AFM – CHAPTER 4

Functions

5 Function Families

• What you need to know:• Name• Equation• Domain• Range

Linear

• Name – Constant

• Equation –

• Domain – (-,)

• Range – [b]

y b

Linear

• Name – Oblique Linear

• Equation –

• Domain – (-,)

• Range – (-,)

y m x b

Power Functions

• Name – Quadratic

• Equation –

• Domain – (-,)

• Range – [0,)

y x 2

Power Functions

• Name – Cubic

• Equation –

• Domain – (-,)

• Range – (-,)

y x 3

Root Functions

• Name – Square root

• Equation –

• Domain – [0,)

• Range – [0,)

y x

Root Functions

• Name – Cube root

• Equation –

• Domain – (-,)

• Range – (-,)

y x 3

Reciprocal Functions

• Name – Rational

• Equation –

• Domain –(-,0) (0,)

• Range – (-,0) (0,)

yx

1

Reciprocal Functions

• Name – Rational Squared

• Equation –

• Domain – (-,0) (0,)

• Range – (0,)

yx

1

2

Absolute Value Function

• Name – Absolute value

• Equation –

• Domain – (-,)

• Range – [0,)

y x

Greatest Integer Function

• Name – Greatest Integer

• Equation –

• Domain – (-,)

• Range – (integers)

y x

Trig Functions

• Name – Sine

• Equation –

• Domain – (-,)

• Range – [-1,1]

y x sin ( )

Trig Functions

• Name – Cosine

• Equation –

• Domain – (-,)

• Range – [-1,1]

y x co s( )

Vertical Line Test

• A curve in the coordinate plane is the graph of a function iff no vertical line intersects the curve more than once.

Piecewise Functions

• Sketch the graph of:

f x x

x

2

2 1x

x

1

1

if

if

Equations That Define Functions

• Does the equation define y as a function of x?

1 4

2 2 6

3

4 1

2 2

2

2

2

.

.

.

.

x y

x y

x y

x y y

Homework 4.2

• Page 228 – 230; 2 – 50 even 60 –70 even

• Know Family of Functions graphs• Name, equation, domain, range

4.4 – Average Rate of Change

• Average rate of change of a function f(x) between x = a and x = b is:

• Slope of secant line drawn between x=a & x=b or line through the points (a,f(a)),(b,f(b)).

f b f a

b a

y

x

( ) ( )

Finding the Average Rate of Change

• Find the average rate of change if

f x x

a x x

b x x

c x a x a b

( )

) ;

) ;

) ;

2 4

2 6

5 1 0

Example 2

• An object is dropped from a height of 3000 feet, its distance above the ground h, after t seconds is given by:

• Find the average speed• Between 1 & 2 seconds• Between 4 & 5 seconds

h t t( ) 3 0 0 0 1 6 2

Increasing and Decreasing Functions

• A function f is increasing if:

• A function f is decreasing if:

f x f x w hen

x x

( ) ( )1 2

1 2

f x f x w hen

x x

( ) ( )1 2

1 2

State the intervals on which the function whose graph is shown is increasing or decreasing.

• Sketch the graph of the function:

• Find the domain and range of the function

• Find the intervals on which f increases and decreases

f x x( ) 2 3

• Determine the average rate of change of the function between the given values of the variable.

• Use a graphing calculator to draw the graph of f. State approximately the intervals on which f is increasing and on which f is decreasing.

f x x x( ) 3 24

f x x x x( ) 3 22 2

• The table gives the population in a small coastal community for the period 1990 – 1999. Figures shown are for January 1 in each year.

• What was the average rate of change of population between 1991 and 1994?

• What was the average rate of change of population between 1995 and 1997?

• For what period of time was the population increasing?

• For what period of time was the population decreasing?

Transformations

• Vertical Shift

• Horizontal Shift

• Reflecting

• Stretching/Shrinking

Exploring transformations• Graph

• Graph

y x 2

y x

y x

y x

y x

y x

y x

2

2

2

2

2

2

3

2

4

3

2

1

2

( )

( )

More transformations

• Graph:

• Graph:

y x

y x

y x

General Rules for Transformations• Vertical shift:

• y=f(x) + c c units up• y=f(x) – c c units down

• Horizontal shift:• y=f(x+c) c units left• y=f(x-c) c units right

• Reflection:• y= – f(x) reflect over x-axis• y= f(-x) reflect over y-axis

• Stretch/Shrink:• y=af(x) (a > 1) Stretch vertically• y=af(x) (0 < a < 1) Shrink vertically

Even & Odd Functions

• Algebraically:• Even – f is even if f(-x) = f(x) • Odd – f is odd if f(-x) = - f(x)

• Graphically:• Even – f is even if its graph is

symmetric to the y-axis• Odd – f is odd if its graph is

symmetric to the origin

Determine Algebraically if the function is even, odd or neither

y x x

y x x

y x x

y x x x

2

6 2

3

3 2

4

3 5

2 4 3 1

Use the rules of transformations to graph the following:

y x

y x

y x

y x

yx

2 3 2

1

24 3

2 6

1 3

1

25

2

3

( )

4.7 Combining Functions

• Combining – Addition, Subtraction, Multiplication, or Division

• Composition of functions – Putting two functions together using the rules of one of the functions

Combining Functions• Addition/Subtraction – f(x) and g(x)

• (f±g)(x) = f(x) ± g(x) Add/Subtract, then combine like terms

• Domain: D:f(x) D:g(x)

• Multiplication – f(x) and g(x)• (fg)(x) = f(x)·g(x) Multiply, then combine like terms• Domain: D:f(x) D:g(x)

• Division – f(x) and g(x)• Divide, then simplify

• Domain: D:f(x) D:g(x), where g(x) 0 f

gx

f x

g x

( )

( )

Examplesf x x g x x

f g f g fgf

g

( ) ; ( )

( ), ( ), ( ),

3

Let

Find

Domain f(x) Domain g(x)

f g f g fgf

g

( ); ( ); ;4 4 4 4Find

Composition of Functions

•

f g x f g x

g f x g f x

ExamplesIf : andf x x( ) 2 g x x( ) 3

f g x

g f x

f f x

g g x

f g

g f

5

7

Find:

Composition of 3 Functions

Find: f g h x If:

f xx

xg x x h x x

131 0; ;

Page 276 - # 23,25,27

Variation

• Direct Variation

• Indirect Variation

• Joint Variation

Direct Variation

• y varies directly as x• y is directly proportional to x• y is proportional to x• Formula (Equation to use)

• y = kx kx

y ;k is constant of proportionality

• During a thunderstorm, the distance between you and the storm varies directly as the time interval between the lightening and thunder.

• Suppose thunder from a storm 5400 ft away takes 5 seconds reach you.

• Determine the constant of proportionality and write the variation equation for the model.

• Sketch the graph. What does the k represent?• If the time interval between the lightening and

thunder is 8 sec. How far away is the storm?

Inverse Variation

• y varies inversely as x• y is inversely proportional to x• Formula (Equation to use)

x

ky kyx

• Boyle’s Law – When a sample of gas is compressed at a konstant temperature the pressure of the gas is inversely proportional to the volume of the gas.

• P – pressure• v – volume• k – constant of proportionality• Suppose the pressure of a sample of air that occupies

0.106 m³ @ 25ºC is 50 kPa. Find the constant of proportionality and write the equation that expresses the inverse proportionality

• If the sample expands to a volume of .3m³, find the new pressure

Joint Variation

• Used when a quantity depends on more than one other quantity. It depends on them jointly.

• z varies jointly as x and y• z is jointly proportional to x and y

• z is proportional to x and inversely proportional to y

kxyz

y

kxz

• Newton’s Law of Gravitation – Two objects with masses m1 and m2 attract each other with a force F that is jointly proportional to their masses and inversely proportional to the square of the distance r between the objects. Express Newton’s Law of Gravitation as an equation.

221

r

mmGF

Examples of Variation

• Write an equation that expresses the statement:

• R varies directly as t.• v is inversely proportional to z.• y is proportional to s and inversely proportional

to t.• R is proportional to j and inversely proportional

to the squares of s and t.

Express the statement as a formula. Use the given information to find the constant of proportionality

• y is directly proportional to x. If x = 4, then y = 72.

• M varies directly as x and inversely as y. If x = 2 and y = 6, then M = 5.

• s is inversely proportional to the square root of t. If s = 100, then t = 25.

• Hooke’s Law states that the force F needed to keep a spring stretched x units beyond its natural length is directly proportional to x. Here the constant of proportionality is called the spring constant.

• Write Hooke’s Law as an equation.• If a spring has a natural length of 10 cm and a force of

40 N is required to maintain the spring stretched to a length of 15 cm, find the spring constant.

• What force is needed to keep the spring stretched to a length of 14 cm?

• The resistance R of wire varies directly as its length L and inversely as the square of its diameter d.

• Write an equation that expresses this joint variation.• Find the constant of proportionality if a wire 1.2 m long and

0.005 m in diameter has a resistance of 140 ohms.• Find the resistance of a wire made of the same material that

is 3 m long and has a diameter of 0.008 m.

Modeling Quadratic & Cubic Functions

• Define the variable

• Find the equation (model)

• Answer the question(s) asked

• A breakfast cereal company manufactures boxes to package their product. The prototype box has the following shape: Its width is three times its depth and its height is five times its depth. Find a function that models the volume of the box in terms of its depth.

• A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at $14, average attendance at recent games has been 9500. A market survey indicates that for each dollar the ticket price is lowered, the average attendance increases by 1000. What price maximizes revenue from ticket sales, and what is the maximum revenue?

• A manufacturer makes a metal can that holds 1 L(liter) of oil. What radius minimizes the amount of metal in the can?

• A gardener has 140 feet of fencing for her rectangular vegetable garden. Find the dimensions of the biggest area she can fence.