# Quadratic Functions

**Section 2.1 Quadratic Functions**

**Objective:** In this lesson you learned how to sketch and analyze graphs

of quadratic functions.

**Important Vocabulary** Define each term or concept.

**Constant function** A polynomial function with degree 0. That is, f(x) =
a, a ≠ 0.

Linear function A polynomial function with degree 1. That is, f(x) = mx + b, m ≠
0.

**Quadratic function** Let a, b, and c be real numbers with a ≠ 0. The
function f(x) =

ax^{2} + bx + c is called a quadratic function.

**Axis of symmetry** A line about which a parabola is symmetric. Also called
simply

the axis of the parabola.

**Vertex** The point where the axis intersects the parabola.

**I. The Graph of a Quadratic Function** (Pages 136-138)

Let n be a nonnegative integer and let be

real numbers with A polynomial function of x
with

degree n is . . .

**the function**

What you should learn

How to analyze graphs of

quadratic functions

A quadratic function is a polynomial function of** second**

degree. The graph of a quadratic function is a special “U”-shaped

curve called a **parabola** .

If the leading coefficient of a quadratic function is positive, the

graph of the function opens **upward** and the vertex of

the parabola is the **minimum** point on the graph. If the

leading coefficient of a quadratic function is negative, the graph

of the function opens **downward** and the vertex of the

parabola is the **maximum** point on the graph.

**II. The Standard Form of a Quadratic Function**

(Pages 139-140)

The standard form of a quadratic function is

For a quadratic function in standard form, the axis of the

associated parabola is **x = h** and the vertex is

**(h, k) **.

What you should learn

How to write quadratic

functions in standard

form and use the results

to sketch graphs of

functions

To write a quadratic function in standard form , . . . **use the
process of completing the square on the variable x.**

To find the x-intercepts of the graph of f (x) = ax^{2} + bx + c , . . .

**solve the equation ax ^{2} + bx + c = 0.**

**Example 1:** Sketch the graph of f (x) = x^{2} + 2x - 8 and

identify the vertex, axis, and x-intercepts of the

parabola.

**(- 1, - 9); x = - 1; (- 4, 0) and (2, 0)**

**III. Applications **(Pages 141-142)

For a quadratic function in the form f (x) = ax^{2} + bx + c , when

a > 0, f has a minimum that occurs at** - b/(2a) .**

When a < 0, f has a maximum that occurs at **- b/(2a) .**

To find the minimum or maximum value, e**valuate the
function at - b/(2a) .**

What you should learn

How to use quadratic

functions to model and

solve real-life problems

**Example 2:** Find the minimum value of the function

f (x) = 3x^{2} -11x +16 . At what value of x does

this minimum occur?

**Minimum function value is 71/12 when x = 11/6**

**Homework Assignment**

Page(s)

Exercises