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ACT β Class Opener: β’ If a circleβs area is 16π square centimeters, what is the length, in centimeters of its diameter? β’ Troy works as a server at a restaurant. At the end 1 of the week he give of his total tips to the 4 1 gives 4 hostess. After that, he of his remaining tip money to the cooks and they split it evenly. If each cook receives $6, how much money did Troy start out with? Recall: Polynomial Function β’ A polynomial function is a function that can be defined by evaluating a polynomial. A function Ζ of one argument is called a polynomial function if it satisfies for all arguments x, where n is a non-negative integer and a0, a1,a2, ..., an are constant coefficients. Degrees of Polynomial Functions β’ Constant function such as f(x)=a has a degree of zero. β’ Linear functions, f(x) = mx+b, have a degree of one. β’ Quadratic functions have a degree of two. Quadratic Functions: β’ Let a, b, and c be real numbers with a β 0. The function given by π π₯ = ππ₯ 2 + ππ₯ + π is called a quadratic function. Quadratic Functions β’ All quadratic Functions will produce: β Parabolas β Axis of Symmetry β Vertex Example 1: β’ Describe how the graph of each function is related to the graph of π¦ = π₯ 2 . 1 2 π π₯ = π₯ 3 π π₯ = 2π₯ 2 π π₯ = π₯+2 2 β3 Standard Form of a Quadratic Function β’ π π₯ =π π₯ββ 2 + π, πβ 0 β’ The graph of f is a parabola whose axis is the vertical line x = h and whose vertex is the point (h,k). If a > 0, the parabola opens upward, if a < 0 the parabola opens downward. Example 2: β’ Describe the graph of the following function and identify the vertex. π π₯ = 2π₯ 2 + 8π₯ + 7 β’ Write the function in standard form. Student Check: β’ Rewrite each quadratic function in standard form and identify the vertex: π₯ 2 β 10π₯ + 25 π₯ 2 β 8π₯ + 16 4π₯ 2 β 4π₯ + 21 Identifying the X intercepts of a Quadratic Equation β’ Find the x intercepts of the following quadratic equation: π π = βππ + ππ β π β’ Rewrite the quadratic equation in standard form. Student Check β’ Find the x intercepts of the following quadratic equations: π π = βππ + ππ + π π π π =π βπ+ π π π π = ππ + ππ + π ACT Class Opener: β’ Mr. Mauro gave his class a test on 25 vocabulary words. Only one of the following percent is possible as the percent of 25 words a student defined correctly. Which one is it. A) 99% B) 80% C) 69% D) 45% E) 26% Writing in Standard Form β’ Write the standard form of the equation of the parabola whose vertex is (1,2) and that passes through the points (3,-6) Student Check: β’ Write the equation of a parabola with vertex (-2,5) and passes through the point (0,9) Finding Minimum and Maximum β’ If a > 0 f has a minimum value at π₯ = π β 2π β’ If a < 0 f has a maximum value at π₯ = π β 2π The Maximum Height of a Baseball β’ A baseball is hit at a point 3 ft above the ground at a velocity of 100 ft/sec and at an angle of 45 degrees with respect to the ground. The path of the baseball is given by the function π π = β. ππππππ + π + π where f(x) is the height of the baseball and x is the horizontal distance from home plate. What is the maximum height the ball reaches? Example: β’ A soft drink company has daily production costs of: πΆ π₯ = 70,000 β 120π₯ + .055π₯ 2 where C is the total cost and x is the number of units produced. Estimate numerically the number of units that should be produced each day to yield a minimum cost. Quick Quiz: β’ The number, g, of grants awarded from the Nation Endowment for Humanities fund from 1999 to 2003 can be approximated by the model: π π₯ = β99.14π₯ 2 + 2201.1π₯ β 10896 where x represents the year, with x = 9 corresponding to 1999. Using this model determine the year in which the number of grants awarded was greatest. Partner Practice: β’ Complete the following: Pg. 99 β 102 #21 β 26 #29 β 34 #55 β 61