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 = 253Simply the Radical √60 = √(2 ∙ 2 ∙ 3 ∙ 5) 4x² = 240√60 = 2√(3 ∙ 5) √60 = 2√15 (exact answer) 4x² = x² = 60 x = √60 or x = - √60 (exact answer) x = 7.75 or x = (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 = 121x = 2 + √(121/9) or 2 - √(121/9) 9 9 x = 2 + [√(121) / √ (9)] or 2 – [√(121) / √(9)] (x – 2)² = 121/9x = 2 + (11/3) or 2 – (11/3) √(x – 2)² = ±√(121/9)x = 17/3 or x = - 5/3 x – 2 = ±√(121/9) x – 2 = √(121/9)

Pythagorean Theorem If ∆ABC is a right triangle with the right angle at C, then a² + b² = c² A c a C B b