Download Chapter 4 Polynomial and Rational Functions

Chapter 4 Linear and Quadratic Functions
4.1 Linear Functions and Their properties
Definition A linear function is a function of the form
f ( x )  mx  b
The graph of a linear function is a line with slope m and y-intercept b. Its domain is the set
of all real numbers.
Example 1 Graph the linear function f ( x)  3x  7
Average rate of change
Linear functions have a constant average rate of change. That is, the average rate of change
y
 m.
of a linear function f ( x )  mx  b is
x
Example 2 Find the average rate of change of f ( x )  3x  7 .
Increasing, decreasing, and constant linear functions
Let f ( x )  mx  b be a linear function.
It is increasing over its domain if its slope is positive, i.e. m>0.
It is decreasing over its domain if its slope is negative, i.e. m<0.
It is constant over its domain if its slope is zero, i.e. m=0.
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Example 3 Determine whether the following linear functions are increasing, decreasing, or
constant.
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(1) f ( x )  5 x  2
(2) g ( x )  3x  8 (3) s (t )  t  4 (4) h( z )  7
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Example 4 Suppose that the quantity supplied, S, and quantity demanded, D, of cell phones
each month are given by the following functions:
S ( p )  60 p  900
D( p)  15 p  2850
where p is the price (in dollars) of the phone.
(1) The equilibrium price of a product is defined as the price at which quantity
supplied equals quantity demanded. That is, the equilibrium price is the price at
which S ( p)  D( p) . Find the equilibrium price of cell phone. What is the
equilibrium quantity, the amount demanded (or supplied) at the equilibrium
price?
(2) Determine the prices for which quantity supplied is greater than quantity
demanded. That is, sole the inequality S ( p)  D( p) .
(3) Graph S  S ( p ) , D  D( p) and label the equilibrium price.
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4.2 Building Linear Functions from Data
Draw and interpret scatter diagrams
Example 1 The data listed below represent the apparent temperature versus the relative
humidity in a room whose actual temperature is 72°
Relative Humidity (%), x
0
10
20
30
40
50
60
70
80
90
100
Apparent Temperature °F, y
64
65
67
68
70
71
72
73
74
75
76
(x, y)
(0, 64)
(10, 65)
(20,67)
(30, 68)
(40, 70)
(50, 71)
(60, 72)
(70, 73)
(80, 74)
(90, 75)
(100, 76)
(1) Draw a scatter diagram by hand treating relative humidity as the independent
variable.
(2) Describe what happens to the apparent temperature as the relative humidity
increase.
Distinguish between linear and nonlinear relations
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Example 2 Determine whether the relation between the two variables below is linear or
nonlinear.
Finding an equation for linearly related date
Example 3 Using the table in Example 1,
(1) Select two points and find an equation of the line containing the points.
(2) Graph the line on the scatter diagram obtained in Example 1(1).
(3) Find the line of best fit using a graphing utility.
(4) Graph the line of best fit on the scatter diagram obtained in Example 1(1).
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4.3Quadratic Functions and Their Properties
Definition A quadratic function is a function of the form
f ( x)  ax 2  bx  c , a  0 .
Its domain is the set of all real numbers.
Properties of the Graph of a Quadratic Function f ( x)  ax 2  bx  c , a  0
 b
b
 b 
Vertex    , f     , Axis of symmetry: line x  
2a
 2a  2a  
Parabola opens up if a  0 ; the vertex is a minimum point.
Parabola opens down if a  0 ; the vertex is a maximum point.
Example 1 Without graphing, locate the vertex and axis of symmetry of the parabola
defined by f ( x)  3x 2  6 x  1 . Does it open up or down?
Example 2 Use the information from Example 1 and the locations of the intercepts to graph
f ( x)  3x 2  6 x  1 .
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Example 3 (1) Graph f ( x)  x 2  6 x  9 by determine whether the graph opens up or
down. Find its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.
(2) Determine the domain and the range of f.
(3) Determine where f is increasing and where it is decreasing.
Example 4 (1) Graph f ( x)  2 x 2  x  1 by determine whether the graph opens up or
down. Find its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.
(2) Determine the domain and the range of f.
(3) Determine where f is increasing and where it is decreasing.
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Find the maximum or minimum value of a quadratic function
For f ( x)  ax 2  bx  c , a  0 ,
 b 
 If a  0 , then f  
 is the maximum value of f .
 2a 
 b 
 If a  0 , then f  
 is the minimum value of f .
 2a 
Example 5 Determine whether the quadratic function f ( x)  x 2  4 x  7 has a maximum
or minimum value. Then find the maximum or minimum value.
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4.4. Quadratic Models; Building Quadratic Functions from Data
Example 1 The marketing department at Texas Instruments has found that, when certain
calculators are sold at a price of p dollars per unit, the number x of calculators sold is
given by the demand equation
x  21000  150 p
(a) Express the revenue R as a function of the price p.
(b) What unit price should be established in order to maximize revenue? ($70.00)
(c) If this price is charged, what is the maximum revenue? ($735,000)
(d) How many units are sold at this price? (10500)
(e) Graph R.
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Example 2 A farmer has 2000 yards of fence to enclose a rectangular field. What are the
dimensions of the rectangle that encloses the most area? (500 by 500)
Example 3 A projectile fired from a cliff 500 feet above the water at an inclination of 45° to
the horizontal, with a muzzle velocity of 400 feet per second. In physics, it is established
that the height h of the projectile above the water is given by
 32 x 2
h( x ) 
 x  500
(400) 2
where x is the horizontal distance of the projectile from the base of the cliff.
(1) Find the maximum height of the projectile. (1750ft)
(2) How far away from the base of the cliff will the projectile strike the water? (5458ft)
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4.5. Inequalities Involving Quadratic Functions
Solve inequalities involving a quadratic function
(1) To solve the inequality ax 2  bx  c  0 , a  0
we graph the function f ( x)  ax 2  bx  c and, from the graph, determine where it is
above x-axis, that is, where f ( x)  0 .
(2) To solve the inequality ax 2  bx  c  0 , a  0
we graph the function f ( x)  ax 2  bx  c and, from the graph, determine where it is
below x-axis, that is, where f ( x)  0 .
(3) If the inequality is not strict, we include the x-intercepts in the solution.
Example 1 Solve the inequality x 2  4 x  12 and graph the solution set.
Example 2 Solve the inequality 2 x 2  x  10 and graph the solution set.
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