Section 2.1 Quadratic Functions and Models - Cengage Learning

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Chapter 2 Polynomial and Rational Functions Section 2.1 Quadratic Functions and Models Objective: In this lesson you learned how to sketch and analyze graphs of functions.

Important Vocabulary

Define each term or concept.

Axis of symmetry A line about which a parabola is symmetric. Vertex The point where the axis intersects the parabola.

I. The Graph of a Quadratic Function (Pages 128−130) Let n be a nonnegative integer and let an, an – 1, . . . , a2, a1, a0 be real numbers with an ≠ 0. A polynomial function of x with degree n is . . . the function f(x) = anxn + an – 1xn – 1 + . . . + a2x2 + a1x + a0. Let a, b, and c be real numbers with a ≠ 0. A quadratic function is . . .

the function given by f(x) = ax2 + bx + c.

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 the parabola is the

upward

and the vertex of

y-value on the graph. If the

minimum

leading coefficient of a quadratic function is negative, the graph of the function opens parabola is the

downward

maximum

and the vertex of the

y-value on the graph. The

absolute value of the leading coefficient a determines widely the parabola opens

how

. If | a | is small,

the parabola opens more widely than if | a | is large.

Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved.

What you should learn How to analyze graphs of quadratic functions

36

Chapter 2

•

Polynomial and Rational Functions

II. The Standard Form of a Quadratic Function (Pages 131−132)

What you should learn How to write quadratic functions in standard form and use the results to sketch graphs of functions

The standard form of a quadratic function is f(x) = a(x – h)2 + k, a ≠ 0

.

For a quadratic function in standard form, the axis of the associated parabola is (h, k)

x=h

and the vertex is

.

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 , . . .

y

solve the equation ax2 + bx + c = 0. Example 1:

Sketch the graph of f ( x) = x 2 + 2 x − 8 and identify the vertex, axis, and x-intercepts of the parabola. (− 1, − 9); x = − 1; (− 4, 0) and (2, 0)

III. Applications of Quadratic Functions (Page 133)

What you should learn How to use quadratic functions to model and solve real-life problems

For a quadratic function in the form f ( x) = ax 2 + bx + c , the − b/(2a)

x-coordinate of the vertex is given as y-coordinate of the vertex is given as Example 2:

and the

f(− b/(2a))

x

.

Find the vertex of the parabola defined by f ( x) = 3 x 2 − 11x + 16 . (11/6, 71/12)

Homework Assignment

Page(s) Exercises

Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved.