Unit 2 – Quadratic Functions & Equations

A quadratic function can be written in the form f(x) = ax 2 + bx + c where a, b, and c are real numbers and a ≠ 0. Important Vocabulary: Quadratic function Parabola Maximum Minimum Vertex Axis of Symmetry Domain Range The Quadratic Function

f(x) = x 2 Vertex Axis of Symmetry f(x) = x f(x) = x Graphing f(x) = x 2 + q Comparing f(x) = x 2 + q with f(x) = x 2

General Rules for y = x 2 + q “q” results in a Vertical Translation: 1.y = x 2 + q produces a graph that is the graph of y = x 2 translated “q” units UP y = x 2 → y = x 2 + q (x, y) → (x, y + q) 2.y = x 2 - q produces a graph that is the graph of y = x 2 translated “q” units DOWN y = x 2 → y = x 2 - q (x, y) → (x, y - q)

x y x y x y f(x) = x 2 f(x) = 2x 2 f(x) = 0.5x 2 Graph on the same grid. Describe the affects in words. Graphing f(x) = ax 2 Comparing f(x) = ax 2 with f(x) = x 2 :

The graph of f(x) = -x 2 is the graph of f(x) = x 2 reflected in the x-axis. Graphing f(x) = -x 2 (0, 0)

General Rules for y = ax 2 “a” results in a Vertical Stretch: 1.y = ax 2 produces a graph that is the graph of y = x 2 vertically stretched by a factor of a. y = x 2 → y = ax 2 (x, y) → (x, ay) If |a| > 1 If 0 < |a| < 1 If a < 0 all y-values are multiplied by a negative number This results in a reflection about the x-axis along with the vertical expansion or compression. all y-values are multiplied by a number less than 1 This results in a vertical compression and a wider parabola all y-values are multiplied by a number greater than 1 This results in a vertical expansion and a narrower parabola.

General Rules for y = (x-p) 2 “p” results in a Horizontal Translation: 1.y = (x + p) 2 produces a graph that is the graph of y = x 2 translated “p” units LEFT y = x 2 → y = (x + p) 2 (x, y) → (x - p, y) 2.y = (x - p) 2 produces a graph that is the graph of y = x 2 translated “p” units RIGHT y = x 2 → y = (x - p) 2 (x, y) → (x + p, y)

The Vertex Form of the Quadratic Function y = a(x-p) 2 +q f(x) = a(x - p) 2 + q Vertical Stretch factor Horizontal translation Vertical translation Indicates direction of opening Coordinates of the vertex are (p, q) Axis of symmetry is x = p If a > 0, the graph opens up and there is a minimum value of y. If a < 0, the graph opens down and there is a maximum value of y.

For graphing - Always apply transformations in the following order: 1.Stretches 2.Reflections 3.Translations Ex. Graph f(x) = -(x - 3) and determine the following: Coordinates of the vertex: Equation of the axis of symmetry: Maximum or minimum value: Domain and range: Coordinates of the y-intercept:

Vertex Form

Vertex Form

Determining a Quadratic Function in Vertex Form Given It’s Graph

Finding Intercepts Given the function y = 2(x - 4) 2 – 12, find the following: The y-intercept The x-intercepts Note: the x-intercepts are also called the zeroes or the roots of the function.

Writing the Equation in Vertex Form Write the equation of the parabola, in vertex form, that has a vertex at (0, 4) and passes through the point (-1, 6).

Writing the Equation in Vertex Form Write the equation of the parabola with vertex (-1, 3), and x-intercepts of 2 and -4.

Writing the Equation in Vertex Form Find the equation of the parabola that passes through (1, 15) and (5, -1) with its axis of symmetry at x = 4.

Homework: Pages : #s1, 3, 5, 8-11,14, 16, 20, 21, 23