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Section 5.1 Introduction to Quadratic Functions Quadratic Function • A quadratic function is any function that can be written in the form f(x) = ax² + bx + c, where a ≠ 0. • It is defined by a quadratic expression, which is an expression of the form as seen above. • The stopping-distance function, given by: d(x) = ⅟₁₉x² + ¹¹̸₁₀x, is an example of a quadratic function. Quadratic Functions • Let f(x) = (2x – 1)(3x + 5). Show that f represents a quadratic function. Identify a, b, and c. • f(x) = (2x – 1)(3x + 5) • f(x) = (2x – 1)3x + (2x – 1)5 • f(x) = 6x² - 3x + 10x – 5 • f(x) = 6x² + 7x – 5 a = 6, b = 7, c = - 5 Parabola • The graph of a quadratic function is called a parabola. Parabolas have an axis of symmetry, a line that divides the parabola into two parts that are mirror images of each other. • The vertex of a parabola is either the lowest point on the graph or the highest point on the graph. Domain and Range of Quadratic Functions • The domain of any quadratic function is the set of all real numbers. • The range is either the set of all real numbers greater than or equal to the minimum value of the function (when the graph opens up). • The range is either the set of all real numbers less than or equal to the maximum value of the function (when the graph opens down). Minimum and Maximum Values • Let f(x) = ax² + bx + c, where a ≠ 0. The graph of f is a parabola. • If a > 0, the parabola opens up and the vertex is the lowest point. The y-coordinate of the vertex is the minimum value of f. • If a < 0, the parabola opens down and the vertex is the highest point. The y-coordinate of the vertex is the maximum value of f. Minimum and Maximum Values • f(x) = x² + x – 6 • Because a > 0, the parabola opens up and the function has a minimum value at the vertex. • g(x) = 5 + 4x - x² • Because a < 0, the parabola opens down and the function has a maximum value at the vertex. Section 5.2 Introduction to Solving Quadratic Equations Solving Equations of the Form x² = a • If x² = a and a ≥ 0, then x = √a or x = - √a, or simply x = ± √a. • The positive square root of a, √a is called the principal square root of a. • Simplify the radical for the exact answer. Solving Equations of the Form x² = a • Solve 4x² + 13 = 253 • 4x² + 13 = 253 - 13 - 13 4x² = 240 Simply the Radical √60 = √(2 ∙ 2 ∙ 3 ∙ 5) √60 = 2√(3 ∙ 5) √60 = 2√15 (exact answer) 4x² = 240 4 4 x² = 60 x = √60 or x = - √60 (exact answer) x = 7.75 or x = - 7.75 (approximate answer) Properties of Square Roots • Product Property of Square Roots: • If a ≥ 0 and b ≥ 0: √(ab) = √a ∙ √b • Quotient Property of Square Roots: • If a ≥ 0 and b > 0: √(a/b) = √(a) ÷ √(b) Properties of Square Roots • Solve 9(x – 2)² = 121 • 9(x – 2)² = 121 9 9 (x – 2)² = 121/9 √(x – 2)² = ±√(121/9) x – 2 = ±√(121/9) x–2 +2 = √(121/9) +2 x = 2 + √(121/9) or 2 - √(121/9) x = 2 + [√(121) / √ (9)] or 2 – [√(121) / √(9)] x = 2 + (11/3) or 2 – (11/3) x = 17/3 or x = - 5/3 Pythagorean Theorem • If ∆ABC is a right triangle with the right angle at C, then a² + b² = c² A c a C B b