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Notes Section 5.1: Modeling Data with Quadratic Functions Example One: Find a quadratic model for the following: a. x -1 0 2 f(x) 1 -1 7 (-1, 1) (0, -1) (2 , 7) 1= a(-1)2 + b(-1) + c -1 = a(0)2 + b(0) + c 7 = a(2)2 + b(2) + c or or or 1=a–b+c -1 = c 7 = 4a + 2b + c Finish: b. (-1, 10) (2, 4) (3, -6) Example Two: Identify the vertex and axis of symmetry of each parabola: Vertex ( _____ , ______) Axis of symmetry :________ Vertex ( _____ , ______) Axis of symmetry :________ Vertex ( _____ , ______) Axis of symmetry :________ Example Three: Determine whether the function is quadratic, linear. Identify the quadratic, linear, and constant terms. a. y=(x-2)(x+4) b. y = 3x(x-7) c. D. Example Four: A toy rocket is shot upward from ground level. The table shows the height of the rocket at different times: Time (seconds) 0 1 2 3 4 Height (feet) 0 256 480 672 832 a. Find a quadratic model for the data b. Use the model to estimate the height of the rocket after 1.5 seconds Height Time Complete the equation of each function using the given point. (Find c) 1) 2) and (3, 10) and (-2, -15) 3) 4) and and (4, 15) Notes 5.2 Properties of Parabolas General Form of a Parabola: Opens Up: and Opens Down: Vertex: Axis of Symmetry: y-intercept: Example One: Graph Vertex: y-intercept: axis of symmetry: Chart of Points: Example Two: Graph Vertex: y-intercept: axis of symmetry: Chart of Points: (0 , c ) Example Three: Graph Vertex: y-intercept: axis of symmetry: Chart of Points: Example Four: The number of weekend get-away packages a hotel can sell is modeled by -0.12p + 60, where p is the price of a get-away package. What price will maximize the revenue? What is the maximum revenue? Revenue = Price # of packages sold Revenue = Example Five: The number of Albert Pujols bobble head figures the St. Louis Cardinals can sell can be modeled by -4p+100, where p is the price of the bobble head. What price will maximize revenue? What is the maximum revenue? Notes 5.3: Transforming Parabolas Vertex Form: _______________ Equation Axis of Symmetry Vertex y-intercept a<0 a>0 k h Examples: Fill out the information, then graph the parabola. 1) Vertex: _______ Axis of symmetry: _______ y-intercept: ________ point: (1, x y ) Its reflection: _______ 2) Vertex: _______ Axis of symmetry: _______ y-intercept: ________ Its reflection: ________ point: (1, ) Its reflection: ________ x y Write the equation of each parabola in vertex form. 3) 4) plug in h = 3 and k=4 plug in (1, -4) Solve for “a” The Equation is: y = _________________ Standard form: ___________________ Vertex form: ___________________ Change each equation from standard form to vertex form. 1) 2) 3 5) 6) 7) 8) Change each equation into standard form. 9) 10) Notes 5.4: Factoring Quadratic Expressions Review: F O I L 1) 2) 3) Factor: 3-types of Quadratics we’ll factor Examples: 1) 2) 3) 4) 5) 6) More Practice: Factor each expression. 1) 2) 2 3) 4 4) 3 5) 6) 4 7) 8) 9) 10) Notes 5.5: Solving Quadratic EQNS by factoring, square roots, and graphing!!!! Factoring: get the equation = 0, factor, ( )( Square roots: rewrite as , take square root of both sides, +/- , isolate )=0, set each individual =0 Graphing: 2nd calc intersect Example One: Solve by factoring x = _____ or x = _____ Example Two: Solve the following by factoring: A. B. C. Example Three: Solve by finding square roots. Solve: Example Four: The function models the height of a heavy object t seconds after it is dropped from a building that is 270 feet tall. How long until it hits the ground. 270 ft t = ____ for a height of 0 t Example Five: Use Square roots to solve the following: A. B. Example Six: Solve by graphing A carpenter wants to cut a piece of plywood in the shape of a right triangle as shown. He wants the hypotenuse of the triangle to be 6 feet long. About how long should the perpendicular sides be? X+1 6 ft x Example Seven: Solve by graphing A. B. Y1 = __________ Y3 = __________ Y1 = __________ Y3 = __________ 2nd calc: Intersect 2nd calc: Intersect x = ____ or x = _____ x = ____ or x = _____ C. Y1 = _________ Y3 = ____________ 2nd Calc Intersect x= ____ or x = _____ Extra Practice (5.1-5.5) Name ___________________ 1) Write the equation of the quadratic function that contains the 3 given points (use elimination). (1, 9) (-1, 3) (-2, 6) 2) Complete the equation of each function using the given point. (Find c) , and (2, -1) 3) Fill in the information about the quadratic, and then graph the parabola. y 2 x 2 6 axis: _______ vertex: ______ point: (1, ) reflection: _______ point: (2, ) reflection: _______ 4) Fill in the information about the quadratic, and then graph the parabola. y 1 2 x 2x 6 2 axis: _______ vertex: ______ y-int: _______ reflection: _______ point: (-2, ) reflection: _______ 5) Fill in the information about the quadratic, and then graph the parabola. y ( x 2) 2 2 axis: ________ vertex: _______ y-int: _______ reflection: ________ point: (1, ) reflection: ________ 6) Change the equation to vertex form. y 2 x 2 8x 3 7) Change the equation to standard form. y 3( x 1) 2 2 Factor each expression completely. 8) x 2 5 x 24 10) Solve the quadratic by factoring. x 2 9 x 18 0 9) 2 x 2 7 x 15 Notes 5.6 Complex Numbers: Square root of a negative real number: a+bi = Imaginary Number: Complex Number: Example One: Simplifying Numbers a. d. b. c. e. f. Example Two: Simplifying Imaginary Numbers. (Rewrite as a + bi ) a. b. c. Example Three: Operations with Complex Numbers a. b. c. d. f. g. h. i. Example Four: Finding Complex Solutions NO b TERM!!!!! a. c. +54=0 b. d. Notes 5.8 Solving Quadratic Equations: Solving using the quadratic formula and our brains!!!! To determine the Possible Number of Real Solutions Graphically-Look at the x- intercepts: Two x-intercepts 2 Real Solutions Zero x-intercepts 0 Real Solutions & 2 Complex One x-intercept 1 Real Solution To determine the Possible Number of Real Solutions look at the Discriminant: discriminant 2 Real Solutions BONUS INFORMATION: 0 Reals, 2 Complex 1 Real Solution PERFECT SQUARE, then we can solve using factoring Example: Solve using Factoring or the Quadratic Formula 3x2-x=4 -2x2=4x+3 4x2=8x-3 x2+4x=41 Example Two: Determine the type and number of solutions of each equation: x2+6x+9 x2+6x+10=0 A player throws a ball up and toward a wall that is 17 feet high. The height h in feet of the ball t seconds after it leaves the player’s hand is modeled by h = -16t2+25t+6. If the ball makes it to where the wall is, will it go over the wall or hit the wall? h = -16t2+25t+6 17= -16t2+25t+6 Example Three: The longer leg of a right triangle is 1 unit longer than the shorter leg. The hypotenuse is 3 units long. What is the length of the shorter leg? 5.6-5.8 Review 1) Solve each equation by factoring or by taking square roots. a) b) c) d) e) f) 2) Simplify each expression. a) b) c) d) e) f) 3) Solve each quadratic equation a) d) b) c) e)