Properties of Functions (2.3)

To find the average rate of change of a function between any two points we calculate the slope of the line containing the two points (secant line).

The Average Rate of Change of a Function is the slope of the secant line containing two points on the graph.

Local (Relative) Maximum and Local (Relative) Minimum

•A function f has a local maximumat c if there is an open interval I containing c so that, for all x = c in I, f(x) < f(c). We call f(c) a local maximum of f.

A function f has a local minimumat c if the

•re is an open interval I containing c so that, for all x = c in I, f(x) > f(c). We call f(c) a local minimum of f.

Use a graphing calculator to graph the function.

Determine where f has a local maximum and/or local minimum.

Determine where f is increasing/decreasing. Write answers rounded to the nearest hundredth.

Even and Odd Functions

A function f is even if for every number x in its domain the number -x is also in its domain and f(-x) = f(x). The graph of every even function is symmetric with respect to the y-axis.

A function f is odd if for every number x in its domain the number -x is also in its domain and

f(-x) = - f(x). The graph of every odd function is symmetric with respect to the origin.

Identify the following functions as even, odd, or neither.

1. f(x) = 2x 3 + 3x 22.f(x) = 4x 3 + 3x

3.f(x) = 3x 2 64.

Complete the graph.

1. A partial graph of f(x) is shown .

Complete the graph if f(-x) = -f(x).

Is f(x) even, odd, or neither?

6.A partial graph of g(x) is shown.

Complete the graph if g(-x) = g(x).

Is g(x) even, odd, or neither?

Library of Functions (2.4)

Directions: Complete each table, plot each point in the table, and graph the function. State the domain and range of the functions.

1.Absolute Value Function:

x / y
-6
-2
0
1
3
4
5

x / y
-1.5
-1
-.6
-.2
0
.5
.8
1
2

3. Square Root Function:

x / y
0
.09
.36
.81
1
9

4.Rational Function:

x / y
-4
-2
-1.5
-1
-.5
0
.2
.7
2
3

5. Cubing Function:

x / y
-2
-1
-.8
-.5
-.2
0
.3
.6
2

6.Identity Function:

x / y
-3
-2
-1
0
2
4

7.Greatest Integer Function:

8.Piecewise Defined Function:

When functions are defined by more than one equation, they are called piece-wise defined functions.

Let

a.f(-1) =b.f(0) =

c.f(2) =d.f(3) =

Transformation of Functions (2.5)

Part 1: Vertical Shifts

Use a graphing utility to display the graphs of the transformations of the function.

a)

b)

c)

Use the results to describe the effect c has on the graph of f;describe the transformation resulting from function .

Part 2: Horizontal Shifts

Use a graphing utility to display the graphs of the transformations of the function.

d)

e)

f)

Use the results to describe the effect c has on the graph of f;describe the transformation resulting from function .

Part 3: Reflections

Use a graphing utility to display the graphs of the transformations of the function.

g)

h)

i)

Use the results to describe the effects on the graph of f;describe the transformation resulting from and .

Part 4: Vertical Stretches and Compressions

Use a graphing utility to display the graphs of the transformations of the function.

j)

k)

l)

Use the results to describe the effect k has on the graph of f;describe the transformation resulting from the function .

Part 5: Horizontal Stretches and Compressions

Use a graphing utility to display the graphs of the transformations of the function.

m)

n)

o)

Use the results to describe the effect k has on the graph of f;describe the transformation resulting from.

1. Consider the function , and graph each of the following. Describe the transformation.

a.-f(x)b.-2f(x)c.2f(x) - 3

1. Suppose f(x) = x 2. Name a sequence of transformations, in a correct order, that will change f(x) into:

a. y = 2x 2 3b.y = (x  4) 2 2

c. y = -(x + 3) 2 + 1d.y = 0.5(x  2) 2 3

Combination and Composition of Functions (2.6)

1. Suppose and . Find the following and determine the domain of each.

a.b.

c.d.

2.Suppose and . Find .

3.Suppose and . Find the domain of

## Mathematical Models: Constructing Functions (2.7)

1.A rectangular pen that borders a river needs 250 m of fencing to enclose the area.

a. Draw a diagram of the situation and label with appropriate information

b. Write an algebraic expression for the perimeter P.

c. Express the area A as a function of x and determine the domain of the function.

d. Use a graphing utility to graph the area function. Approximate the length and width of the rectangle of maximum area.

2.An open box with locking tabs is made from a 12-inch-square piece of material. This is done by cutting equal squares from each corner and turning up the sides.

a) Construct a paper model.

b) Draw a diagram and label with the appropriate information

c) Write a function that expresses the volume of the box V in terms of the height.

(Volume = length  width  height).

d) Determine the algebraic domain of the function V.

e)Determine the practical/implied domain of the function V.

f)Estimate the height that maximizes volume.

3.A media company is going to install cable from a house to their connection box B. The house is located at one end of a driveway 7 miles back from a road (see diagram). The other end of the driveway and the nearest connection box are on the same road, 25 miles apart. The cost of installing the cable is \$656 per mile off the road and \$375 per mile along the road.

Let x be the distance from where the driveway meets the road to where the cable comes to the road. Develop a function C(x) that expresses the total installation cost as a function of x.

Now use your calculator to graph C. Use the graph to determine the value of x that will produce the minimum cost. Round to the nearest thousandth of a mile.

(Tip: use a window [0, 20, 1] x [ 12,000 , 20,000 , 1000 ].)

State the minimum cost for that installation, rounded to the nearest cent.

1. Let x be the amount (in hundreds of dollars) a company spends on advertising, and let P be the profit, where P = 230 + 20x  0.5x 2. What expenditure for advertising results in the maximum profit?

5.Let be a point on the graph of .

a)Express the distance d from P to the point (0, -1) as a function of x.

b)What is d if ? ?

c)Use a graphing utility to graph .

d)For what values of x is d smallest?

Properties of Functions (2.3)