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Unit 2 – Quadratic, Polynomial, and Radical Equations and Inequalities Chapter 5 – Quadratic Functions and Inequalities 5.1 – Graphing Quadratic Functions

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Quadratic function – described by an equation in the following form: f(x) = ax 2 + bx + c, where a ≠ 0 ax 2 – quadratic term bx – linear term c – constant term The graph of a quadratic function is called a parabola

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5.1 – Graphing Quadratic Functions Example 1 Graph f(x) = x 2 + 3x – 1

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5.1 – Graphing Quadratic Functions Axis of symmetry – imaginary line that creates a mirror image of the graph on either side of the line Vertex – the point where the axis of symmetry intersects the parabola

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5.1 – Graphing Quadratic Functions y = ax 2 + bx + c, where a ≠ 0 Y-intercept: a(0) 2 + b(0) + c, or simply c Axis of symmetry: x = -b/2a Vertex (x-coordinate): x = -b/2a

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5.1 – Graphing Quadratic Functions Example 2 Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex of f(x) = 2 – 4x + x 2

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5.1 – Graphing Quadratic Functions Example 2 (cont.) Make a table of values that includes the vertex. Use this information to graph the functions.

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5.1 – Graphing Quadratic Functions The y-coordinate of the vertex is either the minimum or the maximum value of the function. If the parabola: Opens up, it has a minimum value when a > 0 Opens down, it has a maximum value when a < 0 The range of a quadratic function is all real numbers greater then or equal to the minimum, OR all real numbers less than or equal to the maximum

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5.1 – Graphing Quadratic Functions Example 3 Consider the function f(x) = -x 2 + 2x + 3 Determine whether the function has a maximum or a minimum value State the maximum or minimum value of the function State the domain and range of the function

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5.1 – Graphing Quadratic Functions Example 4 A souvenir shop sells about 200 coffee mugs each month for $6 each. The shop owner estimates that for each $0.50 increase in the price, he will sell about 10 fewer coffee mugs per month. How much should the owner charge for each mug in order to maximize the monthly incomes from their sales? What is the maximum monthly income the owner van expect to make from the mugs?

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5.1 – Graphing Quadratic Functions HOMEWORK Page 241 #13 – 31 odd, 32 – 36 all

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