Sections 4.3 and 4.4 Quadratic Functions and Their Properties
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Transcript of 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, )
Graph of a Quadratic Function
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
Graph of a Quadratic Function
Example 2: Sketch the graph of 2( ) 4 12 9f x x x
Graph of a Quadratic Function
Example 2: Sketch the graph of 2( ) 4 12 9f x x x
Absolute Maximum of f is:
Range of f is:
(3 / 2) 0 at 3 / 2 2
by f x
a
Does not exist
[0, )
Absolute Minimum of f is:
Vertex:
x – intercepts
y – intercept
4 (4) 42
bx y f
a
0 12x y
214 12 0
2x x
Graph of a Quadratic Function
Example 3: Sketch the graph of 21( ) 4 12
2g x x x
no solutions
Graph of a Quadratic Function
Example 3: Sketch the graph of 21( ) 4 12
2g x x x
Absolute Maximum of f is:
Range of f is:
(4) 4 at 42
by f x
a
Does not exist
( , 4]
Absolute Minimum of f is:
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
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 Form of a Parabola
Vertex formStandard form
Vertex Form of a Parabola
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
Vertex Form of a Parabola
Use the vertex form of to graph the parabola
2( ) 2 8 5f x x x
2( ) 2 2 3f x x
Vertex Form of a Parabola
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
Vertex Form of a Parabola
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
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?
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
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
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
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
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.
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
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
The Golden Gate Bridge
The Golden Gate Bridge