ACT – Class Opener: If a circle’s area is 16𝜋 square centimeters, what is the length, in centimeters of its diameter? Troy works as a server at a restaurant. At the end of the week he give 1 4 of his total tips to the hostess. After that, he gives 1 4 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 A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ 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 Constant function such as f(x)=a has a degree of zero. Linear functions, f(x) = mx+b, have a degree of one. Quadratic functions have a degree of two.

Quadratic Functions: Let a, b, and c be real numbers with a ≠0. The function given by 𝑓 𝑥 =𝑎 𝑥 2 +𝑏𝑥+𝑐 is called a quadratic function.

Quadratic Functions All quadratic Functions will produce: Parabolas Axis of Symmetry Vertex

Example 1: Describe how the graph of each function is related to the graph of 𝑦= 𝑥 2 . 𝑓 𝑥 = 1 3 𝑥 2 𝑓 𝑥 =2 𝑥 2 𝑘 𝑥 = 𝑥+2 2 −3

Standard Form of a Quadratic Function 𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘, 𝑎≠0 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: Describe the graph of the following function and identify the vertex. 𝑓 𝑥 =2 𝑥 2 +8𝑥+7 Write the function in standard form.

Student Check: Rewrite each quadratic function in standard form and identify the vertex: 𝑥 2 −10𝑥+25 𝑥 2 −8𝑥+16 4 𝑥 2 −4𝑥+21

Identifying the X intercepts of a Quadratic Equation Find the x intercepts of the following quadratic equation: 𝒇 𝒙 =− 𝒙 𝟐 +𝟔𝒙−𝟖 Rewrite the quadratic equation in standard form.

Student Check Find the x intercepts of the following quadratic equations: 𝒇 𝒙 =− 𝒙 𝟐 +𝟐𝒙+𝟓 𝒇 𝒙 = 𝒙 𝟐 −𝒙+ 𝟓 𝟒 𝒇 𝒙 = 𝒙 𝟐 +𝟐𝒙+𝟏

ACT Class Opener: 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. 99% 80% 69% 45% 26%

Writing in Standard Form 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: Write the equation of a parabola with vertex (-2,5) and passes through the point (0,9)

Finding Minimum and Maximum If a > 0 f has a minimum value at 𝑥=− 𝑏 2𝑎 If a < 0 f has a maximum value at 𝑥=− 𝑏 2𝑎

The Maximum Height of a Baseball 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 𝒇 𝒙 =−.𝟎𝟎𝟑𝟐 𝒙 𝟐 +𝒙+𝟑 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: A soft drink company has daily production costs of: 𝐶 𝑥 =70,000−120𝑥+.055 𝑥 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: The number, g, of grants awarded from the Nation Endowment for Humanities fund from 1999 to 2003 can be approximated by the model: 𝑓 𝑥 =−99.14 𝑥 2 +2201.1𝑥−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: Complete the following: Pg. 99 – 102 #21 – 26 #29 – 34 #55 – 61