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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 of his total tips to the hostess. After that, he gives 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?

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**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.

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**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.

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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.

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**Quadratic Functions All quadratic Functions will produce: Parabolas**

Axis of Symmetry Vertex

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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

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**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.

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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.

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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

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**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.

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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

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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%

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**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)

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**Student Check: Write the equation of a parabola with vertex**

(-2,5) and passes through the point (0,9)

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**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

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**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?

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**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.

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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 𝑥−10896 where x represents the year, with x = 9 corresponding to Using this model determine the year in which the number of grants awarded was greatest.

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**Partner Practice: Complete the following: Pg. 99 – 102 #21 – 26**

#29 – 34 #55 – 61

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Quadratic Functions; An Introduction

Quadratic Functions; An Introduction

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