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8-5-1

Chapter 1

Section 8-5Quadratic Functions, Graphs, and

Models

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8-5-2

Quadratic Functions, Graphs, and Models

• Quadratic Functions and Parabolas

• Graphs of Quadratic Functions

• Vertex of a Parabola

• General Graphing Guidelines

• A Model for Optimization

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8-5-3

Quadratic Functions

A function f is a quadratic function if

where a, b, and c are real numbers, with

2( ) ,f x ax bx c

0.a

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8-5-4

Graph of

y

x(0, 0)

(1, 1)(-1, 1)

(2, 4)(-2, 4)

The graph is called a parabola. Every quadratic function has a graph that is a parabola.

Vertex

Axis (line of symmetry)

2( )f x x

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8-5-5

Example: Graphing Quadratic Functions

2a) ( )g x x

2( ( ) )g x ax21

b) ( )2

g x x

y

x

y

x

2( )g x x

21( )

2g x x

2( )f x x2( )f x x

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8-5-6

Example: Graphing Quadratic Functions (Vertical Shift)

2a) ( ) 2g x x 2b) ( ) 1g x x

y

x

y

x

2( )f x x2( )f x x

2( ) 2g x x

2( ) 1g x x

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8-5-7

Example Graphing Quadratic Functions (Horizontal Shift)

2a) ( ) ( 2)g x x 2b) ( ) ( 1)g x x

y

x

y2( )f x x

2( )f x x

2( ) ( 2)g x x

2( ) ( 1)g x x

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8-5-8

General Principles for Graphs of Quadratic Functions

1. The graph of the quadratic function defined by

is a parabola with vertex (h, k), and the vertical line x = h as axis.

2. The graph opens upward if a is positive and downward if a is negative.

3. The graph is wider than that of f (x) = x2 if 0 < |a| < 1 and narrower if |a| > 1.

2( ) ( ) , 0,f x a x h k a

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8-5-9

Example: Graphing Quadratic Functions (Vertical Shift)

2( ) 2( 3) 1.f x x

y

2( ) 2( 3) 1f x x

Graph

Solution

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8-5-10

Example: Finding the Vertex by Completing the Square

Find the vertex of the graph of2( ) 2 3.f x x x

We need to put the function in the form (x – h)2 + k by completing the square.

2( ) 3 2f x x x 2( ) 3 1 2 1f x x x

2( ) ( 1) 2f x x

Solution

2( ) 2 ( 1)f x x The vertex is at (–1, 2).

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8-5-11

Vertex Formula

The vertex of the graph of2( ) ( 0)f x ax bx c a

has coordinates

, .2 2

b bf

a a

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Example: Vertex Formula

Find the vertex of the graph of

21

2(1)x

2( 1) ( 1) 2( 1) 3 4f

Solution

The vertex is at (–1, –4).

2( ) 2 3.f x x x

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General Guidelines for Graphing a Quadratic Function 2( )f x ax bx c

Step 1 Decide whether the graph opens upward or downward.

Step 2 Find the vertex.

Step 3 Find the y-intercept by evaluating f (0).

Step 4 Find the x-intercepts, if any, by solving f (x) = 0.

Step 5 Complete the graph. Plot additional points as needed.

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Example: Graph Using General Guidelines

Graph

SolutionStep 1 a = 1 > 0 so it opens upward.

Step 2 The vertex is at (–1, –4).

Step 3 f (0) = –3, so the y-intercept is (0, –3)

Step 4

2( ) 2 3.f x x x

2 2 3 0x x ( 3)( 1) 0x x

3 or 1x

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8-5-15

Graphing Quadratic Functions (Vertical Shift)

y

2( ) 2 3f x x x

Step 5

The x-intercepts are (–3, 0) and (1, 0).

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8-5-16

A Model for Optimization

The y-value of the vertex gives the maximum or minimum value of y, while the x-value tells us where that maximum or minimum occurs. Often a model can be constructed so that y can be optimized.

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8-5-17

Example: Finding a Maximum Area

A farmer has 132 feet of fencing. He wants to put a fence around three sides of a rectangular plot of land, with the side of a barn forming the fourth side. Find the maximum are that he can enclose. What dimensions give this area?

SolutionLet x represent the width (see picture on the next slide). If we have 2 widths, that leaves 132 – 2x for the length.

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8-5-18

Example: Finding a Maximum Area

Area

Barn

x x132 - x

The area: A(x) = x(132 – 2x) = –2x 2 + 132x

The vertex is (33, 2178).

The farmer can enclose 2178 square feet, when the width is 33 feet and the length is 132 – 2(33) = 66 feet.