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# Math 142 - Chapter 2 Lecture Notes

The purpose of this handout is to facilitate your
note-taking during lecture. This handout is not a comprehensive summary of the
chapter; I will be giving additional notes during lecture which you will also be
responsible

to know for exams and quizzes. Some of these examples come from examples and
exercises in Calculus for Business, Economics, Life Sciences, and Social
Sciences, 11th ed. by Barnett, Ziegler, and Byleen.

## Section 2.1 - Functions

**Function -**

**Domain** - The set of _______ of a function; i.e.,
the set of values for which it makes sense to plug into the function.

**Range** - The set of _______ of a function; i.e., the
set of values obtained by substituting all input values into the

function.

**NOTE:** When graphing a function, x represents
____________________________and y represents ____________________________

**Functions Specified by Equations**

The correspondence between domain and range elements is often specified by an
equation in two variables. For example,

consider y = 3x+7. In equations like these, we refer to x as
____________________________ the (since values

can be ____________________________ from the domain), and we refer to y as
the____________________________

____________________________(since the value of y____________________________

If in an equation in two variables, we get exactly one
output (value for the dependent variable) for each input (value for

the independent variable), then the equation specifies a function.

**Example 1 - Determining Algebraically Whether an
Equation Specifies a Function**

Determine which of the following equations specify functions with independent
variable x. (Assume x represents a real

number in each equation.)

**Identifying Functions - Vertical Line Test**

Suppose that a graph has inputs located on the x-axis and outputs located on the
y-axis. If at any input you can draw a

vertical line that crosses the graph in two or more
places,____________________________

**Example 2 - Identifying Functions**

Which of the following graphs are functions?

**NOTE:** If a function is specified by an equation and
the domain is not indicated, then we will assume that the domain

is the set of all real numbers that, when substituted for the independent
variable, produce real output values for the

dependent variable.

**Example 3 - Warm-up: Finding Domains**

Find the domain of each of the following functions, assuming that x is the
independent variable.

**Finding Domains**

The domain of a function is all real numbers unless

• the function involves a ____________________________. In this case, exclude____________________________

• the function involves an ____________________________. In this case, exclude____________________________

• the function involves a ____________________________. In this case, exclude____________________________

• the function represents a ____________________________. In this case, the domain should be restricted to____________________________

**Example 4
**Find the domain of each of the following functions, assuming that x is the
independent variable.

NOTE: The vertical line test implies that equations of the form y=mx+b, where m≠0, specify functions; they are called ____________________________. Similarly, equations of the form y=b specify what are called ____________________________

**Function Notation**

It is often useful to assign names to functions using letters. For example, if f
denotes the function specified by the

equation y = 3x^3 −9, then we can write
__________________________. We use the symbol
__________________________in place of y to designate the output value in
the range that is obtained when the input value x is substituted into the
function.

**Example 5**

Let f(x) = x^2 −2x+1 and Find each of the following.

**Difference Quotient**

The expression
,
where x and x+h are in the domain of f and h ≠ 0, is
referred to as the difference

quotient and is studied extensively in calculus.

**Example 6
**Find the difference quotient for f (x) = x^2−4x+1 by finding each of the
following.

**Applications: Break-Even and Profit-Loss Analysis**

**• Cost function
• Price-Demand Function
• Revenue Function
• Profit Function**

• A company will have a loss if __________, will break even if __________, and will have a profit if__________

**Example 7**

Acme Computer Company has a variable production cost per computer of $180, and
has fixed costs that amount to

$45,000. Acme’s sales research department has established the price-demand
function for its computers to be p =

−0.23x+574 dollars per computer. Using this information, find each of the
following.

(a) Cost Function

(b) Revenue Function

(c) Profit Function

(d) Graph these three functions on the same coordinate axes. What do you notice?

## Section 2.2 - Elementary Functions and Piecewise Functions

**Elementary Functions**

**Graph Transformations of y = f (x)**

1. Vertical Translation

• If y = f (x)+k, shift the graph of y = f (x) _____________________ if k > 0
and _____________________ if

k < 0.

2. Horizontal Translation

• If y = f (x+h), shift the graph of y = f (x) _____________________ if h > 0
and _____________________ if

h < 0.

3. Reflection

• If y = −f (x), reflect the graph of y = f (x) _____________________.

4. Vertical Stretch and Shrink

• If y = A f (x) where A > 1, stretch the graph of y = f (x) vertically by
multiplying each y coordinate value by

A.

• If y = A f (x) where 0 < A < 1, shrink the graph of y = f (x) vertically by
multiplying each y coordinate value

by A.

**Example 1**

Sketch the graph of g(x) = − | x+3 | +4.

**Example 2 **Suppose that we are given the graph of a function f . How
must we transform the graph of f to obtain the

graph of g if g(x) = 6−4 f (x−3)?

**Definition:** A function whose definition involves different rules for
different parts of its domain is called a_____________________

Recall: The absolute value function:

**Example 6**

Consider the function

(a) Find f (−2), f (−1), and f (10).

(b) Sketch the graph of f .

**Example 7**

AB&C’s monthly cell phone service charges are as follows: All customers pay a
flat rate of $35 per month, and then

additional charges are added based on the number of minutes of air time used.
The first 100 minutes are free, but for the

next 100 minutes, a fee of $0.05 per minute is added. Any minutes used beyond
this are charged at a rate of $0.15 per

minute. Let x represent the number of minutes of air time used by a customer in
one month.

(a) Write a function f in terms of x that gives a customer’s monthly bill.

(b) Find and interpret f (75), f (150), and f (400).

(c) Sketch the graph of f .

Read Example 6 and Matched Problem 6 on pages 70 and 71.

Suggested Homework: 1-17 odd, 23, 25, 29-41 odd, 43-48 (all), 65-68 (all)

## Section 2.3 - Quadratic Functions

**Quadratic Function**

If a, b, and c are real numbers with a ≠ 0, then the
function _____________________ is a quadratic function.

**Finding Intercepts of a Quadratic Function**

• To find the y-intercept of a quadratic function f , calculate_____________________

• To find the x-intercept(s) (if any exist) of a quadratic function f , you have three options:

1.

2.

3.

**Example 1**

Find the x and y-intercepts of the following quadratic functions.

**Examples: Graphs of Quadratic Functions**

**Properties of the Graph of a Quadratic Function y = ax^2 +bx+c**

• The graph of a quadratic function is a _____________________.

• The lowest or highest point on the _____________________, whichever exists, is
called the _____________________.

• If a>0, then the parabola opens _____________________, and the
_____________________ value of the quadratic function

occurs at the vertex.

• If a<0, then the parabola opens _____________________, and
the_____________________ value of the quadratic function

occurs at the vertex.

• The vertical line passing through the vertex of a parabola is called the
_____________________ of

the parabola, or simply the _____________________ .

**Example 2**

Use graph transformations to graph f (x) = −(x−2)^2+3

**Example 3**

Find the vertex, axis of symmetry, maximum or minimum value (whichever exists),
x and y intercepts, domain, and range

of the quadratic function given in Example 1.

**NOTE: **Carefully examine the vertex you found in Example 3 for the
function f (x) = −(x−2)^2+3. Notice anything?

**The Vertex Form of a Quadratic Function**

Using a process called _____________________, a quadratic function f
(x)=ax^2+bx+c can be rewritten

into the form

which is called the vertex form of a quadratic function. When a quadratic
function is written in the vertex form, we have

the following general properties:

1. The vertex of the parabola is _______________ . (Note k = ________ )

2. The axis of symmetry is the vertical line _______________ .

3. f (h) = k is the _______________ if a > 0 and the _______________
if a < 0.

4. The domain of a quadratic function is _______________ .

5. The range of a quadratic function is _______________ if a > 0 or
_______________ if a < 0.

**Another Method for Finding the Vertex of a Parabola**

It can be shown using the method of completing the square that, given a
quadratic function f (x) = ax^2 +bx+c,

• the x coordinate of the vertex (h,k) is given by _______________ .

• the y coordinate of the vertex (h,k) is given by _______________ .

• If a > 0, then the _______________ value of f occurs at
and
has value

• If a < 0, then the _______________ value of f occurs atand
has value

**Example 4**

Find the minumum value of g(x) = 5x^2 −7x+2.

**Example 5 **(adaptation of #58, pg. 92 of Barnett, Ziegler, and Byleen)

The market research department for a company that manufactures widgets
established the price-demand function to be

p(x) = 2000−60x, where p(x) is the wholesale price per widget in dollars at
which x widgets can be sold. This company

has fixed costs that amount to $4,000, and the variable production cost per
widget is $500.

(a) Find this company’s break-even point(s).

(b) Find the output that will produce the maximum revenue, and find the maximum revenue.

(c) Find the maximum profit.

(d) What price should the company charge per widget to achieve maximum profit?

**Polynomial Functions**

A polynomial function is a function that can be written in the form

where n is a nonnegative integer called the
_______________ of the polynomial. The
coefficients a_{0},a_{1}, . . . ,a_{n} are real

numbers with a_{n} ≠ 0 (a_{n} is called the
_______________). The domain of a polynomial is _______________

**Examples of Polynomials, Degree 1 through Degree 6, a _{n}
> 0**

**Properties of Polynomials**

• For even-degree polynomials, both “ends” of the graph
will go to if the leading coefficient is
positive, and both

“ends” of the graph will go to if the
leading coefficient is negative.

• For odd-degree polynomials, the graph starts negative and ends positive if the
leading coefficient is positive, or the

graph starts positive and ends negative if the leading coefficient is negative.

• An odd-degree polynomial will cross the x-axis at least once.

• An even-degree polynomial may never cross the x-axis.

• In general, a polynomial of degree n can have at most n linear factors.
Therefore, the graph of a polynomial of

positive degree n may intersect the x axis at most n times.

• The graph of a polynomial is continuous. That is, the graph can be drawn
without removing a pen from the paper.

• The graph of a polynomial is smooth. That is, the graph has no sharp corners.

**Rational Functions**

A **rational function** is any function that can be written in the form

where n(x) and d(x) are polynomials. The domain is the set of all real numbers such that _______________

**Examples of Rational Functions**

**Example 6**

Find the domain of each of the following and state whether or not each is a
rational function.