Sections 4.3 and 4.4 Quadratic Functions and Their Properties

Sections 4.3 and 4.4

Quadratic Functions and Their Properties

Quadratic Function

2( ) 0f x ax bx c a

A quadratic function of the variable x is a function that can be written in the form

Example:

where a, b, and c are fixed real numbers. Also, the domain of f is all real numbers, namely, (, ).

2( ) 12 3 1f x x x

The graph of a quadratic function is a parabola.

a > 0 a < 0

Quadratic Function

2( ) 0f x ax bx c a

Vertex coordinates are:

x – intercepts are solutions of

y – intercept is:

symmetry

,2 2

b bx y f

a a

2 0ax bx c 2

bx

a

0x y c

Vertex, Intercepts, Symmetry

Vertex:

x – intercepts

y – intercept

21 ( 1) 9

2 2

bx y f

a

0 8x y

2( ) 2 8f x x x

2 2 8 0x x 4,2x

Graph of a Quadratic Function

Example 1: Sketch the graph of

Absolute Maximum of f is:

Range of f is:

( 1) 9 at 1 2

by f x

a

Does not exist

2( ) 2 8f x x x

[ 9, )


Example 1: Sketch the graph of

Absolute Minimum of f is:

Vertex:

x – intercepts

y – intercept

12 3 (3 / 2) 0

2 2 4 2

bx y f

a

0 9x y

24 12 9 0x x 3/ 2x


Example 2: Sketch the graph of 2( ) 4 12 9f x x x


Example 2: Sketch the graph of 2( ) 4 12 9f x x x


Range of f is:

(3 / 2) 0 at 3 / 2 2

by f x

a

Does not exist

[0, )


Vertex:

x – intercepts

y – intercept

4 (4) 42

bx y f

a

0 12x y

214 12 0

2x x


Example 3: Sketch the graph of 21( ) 4 12

2g x x x

no solutions


Example 3: Sketch the graph of 21( ) 4 12

2g x x x


Range of f is:

(4) 4 at 42

by f x

a

Does not exist

( , 4]


Example: For the demand equation below, express the total revenue R as a function of the price p per item and determine the price that maximizes total revenue.

( ) 3 600q p p

( ) 3 600R p pq p p 23 600p p

Maximum is at the vertex, p = $100

Applications

Example: As the operator of Workout Fever Health Club, you calculate your demand equation to be q 0.06p + 84 where q is the number of members in the club and p is the annual membership fee you charge.

1. Your annual operating costs are a fixed cost of $20,000 per year plus a variable cost of $20 per member. Find the annual revenue and profit as functions of the membership price p.

2. At what price should you set the membership fee to obtain the maximum revenue? What is the maximum possible revenue?

3. At what price should you set the membership fee to obtain the maximum profit? What is the maximum possible profit? What is the corresponding revenue?

Applications

The annual revenue is given by

Solution

( ) 0.06 84R p pq p p 20.06 84p p

The annual cost as function of q is given by

( ) 20000 20C q q The annual cost as function of p is given by

( ) 20000 20 20000 20 0.06 84

1.2 21680

C p q p

p

Thus the annual profit function is given by

Solution

2

2

( ) ( 0.06 84 ) 1.2 21680

0.06 85.2 21680

P p R C p p p

p p

The graph of the revenue function is

84Maximum is at the vertex $700

2 2( 0.06)

bp

a

20.06 84R p p

The graph of the revenue function is

Maximum revenue is (700) $29,400R

20.06 84R p p

The profit function is

85.2Maximum is at the vertex $710

2 2( 0.06)

bp

a

2( ) 0.06 85.2 21680P p p p

The profit function is

Maximum profit is (710) $8,566

Corresponding Revenue is (710) $29,394

P

R

2( ) 0.06 85.2 21680P p p p

Vertex Form of a Parabola


To get the vertex form of the parabola we complete the square in x as indicated in the next steps:


Vertex formStandard form


Example: Find the vertex form of 2( ) 2 8 5f x x x

2( ) 2 4 __ 5 2 __f x x x

2( ) 42 4 2 45f x x x

2( ) 2 2 3f x x


Use the vertex form of to graph the parabola

2( ) 2 8 5f x x x

2( ) 2 2 3f x x


2( ) 2 2 3f x x

Use the vertex form of to graph the parabola

2( ) 2 8 5f x x x

Find a Quadratic Function Given Its Vertex and One

Other Point

Determine the quadratic function whose vertex is (2, 3) and whose y-intercept is 1.

2 22 3f x a x h k a x

2

Using the fact that the y-intercept

is 1: 1 0 2 3a 1

Thus 1 4 3 and 2

a a

212 3

2f x x

x

y


More Examples

The marketing department at Widgets Inc. found that, when certain widgets are sold at a price of p dollars per unit, the number x of widgets sold is given by the demand equation

x = 1500 30p

1. Find a model that expresses the revenue R as a function of the price p.

2. What is the domain of R?3. What unit price should be used to maximize revenue?4. If this price is charged, what is the maximum revenue?

Maximizing Revenue


x = 1500 30p

1. Find a model that expresses the revenue R as a function of the price p.

2. What is the domain of R?

Maximizing Revenue

2 Revenue 1500 30 30 +1500. 1 R xp p p p p

0 so 1500 302. 0 x p 30 1500 p 50p

The domain of is 0 50 [0,50]R p p


x = 1500 30p

3. What unit price should be used to maximize revenue?4. If this price is charged, what is the maximum revenue?

Maximizing Revenue

153.

00 $25

2 2( 30)

bp

a

24. (25) 30 25 1500 25 $18,750R


x = 1500 30p

5. How many units are sold at this price?6. Graph R.7. What price should Widgets Inc. charge to collect at least $12,000

in revenue?

Maximizing Revenue

So the company should charge between $10 and $40 to earn at least $12,000 in revenue.


x = 1500 30p

Maximizing Revenue

1500 30 25 = 7505. x 2 12000 30 15007. p p

230 1500 12000 0p p

230 50 40 0p p

30( 10)( 40) 0p p 10 or 40p p

The farmer should make the rectangle 400 yards by 400 yards to enclose the most area.

A farmer has 1600 yards of fence to enclose a rectangular field. What are the dimensions of the rectangle that encloses the most area?

2 2 1600x w A xw

800w x 2(800 ) 800A x x x x

800400

2 2( 1)

bx

a

800 400 400w

Maximizing Area

1. Find the maximum height of the projectile.

2

1 1 50002500

32 12 22 2400 5000

bx

a

212500 2500 2500 500 1750 ft

5000h

Projectile Motion

2. How far from the base of the cliff will the projectile strike the water?

Projectile Motion

21500 0

5000h x x x

2 11 1 4 500

50001

25000

x

458 or 5458x

Solution cannot be negative so the projectile will hit the water about 5458 feet from the base of the cliff.

The Golden Gate Bridge



Documents

Sections 4.3 and 4.4 Quadratic Functions and Their Properties