Unformatted text preview: MAC 1105
Module 1
Introduction to Functions and
Graphs
Rev.S08 Learning Objectives
Upon completing this module, you should be able to:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. Recognize common sets of numbers.
Understand scientific notation and use it in applications.
Find the domain and range of a relation.
Graph a relation in the xyplane.
Understand function notation.
Define a function formally.
Identify the domain and range of a function.
Identify functions.
Identify and use constant and linear functions.
Interpret slope as a rate of change. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 2 Learning Objectives
11.
12. Write the pointslope and slopeintercept forms for a line.
Find the intercepts of a line. 13. Write equations for horizontal, vertical, parallel, and
perpendicular lines. 14. Write equations in standard form. 15.
16.
17.
18. Identify and use nonlinear functions.
Recognize linear and nonlinear data.
Use and interpret average rate of change.
Calculate the difference quotient. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 3 1 Introduction to
Functions and Graphs
There are four major topics in this module:
 Functions and Models  Graphs of Functions  Linear Functions  Equations of Lines Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 4 Let’s get started by recognizing some
common set of numbers. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 5 What is the difference between Natural
Numbers and Integers?
•Natural Numbers (or counting numbers)
are numbers in the set N = {1, 2, 3, ...}.
are
•Integers are numbers in the set
I = {… −3, − 2 , − 1, 0, 1, 2, 3, ...}.
3, 2, Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 6 2 What are Rational Numbers?
Rational Numbers are real numbers which can be expressed as the
ratio of two integers p/ q where q ≠ 0
ratio
Examples:
0.5 = ½ 3 = 3/1
3/1 − 5 = − 10/2 0.52 = 52/100 0 = 0/2 0.333… = 1/3 Note that:
• Every integer is a rational number.
E very
• Rational numbers can be expressed as decimals
R ational
which either terminate (end) or repeat a sequence
of digits. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 7 What are Irrational Numbers?
• Irrational Numbers are real numbers which are not rational
numbers.
• Irrational numbers Cannot be expressed as the ratio of two
Cannot
integers.
integers.
• Have a decimal representation which does not
does
terminate and does not repeat a sequence of digits.
terminate
does
Examples: Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 8 Classifying Real Numbers
Classify each number as one or more of the following:
Classify
natural number, integer, rational number, irrational
number. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 9 3 Let’s Look at Scientific Notation
A real number r is in scientific notation
real
scientific
when r is written as c x 10n, where
when
and n is an integer.
and
Examples:
The distance to the sun is 93,000,000 mi.
In scientific notation for this is 9.3 x 107 mi.
The size of a typical virus is .000005 cm.
In scientific notation for this is 5 x 10−6 cm.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 10 What is a Relation?
What are Domain and Range?
A relation is a set of ordered pairs.
relation is
If we denote the ordered pairs by (x , y)
The s et of all x − values is the DOMAIN.
The
The s et of all y − values is the RANGE.
The
Example
The relation {(1, 2), (− 2, 3), (− 4, − 4 ), (1, − 2), (− 3 ,0), (0, − 3)}
The r elation
4, 4),
3,0),
h as domain D = {− 4 , − 3 , − 2, 0, 1}
has d omain
4, 3,
a nd range R = {− 4 , − 3, − 2, 0, 2, 3}
and range
4, 3, Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 11 How to Represent
a Relation in a Graph?
The relation {(1, 2), (− 2, 3), (− 4, − 4 ), (1, − 2), (− 3 , 0), (0, − 3)}
The r elation
4, 4),
3,
has the following graph: Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 12 4 Is Function a Relation?
Recall that a relation is a set of ordered pairs (x , y) .
Recall
set
If we think of values of x as being inputs and values of y
If
and
as being outputs, a function is a relation such that
as
outputs function
for each input there is exactly one output.
for
input
exactly
o utput.
This is symbolized by output = f (input) or
o utput
y = f (x ) Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 13 Function Notation
y = f( x)
–
Is pronounced “ y is a function of x.”
Is
–
Means that given a value of x (input), there is exactly one
Means
( input),
e xactly
corresponding v alue of y (output).
corresponding
(output).
– – Rev.S08 x is called the independent variable as it represents
is
i ndependent
iinputs, a nd y is called the dependent variable as it
nputs,
dependent
represents outputs.
represents outputs
Note that: f(x ) iis NOT f multiplied by x . f is NOT a
s NOT
Note
is
variable, but the name of a function (the name of a
variable,
t he
relationship between variables). http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 14 What are Domain and Range?
The set of all meaningful inputs is called the DOMAIN
The
inputs
DOMAIN
of the function.
The set of corresponding outputs is called the RANGE
The
outputs
RANGE
of the function. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 15 5 What is a Function?
A function is a relation in which each element of the
function
domain corresponds to exactly one element in the
range.
The function may be defined by a set of ordered pairs,
a table, a graph, or an equation. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 16 Here is an Example
• Suppose a car travels at 70 miles per hour. Let y be the
Suppose
distance the car travels in x hours. Then y = 70 x.
distance • Since for each value of x (that is the time in hours the car
Since
travels) there is just one corresponding value of y (that is the
travels)
distance traveled), y is a function of x and we write
distance
y = f ( x ) = 70x • Evaluate f(3) and interpret.
–
f(3) = 70(3) = 210. This means that the car travels 210
miles in 3 hours. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 17 Here is Another Example
• Given the following data, is y a function of x ?
Given
– Input
x
3
4
8
Input
– Output y
6
6
−5
Output • Note: The data in the table can be written as the set of
ordered pairs {(3, 6), (4 ,6), (8, − 5)}. • Yes, y is a function of x, because for each value of x , there is
because
jjust one corresponding value of y . Using function notation we
ust
write f(3) = 6 ; f( 4) = 6; f ( 8) = − 5.
write Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 18 6 One More Example
• • Undergraduate Classification at StudyHard University (SHU)
Undergraduate
at
iis a function of Hours Earned. We can write this in function
s
H ours
notation as C = f ( H).
notation
Why is C a function of H?
Why
For each value of H there is exactly one corresponding
For
exactly
value of C.
–
In other words, for each input there is exactly one
In
e ach
exactly
corresponding output.
corresponding o utput
– Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 19 One More Example (Cont.) • Here is the classification of students at SHU (from catalogue): • No student may be classified as a sophomore until after earning at
least 30 semester hours. • No student may be classified as a junior until after earning at least
60 hours. • No student may be classified as a senior until after earning at least
90 hours. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 20 One More Example (Cont.)
• Remember C = f (H)
Remember • Evaluate f ( 20), f ( 30), f (0 ),
Evaluate
),
),
),
–
–
–
– •
• f ( 20) and f(61):
and f( 20) = Freshman
Freshman
f( 30) = Sophomore
Sophomore
f( 0 ) = Freshman
F reshman
f( 61) = Junior
Junior What is the domain of f?
What
domain
What is the range of f?
What
range Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 21 7 One More Example (Cont.)
Domain of f is the set of nonnegative integers
n onnegative
Alternatively, some individuals say the domain is the set of
positive rational numbers, since technically one could
earn a fractional number of hours if they transferred in
some quarter hours. For example, 4 quarter hours = 2
2/3 semester hours.
Some might say the domain is the set of nonnegative real
Some
n onnegative r eal
numbers
, but this set includes irrational numbers.
It is impossible to earn an irrational number of credit
hours. For example, one could not earn
hours.
Range of f is {Fr, Soph, Jr, Sr}
R ange
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 22 Identifying Functions
Referring to the previous example concerning SHU, is
hours earned a function of classification? That is,
is H = f (C )? Explain why or why not.
Is classification a function of years spent at SHU? Why
or why not?
•
•
• Given x = y 2, iis y a function of x? Why or why not?
s
Given
Given x = y 2, iis x a function of y? Why or why not?
s
Given
Given y = x 2 +7, iis y a function of x? Why, why not?
s
Given http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 23 Identifying Functions (Cont.)
• Is hours earned a function of classification? That is, is H =
Is
f( C )? • That is, for each value of C is there just one corresponding
That
is
value of H?
–
No. One example is
• Rev.S08 if C = Freshman, then H could be 3 or 10 (or lots of
other values for that matter) http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 24 8 Identifying Functions (Cont.)
•
• Is classification a function of years spent at SHU? That is, is
C = f(Y )?
That is, for each value of Y is there just one corresponding
That
is
value of C?
–
No. One example is
• Rev.S08 iif Y = 4, then C could be Sr. or Jr. It could be Jr if a
f
student was a part time student and full loads were not
taken. http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 25 Identifying Functions (Cont.)
•
• Given x = y 2 , iis y a function of x ?
s
Given
That is, g iven a value of x , is there just one corresponding
That
value of y ?
value
–
No, if x = 4, then y = 2 or y = − 2.
No, Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 26 Identifying Functions (Cont.)
• Given x = y 2 , iis x a function of y ?
s
Given • That is, given a value of y, is there just one corresponding
value of x ?
value
– Rev.S08 Yes, given a value of y, there is just one corresponding
value of x , namely y 2.
value
namely http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 27 9 Identifying Functions (Cont.)
• Given y = x 2 + 7, iis y a function of x ?
s
Given • That is, given a value of x, is there just one corresponding
value of y ?
value
– Yes, given a value of x , there is just one corresponding
Yes,
value of y , namely x 2 + 7 .
value
namely http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 28 Five Ways to Represent
a Function
•
• Verbally
Numerically •
•
• Diagrammaticly
Symbolically
Graphically http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 29 Verbal Representation
• Referring to the previous example:
If you have less than 30 hours, you are a freshman. –
–
–
– Rev.S08 If you have 30 or more hours, but less than 60 hours,
you are a sophomore.
If you have 60 or more hours, but less than 90 hours,
you are a junior.
If you have 90 or more hours, you are a senior. http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 30 10 Numeric
Representation H
0
1
?
?
?
?
29
30
31
?
?
?
59
60
61
?
?
?
89
90
91
?
?
? C
Freshman
Freshman Freshman
Sophomore
Sophomore Sophomore
Junior
Junior Junior
Senior
Senior http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 31 Symbolic Representation http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 Di
R e ag ra
p re m m
sen at
ta t ic
io n H Rev.S08 0
1
2
•
•
•
29
30
31
•
•
•
59
60
61
•
•
•
89
90
91
•
•
• 32 C
Freshman Sophomore Junior Senior http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 33 11 Graphical Representation
• In this graph the domain is considered to be
In
domain • instead of {0,1,2,3… }, and note that inputs are typically
inputs
graphed on the horizontal axis and outputs are typically
graphed
horizontal
o utputs
graphed on the vertical axis.
graphed
vertical http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 34 Vertical Line Test
• Another way to determine if a graph represents a function,
simply visualize vertical lines in the xy plane. If each vertical
simply
xyplane. If
line intersects a graph at no more than one point , then it is
the graph of a function. http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 35 What is a Constant Function?
A function f represented by f ( x ) = b,
function
w here b is a constant (fixed number), is a
where
constant function.
f(x ) = 2 Examples: Note: Graph of a constant function is a horizontal line .
Note:
constant
horizontal
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 36 12 What is a Linear Function?
A function f represented by f ( x ) = ax + b,
function
ax
w here a and b are constants, iis a linear function.
where
constants s linear
(It will be an identity function, if constant a = 1 and constant b = 0.)
Examples:
f(x ) = 2x + 3 Note that a f(x ) = 2 iis both a linear function and a constant function.
s
Note
linear
constant
A constant function is a special case of a linear function .
constant
special
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 37 Rate of Change
of a Linear Function
x y −2 −1
−1 1 0
1
2
3 3
5
7
9 Rev.S08 Table of values for f ( x ) = 2x + 3.
Table
• Note throughout the table, as x
increases by 1 unit, y increases by 2
units. In other words, the RATE OF
units.
RATE
CHANGE of y with respect to x is
constantly 2 throughout the table.
Since the rate of change of y with
respect to x is constant, the function is
LINEAR. Another name for rate of
LINEAR. A nother
change of a linear function is SLOPE. http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 38 The Slope of a Line
• The s lope m of the line passing through the points (x 1 , y 1) and
The
of
(x2, y2) is Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 39 13 Example of Calculation of Slope
• Find the slope of the line passing through the
Find
slope
points (− 2, − 1) and (3, 9).
2, (3, 9) (2, 1) • The slope being 2 means that for each unit x increases, the
The s lope
corresponding increase in y is 2. The rate of change of y with
corresponding
rate
respect to x is 2/1 or 2.
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 40 How to Write the PointSlope Form of the
Equation of a Line?
The line with slope m passing through the point (x 1, y 1 ) has equation
The
p oint
y = m ( x − x1) + y1
or
or
y − y 1 = m ( x − x1) Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 41 How to Write the Equation of the Line Passing
Through the Points (−4, 2) and (3, − 5)?
Through
4,
To write the equation of the line using pointslope form
To
pointslope
y = m ( x − x 1) + y 1
t he slope m and a point ( x 1, y 1 ) are needed.
the slope
point
Let (x 1, y 1) = (3, − 5).
Calculate m using the two given points.
Calculate Equation is
Equation
This simplifies to
This Rev.S08 y = − 1 (x − 3 ) + (− 5)
y = −x + 3 + (− 5)
y = −x − 2
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 42 14 SlopeIntercept Form
The line with s lope m and yintercept b is given by
The
intercept
– y=mx+b Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 43 How to Write the Equation of a line passing
through the point (0,2) with slope ½? Since the point (0, − 2) has an x coordinate of 0, the point is a y Since
2)
coordinate
iintercept. Thus b = − 2
ntercept.
Using slopeintercept form
Using slopeintercept
y=mx+b
the equation is
y = (½ ) x − 2 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 44 How to Write an Equation of a Linear
Function in SlopeIntercept Form?
•
– • What is the slope?
As x increases by 4 units,
y decreases by 3 units so
the slope is − 3/4 What is the yintercept?
–
The graph crosses the
y− axis at (0,3) so the
y− iintercept is 3.
ntercept •
–
– What is the equation?
Equation is
f ( x ) = (− ¾ ) x + 3 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 45 15 What is the Standard Form for the
Equation of a Line?
ax + by = c
iis standard form (or general form) for the equation of
s standard
general
a line. http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 46 How to Find xIntercept and
yintercept?
• • To find the xintercept, let
intercept, l et
y = 0 and solve for x.
– 2x – 3(0) = 6
– 2x = 6
– x=3
To find the yintercept, let
intercept, l et
x = 0 and solve for y.
– 2(0) – 3y = 6
2(0)
– –3 y = 6
– y = –2 (0, 2)
(3, 0) http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 47 What are the Characteristics of
Horizontal Lines?
Slope is 0, since Δy = 0 and m = Δy / Δx
since
Equation is: y = mx + b
Equation
mx y = (0)x + b
y = b w here b is the yintercept
Example: y = 3 (or 0x + y = 3)
(or
(3, 3) Rev.S08 (3, 3) Note that regardless of
the value of x, the value
the
of y is always 3.
of http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 48 16 What are the Characteristics of
Vertical Lines?
•
• Slope is undefined, since Δ x = 0 and m = Δy /Δ x
since
Example:
• Note that regardless of the value
of y, the value of x is always 3.
of the
• Equation is x = 3 (or x + 0 y = 3)
Equation
• Equation of a vertical line is x = k
Equation
where k is the x intercept.
where Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 49 What Are the Differences Between
Parallel and Perpendicular Lines?
• Parallel lines have the
same slant, thus they have
the same slopes.
the s ame Rev.S08 Perpendicular lines have slopes
Perpendicular
slopes
which are negative reciprocals
(unless one line is vertical!) http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 50 How to Find the Equation of the Line
Perpendicular to y = 4x  2
Through the Point (3,1)?
The
The slope of any line perpendicular to y = −4x – 2 is ¼
(−4 a nd ¼ are negative reciprocals)
and are
Since
Since we know the slope of the line and a point on the line we can use
slope
point
pointslope form of the equation of a line:
y = m (x − x 1 ) + y1
y = ( 1/4) ( x − 3) + (− 1)
y = − 4x – 2
In s lopeintercept form:
slopeintercept
y = ( 1/4)x − (3/4) + (− 1)
y = ( 1/4)x − 7/4
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. y = (1/4)x − 7/4 51 17 Example of a Linear Function
The table and corresponding graph show the price y of x
The
tons of landscape rock.
X (tons) y (price in dollars)
25
5
75
4 100 y is a linear function of x and the slope is
The rate of change of price y with respect to tonage x is 25 to 1.
The r ate
This means that for an increase of 1 ton of rock the price
increases by $25.
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 52 Example of a Nonlinear Function
x y 0 0 1
2 1
4 Table of values for f ( x ) = x2
Table
Note that as x increases from 0 to 1, y increases by 1 unit; while as x
Note
iincreases from 1 to 2, y increases by 3 units. 1 does not equal 3.
ncreases
This function does NOT have a CONSTANT RATE OF CHANGE of
This
does
C ONSTANT
y with respect to x , so the function is NOT LINEAR.
so
N OT
Note that the graph is not a line.
Note
n ot
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 53 Average Rate of Change
Let (x1, y1 ) and (x2, y 2 ) be distinct points on the
g raph
graph of a function f. The average rate of
The average
c hange
change of f from x 1 to x2 is Note
Note that the average rate of change of f from x1 to x 2
is
is the s lope of the line passing through
slope
(x1, y 1 ) and (x2 , y 2 ) Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 54 18 What is the Difference Quotient?
The difference quotient of a function f is an
The d ifference
expression of the form
w here h is not 0.
where Note that a difference quotient is actually
Note
difference
an average rate of change.
an a verage http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 55 What have we learned?
We have learned to:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. Recognize common sets of numbers.
Understand scientific notation and use it in applications.
Find the domain and range of a relation.
Graph a relation in the xyplane.
Understand function notation.
Define a function formally.
Identify the domain and range of a function.
Identify functions.
Identify and use constant and linear functions.
Interpret slope as a rate of change. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 56 What have we learned? (Cont.)
11.
12. Write the pointslope and slopeintercept forms for a line.
Find the intercepts of a line. 13. Write equations for horizontal, vertical, parallel, and
perpendicular lines. 14. Write equations in standard form. 15.
16.
17.
18. Identify and use nonlinear functions.
Recognize linear and nonlinear data.
Use and interpret average rate of change.
Calculate the difference quotient. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 57 19 Credit
Some of these slides have been adapted/modified in part/whole from the slides of
the following textbook:
• Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 58 20 ...
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