Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Fundamental theorem of algebra wikipedia , lookup
System of linear equations wikipedia , lookup
System of polynomial equations wikipedia , lookup
Cubic function wikipedia , lookup
Median graph wikipedia , lookup
Factorization wikipedia , lookup
Elementary algebra wikipedia , lookup
Signal-flow graph wikipedia , lookup
History of algebra wikipedia , lookup
Quartic function wikipedia , lookup
1 Name: Date: Quadratic Equations and Functions Essential Question for Chapter 10: How can we model and evaluate real world situations using quadratic functions? Section 1: Exploring Quadratic Graphs Objectives: To graph quadratic functions of the form y = ax2 & y = ax2 + c 1 Activity: Plotting Quadratic Curves Graph the equations y = x2 and y = 3x2 on the same coordinate plane 2a Describe how the graphs are alike. 2b Describe how the graphs are different. 3 Predict how the graph of y = (1/3)x2 will be similar to and different from the graph of y = x2 2 4 Graph y = (1/3)x2. Where your predictions correct? Explain. Quadratic Function: _____________________________________________________________ _____________________________________________________________ Standard form of a Quadratic Function: _____________________________________________________________ _____________________________________________________________ Standard Form of a Quadratic Function Quadratic Parent Function: _______________________________________ Parabola: _____________________________________________________ Real World Examples: Axis of Symmetry: _____________________________________________________________ Vertex: _______________________________________________________ 3 Minimum Maximum If a > 0 in y = ax2 + bx + c If a < 0 in y = ax2 + bx + c The parabola opens upward The parabola opens downward Identify the Vertex 3) 1) 2) 4) 4 Graphing y = ax2 1) Make a table of values and graph the quadratic function y = ½x2. x y = ½x2 (x, y) 2) Make a table of values and graph the quadratic function y = -2x2. x y = ½x2 (x, y) 5 Comparing Widths of Parabolas on the Calculator Put all 3 graphs into Y= in your calculator, then press Zoom Standard. Graph the quadratic functions and order from widest to narrowest. 1) f(x) = - 4x2; f(x) = (1/4)x2, and f(x) = x2 2) y = x2; y = (1/2)x2, and y = -2x2 Note: When m n the graph of y = mx2 is wider than the graph of y = nx2 Graphing y = ax2 + c 1) How do the graphs below differ? x y = 2x2 y = 2x2 + 3 6 2) How do the graphs below differ? x y = x2 y = x2 – 4 3) Suppose you see an eagle flying over a canyon. The eagle is 30 feet above the level of the canyon’s edge when it drops a stick from its claws. The force of gravity causes the stick to fall toward Earth. The function h = -16t2 + 30 gives the height of the stick h in feet after t seconds. Graph this quadratic function. x h = -16t2 + 30 7 4) Suppose a squirrel is in a tree 24ft above the ground. She drops an acorn. The function h = -16t2 + 24 gives the height of the acorn in feet after t seconds. Graph this quadratic function. x h = -16t2 + 30 Homework: Practice 10-1 (Multiples of 3) 8 Section 2: Quadratic Functions Objectives: To graph quadratic functions of the form y = ax2 + bx + c y = 2x2 + 2x y = 2x2 + 4x Property: Graph of a Quadratic Function y = 2x2 + 6x 9 Graphing y = ax2 + bx + c 1) Graph: y = -3x2 + 6x + 5 Step 1: Find the equation of the axis of symmetry and the coordinates of the vertex. a = -3 b = 6 c = 5 x b 6 6 1 The axis of symmetry is x = 1. 2a 2(3) 6 To find the y-coordinate substitute 1 in for x. y = -3(1)2 + 6(1) + 5 = 8 The vertex is (1, 8) Step 2: Find two other points on the graph. Set up a table with x & y. x -1 0 1 2 3 Step 3: Graph y -4 5 8 5 -4 Set up a table with five ordered pairs. Put the vertex in the middle Pick two points above and below the vertex Find your five ordered pairs Notice how the points will reflex each other Graph 10 2) f(x) = x2 – 6x + 9 x -1 0 1 2 3 y -4 5 8 5 -4 11 3) In professional fireworks display, aerial fireworks carry “star” 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 feet with an initial upward velocity of 72 ft./s. How long will it take the star to reach its maximum height? How far above the group will it be? The equation h = -16t2 + 72t + 520 gives the star’s height h in feet at time t in seconds. Since the coefficient of t2 is negative, the curve opens downward, and the vertex is the maximum point. 4) A ball is thrown into the air with an initial upward velocity of 48 ft./sec. Its height h in feet after t seconds is given by the function h = -16t2 + 48t + 4. a. In how many seconds will the ball reach its maximum height? b. What is the ball’s maximum height? Homework: Practice 10-2 Worksheet (Multiples of 3) 12 Section 3: Solving Quadratic Equations Objectives: To solve quadratic equations by graphing and by using the roots Activity: Finding x-intercepts 1 Find the x-intercepts of each graph. a b 2a Solve: 2x – 3 = 0 2b Is the solution of 2x – 3 = 0 the same as the x-intercept of y = 2x – 3? 3 4a Do the x-intercepts that you found in Question 1b satisfy the equation x2 + 3x – 4 = 0? Graph y = x2 + x – 6 4b Find the x-intercepts of the graph of y = x2 + x – 6. 4c Do the values you found in part (b) satisfy the equation x2 + x – 6? 13 Definition: Standard Form of a Quadratic Equation Roots of the equation or Zeros of the Function: _____________________________________________________________ _____________________________________________________________ Solving by Graphing 1) x2 – 4 = 0 y = x2 – 4 2) x2 = 0 y = x2 14 3) x2 + 4 = 0 4) x2 – 1 = 0 5) 2x2 + 4 = 0 y = x2 + 4 15 6) x2 – 16 = -16 Using Square Roots 1) 2x2 – 98 = 0 3) 3n2 + 12 = 12 2) t2 – 25 = 0 4) 2g2 + 32 = 0 5) A city is planning a circular duck pond for a new park. The depth of the pond will be 4 ft and the volume will be 20,000ft3. Find the radius of the pond to the nearest tenth of a foot. Use the equation V = πr2h, where V is the volume, r is the radius, and h is the depth. Calculator Activity with Roots Homework: Page 567: # 1-21 16 Section 4: Factoring to Solve Quadratic Equations Objectives: To solve quadratic equations by factoring Zero-Product Property Using the Zero-Product Property 1) (x + 5)(2x – 6) = 0 2) (x + 7)(x – 4) = 0 3) (3y – 5)(y – 2) = 0 4) (6k + 9)(4k – 11) = 0 5) (5h + 1)(h + 6) = 0 Solving by Factoring 1) x2 + 6x + 8 = 0 2) x2 – 8x – 48 = 0 17 3) 2x2 – 5x = 88 4) x2 – 12x = -36 5) The diagram shows a pattern for an open-top box. The total area of the sheet of material used to manufacture the box is 288 in 2. The height of the box is 3-in. Therefore, 3-in x 3-in squares are cut from each corner. Find the dimensions of the box. 6) Suppose that a box has a vase with a width of x, a length of x + 1, and a height of 2 in. It is cut from a rectangular sheet of material with an area of 182 in2. Find the dimensions of the box. Homework: Page 574: # 1-25 18 Section 5: Completing the Square Objectives: To solve quadratic equations by completing the square *Completing the Square Activity* Completing the Square: _____________________________________________________________ _____________________________________________________________ Finding n to Complete the Square 1) x2 – 12x + n 2) x2 + 22x + n Solving x2 + bx = c 1) x2 + 9x = 36 Solving x2 + bx + c = 0 3) x2 – 20x + 32 = 0 4) x2 + 5x + 30 = 0 5) x2 – 14x + 16 = 0 2) m2 – 6m = 247 19 6) Suppose a woodworker wants to build a tabletop like the one shown at the right. If the surface area is 26ft2, what is the value of x? 7) 4a2 – 8a = 24 8) 5n2 – 3n – 15 = 10 Homework: Page 582: # 1-25 20 Section 6: Using the Quadratic Formula Objectives: To use the quadratic formula when solving equations and choosing an appropriate method for solving a quadratic eqation. Quadratic Formula: _____________________________________________________________ _____________________________________________________________ Deriving the Quadratic Formula: 21 Quadratic Formula *Quadratic Formula Song!* Using the Quadratic Formula 1) x2 + 6 = 5x 2) x2 – 2x – 8 = 0 3) x2 – 4x = 117 4) 2x2 + 4x – 7 = 0 5) -3x2 + 5x – 2 = 0 6) 7x2 – 2x – 8 = 0 22 7) Suppose a football player kicks a ball and gives it an initial upward velocity of 47 ft/s. The starting height of the football is 3ft. If no one catches the football, how long will it be in the air? (Hint: The vertical motion formula: h = -16t2 + vt + c) 8) A football player kicks a ball with an initial upward velocity of 38.4 ft/sec from a starting height of 3.5 ft. a) Substitute the values into the vertical motion fomula. Let h = 0. b) Sovle. If no one catches the ball, how long will it be in the air? Round to the nearest tenth of a second. 23 Choosing an Appropriate Method There are many methods for solving a quadratic equation. You can always use the quadratic formula, but sometimes another method may be easier. Method Graphing Square Roots Factoring Completing the Square Quadratic Formula When to Use Use if you have a graphing calculator handy Use if the equation has not x term Use if you can factor the equation easilty Use if the x2 term is 1, but you cannot factor the equation easily Use if the equation cannot be factored easily or at all 1) 2x2 – 6 = 0 2) 6x2 + 13x – 17 = 0 3) x2 + 2x – 15 = 0 4) 16x2 – 96x + 45 = 0 5) x2 – 7x + 4 = 0 6) 13x2 – 5x + 21 = 0 7) x2 – x – 30 = 0 8) 144x2 = 25 Homework: Page 588: # 1-17 24 Section 7: Using the Discriminant Objective: To find the number of solutions of a quadratic equation Discriminant: _____________________________________________________________ _____________________________________________________________ b b 2 4ac x 2a Find the discriminant for each quadratic equation y = x2 – 6x + 3 y = x2 – 6x + 9 y = x2 – 6x + 12 25 Discriminate is negative. Discriminate is zero. Discriminate is positive. Property of Discriminant For the quadratic equation ax2 + bx + c = 0, where a ≠ 0, you can use the value of the discriminant to determine the number of solutions. If b2 – 4ac > 0, there are two solutions If b2 – 4ac = 0, there is one solution If b2 – 4ac < 0, there are no solutions Using the discriminant Find the number of solutions 1) 3x2 – 5x = 1 2) x2 = 2x – 3 3) 3x2 – 4x = 7 26 4) 5x2 + 8 = 2x 5) A constuction worker on the ground tosses an apple to a fellow worker who is 20 ft above the ground. The starting height of the apple is 5ft. Its initial upward velocity is 30 ft/s. Will the apple reach the second worker? 6) Suppose the same construction worker tosses an apple with an initial upward velocity of 32 ft/s. Wil the apple reach the second worker. Homework: Page 594: # 1- 15 & # 17