IB STUDIES Graphing Quadratic Functions

is called a quadratic function. Let a, b, and c be real numbers a  0. The function f (x) = ax2 + bx + c is called a quadratic function. The graph of a quadratic function is a parabola. Every parabola is symmetrical about a line called the axis (of symmetry). x y The intersection point of the parabola and the axis is called the vertex of the parabola. f (x) = ax2 + bx + c vertex axis Quadratic function

The leading coefficient of ax2 + bx + c is a. y a > 0 opens upward When the leading coefficient is positive, the parabola opens upward and the vertex is a minimum. f(x) = ax2 + bx + c vertex minimum x y vertex maximum When the leading coefficient is negative, the parabola opens downward and the vertex is a maximum. f(x) = ax2 + bx + c a < 0 opens downward Leading Coefficient

Simple Quadratic Functions The simplest quadratic functions are of the form f (x) = ax2 (a  0) These are most easily graphed by comparing them with the graph of y = x2. Example: Compare the graphs of , and 5 y x -5 Simple Quadratic Functions

Example: Graph f (x) = (x – 3)2 + 2 and find the vertex and axis. f (x) = (x – 3)2 + 2 is the same shape as the graph of g (x) = (x – 3)2 shifted upwards two units. g (x) = (x – 3)2 is the same shape as y = x2 shifted to the right three units. - 4 x y 4 f (x) = (x – 3)2 + 2 g (x) = (x – 3)2 y = x 2 vertex (3, 2) Example: f(x) = (x –3)2 + 2

Quadratic Function in Standard Form Example: Graph the parabola f (x) = 2x2 + 4x – 1 and find the axis of symmetry, x and y-intercepts and vertex. f (x) = 2x2 + 4x – 1 Use your GDC to find the TP axis of symmetry x-intercepts (-2.22,0) and (0.225,0) y-intercept (0,-1) Quadratic Function in Standard Form

Vertex and x-Intercepts Example: Graph and find the vertex, axis of symmetry and x and y-intercepts of f (x) = –x2 + 6x + 7. a < 0  parabola opens downward. vertex (3,16) y-intercept (0,7) x-intercepts (7, 0), (–1, 0) axis of symmetry x=3 equivalent forms: Vertex and x-Intercepts

The vertex of the graph of f (x) = ax2 + bx + c (a  0) Vertex of a Parabola The vertex of the graph of f (x) = ax2 + bx + c (a  0) Example: Find the vertex of the graph of f (x) = x2 – 10x + 22. f (x) = x2 – 10x + 22 original equation a = 1, b = –10, c = 22 At the vertex, So, the vertex is (5, -3). Vertex of a Parabola

The maximum height of the ball is 15 feet. Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is: The path is a parabola opening downward. The maximum height occurs at the vertex. At the vertex, So, the vertex is (9, 15). The maximum height of the ball is 15 feet. Example: Basketball

Let x represent the width of the corral and 120 – 2x the length. Example: A fence is to be built to form a rectangular corral along the side of a barn 65 feet long. If 120 feet of fencing are available, what are the dimensions of the corral of maximum area? barn corral x 120 – 2x Let x represent the width of the corral and 120 – 2x the length. Area = A(x) = (120 – 2x) x = –2x2 + 120 x The graph is a parabola and opens downward. The maximum occurs at the vertex where a = –2 and b = 120 120 – 2x = 120 – 2(30) = 60 The maximum area occurs when the width is 30 feet and the length is 60 feet. Example: Maximum Area

Substitute point (0,1) to find c. Example: Find an equation for the parabola with vertex (2, –1) passing through the point (0, 1). y x y = f(x) (0, 1) (2, –1) Substitute point (0,1) to find c. Example: Parabola