¼ x2 is the widest f(x) = x2 f(x) = -4x2 is the narrowest So the order from widest to narrowest is: f(x) = -x2, f(x) = 3x2, and f(x) = -1/3 x2

  12. You Do! What is the order, from widest to narrowest, of the graphs of the functions: f(x) = -x2, f(x) = 3x2, and f(x) = -1/3 x2 f(x) = -1/3 x2 , f(x) = -x2 , f(x) = 3x2

  13. Graphing y = ax2 + c How is the graph of y = 2x2 + 3 different from the graph of y = 2x2 ? The graph of y = 2x2 + 3 has the same shape as the graph of y = 2x2 but is shifted up 3 units.

  14. You Do!! Graph y = x2 and y = x2 – 3. How are the graphs related? The 2 graphs have the shape But the second parabola is Shifted down 3 units.

  15. Falling Object As an object falls, its speed continues to increase, so its height above ground decreases at a faster rate. You can model the object’s height with the function: h = -16t2 + c. The height h is in feet, the time t is in seconds, and the object’s initial height c is in feet.

  16. Using the Falling Object Model An acorn drops from a tree branch 20 ft above the ground. The function h = -16t2 + 20 gives the height h of the acorn (in feet) after t seconds. What is the graph of this quadratic function? At about what time does the acorn hit the ground?

  17. Using the Falling Object Model An acorn drops from a tree branch 20 ft above the ground. The function h = -16t2 + 20 gives the height h of the acorn (in feet) after t seconds. What is the graph of this quadratic function? At about what time does the acorn hit the ground? Plan: Use a table of values to graph the function. Use the graph to estimate when the acorn hits the ground. • Know: • The function for the acorn’s height • The initial height is 20 ft Need: The function’s graph and the time the acorn hits the ground

  18. Using the Falling Object Method The acorn hits the ground when its height above the ground is 0 ft. From the graph you can see that the acorn hits the ground after slightly more than 1 s.

  19. You Do! In the previous example, suppose the acorn drops from a tree branch 70 ft above the ground. The function h = -16t2 + 70 gives the height h of the acorn (in feet) after t seconds. What is the graph of this function? At about what time does the acorn hit the ground? About 2 seconds

  20. Homework Workbook Pages: 9-1 pg. 255-256 1 – 27 odd

  21. 9-2 Quadratic Functions The graph of y = ax2 + bx + c, where a ≠ 0, has the line x = -b/2a as its axis of symmetry. The x-coordinate of vertex is –b/2a.

  22. Why is it important that a is not equal to zero? If a = 0, then the equation does not have a quadratic term and is not a parabola. If a and b are both positive, will the axis of symmetry be to the left or to the right of the y-axis? It will be to the left of the y-axis. If a and b are both negative, will the axis of symmetry be to the left or to the right of the y axis? It will be to the left of the y-axis

  23. Graphing y = ax2 + bx + c What is the graph of the function: y = ax2 + bx + c Step 1: Find the axis of symmetry and the coordinates of the vertex. x = -b/2a = -(-6)/2(1) = 3 Use the equation to find the axis of symmetry The axis of symmetry is x = 3. So the x-coordinate of the vertex is 3. y = x2 – 6x + 4 Rewrite function = 32 – 6(3) + 4 Substitute 3 for x y = -5 Simplify The vertex is (3, -5)

  24. Graphing y = ax2 + bx + c Step 2: Find two other points on the graph. Find the y-intercept. When x = 0, y = 4, so one point is (0, 4). Find another point by choosing a value for x on the same side of the vertex as the y-intercept. Let x = 1. y = x2 – 6x + 4 Rewrite function. = 12 – 6 (1) + 4 = -1 Substitute 1 and simplify When x = 1, y = - 1, so another point is (1, -1).

  25. Graphing y = ax2 + bx + c Step 2: Find two other points on the graph. Find the y-intercept. When x = 0, y = 4, so one point is (0, 4). Find another point by choosing a value for x on the same side of the vertex as the y-intercept. Let x = 1. y = x2 – 6x + 4 Rewrite function. = 12 – 6 (1) + 4 = -1 Substitute 1 and simplify When x = 1, y = - 1, so another point is (1, -1).

  26. Graphing y = ax2 + bx + c Step 3: Graph the vertex and the points you found in Step 2, (0, 4) and (1, -1). Reflect the points from Step 2 across the axis of symmetry (x = 3) to get two more points on the graph. Then connect the points with a parabola.

  27. Graphing y = ax2 + bx + c How do you determine the reflection of the point (1, -1) across the axis of symmetry? You determine that the horizontal distance from (1, -1) to the axis of symmetry is 2 units left, and then count 2 horizontal units right from the axis of symmetry to get to point (5, -1).

  28. You Do! What is the graph of the function: y = -x2 + 4x – 2? Label the Vertex and tell if it’s a minimum or maximum. Vertex: (2,2) Maximum

  29. Note! In lesson 9-1 we used h = -16t2 + c to find he height h above the ground of an object falling from an initial eight c at a time t. if an object projected into the air given an initial upward velocity v continues with no additional force of its own, the formula: h = -16t2 + vt + c gives its approximate height above the ground.

  30. Using the Vertical Motion Model During halftime of a basketball game, a sling shot launches T-shirts at the crowd. A T-shirt is launched with an initial upward velocity of 72 ft/s. The T-shirt is caught 35 ft above the court. How long will it take the T-shirt to reach its maximum height? What is the maximum height? What is the range of the function that models the height of the T-shirt over time?

  31. Using the Vertical Motion Model • The function h = -16t2 + 72t + 5 gives the T-shirt’s height h, in feet, after t seconds. Since the coefficient of t2 is negative, the parabola opens downward, and the vertex is the maximum point. Method 1: Use a formula. t = -b/2a = -72/2(-16) = 2.25 Find t h= -16(2.25)2 + 72(2.25) + 5 = 86 Find h The T-shirt will reach its maximum height of 86 ft after 2.25 s. The range describes the height of the T-shirt during its flight. The T-shirt starts at 5ft, peaks at 86 ft, and then is caught at 35 ft. The height of the T-shirt at any time is between 5 ft and 86 ft, inclusive, so the range is 5 ≤ h ≤ 86.

  32. Using the Vertical Motion Model Method 2: Use a graphing calculator

  33. You do! In the previous example, suppose a T-shirt is launched with an initial upward velocity of 64 ft/s and is caught 35 ft above the court. How long will it take the T-shirt to reach its maximum height? How far above court level will it be? What is the range of the function that models the height of the T-shirt over time? 2 seconds; 69 feet; 5≤ h ≤ 69

  34. Homework! Workbook pages: 9-2 pg. 259-260 1 – 27 odd pg. 261 1 – 6 all

  35. Do Now! In your composition notebook!! Explain how you can use the y- intercept, vertex, and axis of symmetry to graph a quadratic function. Assume the vertex is not on the y – axis.

  36. 9-3 Solving Quadratic Equations Vocabulary: Quadratic equation Standard Form of a Quadratic Equation Root of an Equation Zero of a Function

  37. 9-3 Solving Quadratic Equations Vocabulary: Quadratic equation: is an equation that can be written in the form of ax2 + bx + c = 0, where a≠0. Standard Form of a Quadratic Equation: ax2 + bx + c = 0, where a≠0. Root of an Equation: the solutions of quadratic equation and the x-intercepts of the graph of ht related function are called Root of an Equation. Zero of a Function: the solutions of quadratic equation and the x-intercepts of the graph of ht related function are called Zero of a Function

  38. Solving by Graphing What are the solutions of each equation? Use a graph of the related function. • x2 – 1 = 0 Graph y = x2 - 1 There are 2 solutions, + 1. (+1, -1) The x-intercepts show the solutions of the equation.

  39. Solving by Graphing What are the solutions of each equation? Use a graph of the related function. b) x2 = 0 Graph y = x2. There is one solution, 0.

  40. Solving by Graphing What are the solutions of each equation? Use a graph of the related function. c) x2 + 1 = 0 Graph y = x2 + 1 There is no real- number solution.

  41. You Do! What are the solutions of each expression? Use a graph of the related function. • x2 – 16 = 0 + 4 2) x2 – 25 = -25 0

  42. Solving Using Square Roots What are the solutions of 3x2 – 75 = 0 ? 3x2 – 75 = 0 3x2 = 75 x2 = 25 x = +√25 x = + 5

  43. You Do! What are the solutions of each equation? • m2 – 36 = 0 • 3x2 + 15 = 0 • 4d2 + 16 = 16 + 6 no solution 0

  44. Choosing Reasonable Solution An aquarium is designing a new exhibit to showcase tropical fish. The exhibit will include a tank that is a rectangular prism with a length l that is twice the width w. The volume of the tank is 420 ft3. What is the width of the tank to the nearest tenth of a foot?

  45. Choosing Reasonable Solution An aquarium is designing a new exhibit to showcase tropical fish. The exhibit will include a tank that is a rectangular prism with a length l that is twice the width w. The volume of the tank is 420 ft3. What is the width of the tank to the nearest tenth of a foot? V = lwh Use Volume Formula 420 = (2w)(w)(3) Substitute 420 = 6w2 Simplify 70 = w2 Divide by √6 +√70 = w Find square root + 8.366600265 = w Use calculator

  46. Choosing Reasonable Solution An aquarium is designing a new exhibit to showcase tropical fish. The exhibit will include a tank that is a rectangular prism with a length l that is twice the width w. The volume of the tank is 420 ft3. What is the width of the tank to the nearest tenth of a foot? + 8.366600265 = w Use calculator A tank cannot have a negative width, so only the positive square root make sense. The tank will have a width of about 8.4 ft.

  47. You Do! Suppose the tank in the previous example will have a height of 4 ft and a volume of 500 ft3. What is the width of the tank to the nearest tenth of a foot? 7.9 ft

  48. Homework! Workbook: 9-3 pg. 263 – 263 1-23 odd; 31 – 35 odd

  49. Do Now What are the solutions of each equation? • m2 – 36 = 0 • 3x2 + 15 = 0 • 4d2 + 16 = 16 + 6 no solution 0

  50. 9-4 Factoring to Solve Quadratic Equations Vocabulary: Zero – Product Property : For any real numbers a and b, if ab = 0, then a = 0 or b = 0. Example: If (x + 3)(x +2) = 0, then x + 3 = 0 or x + 2 = 0

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