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.
Rev.S08 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.
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 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 What are Rational Numbers?
Rational Numbers are real numbers which can be expressed as the
Rational
ratio of two integers p/q where q ≠ 0
Examples:
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.
• Rational numbers can be expressed as decimals
which either terminate (end) or repeat a sequence
of digits.
of 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
Irrational
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.
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
natural
number.
number. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 9 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
and n is an integer.
and
Examples:
Examples:
The distance to the sun is 93,000,000 mi.
The
In scientific notation for this is 9.3 x 107 mi.
In
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
If we denote the ordered pairs by (x, y)
If
The set of all x − values is the DOMAIN.
The
The set of all y − values is the RANGE.
The
Example
The relation {(1, 2), (− 2, 3), (− 4, − 4), (1, − 2), (− 3,0), (0, − 3)}
The relation
4, 4),
3,0),
has domain D = {− 4, − 3, − 2, 0, 1}
has domain
4, 3,
and 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 relation
4, 4),
3,
3)}
has the following graph:
has Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 12 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
outputs function
for each input there is exactly one output.
input
exactly
output.
This is symbolized by output = f(input) or
This
output
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.”
.”
–
Means that given a value of x (input), there is exactly one
Means
(input),
exactly
corresponding value of y (output).
(output).
–
x is called the independent variable as it represents
is
independent
inputs, and y is called the dependent variable as it
dependent
represents outputs.
outputs
–
Note that: f(x) iis NOT f multiplied by x. f is NOT a
s NOT
Note
variable, but the name of a function (the name of a
the
relationship between variables).
relationship Rev.S08 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.
of
The set of corresponding outputs is called the RANGE
The
outputs
RANGE
of the function.
of Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 15 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.
range.
The function may be defined by a set of ordered pairs,
The
a table, a graph, or an equation.
table, 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. • 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
distance traveled), y is a function of x and we write
y = f(x) = 70x • Evaluate f(3) and interpret.
–
f(3) = 70(3) = 210. This means that the car travels 210
f(3)
miles in 3 hours.
miles 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
Note:
ordered pairs {(3,6), (4,6), (8, − 5)}.
ordered • Yes, y is a function of x, because for each value of x, there is
because
there
just one corresponding value of y. Using function notation we
Using
write f(3) = 6; f(4) = 6; f(8) = − 5. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 18 One More Example
• • Undergraduate Classification at StudyHard University (SHU)
Undergraduate
is a function of Hours Earned. We can write this in function
Hours
We
notation as C = f(H).
).
Why is C a function of H?
Why
–
For each value of H there is exactly one corresponding
For
exactly
value of C.
value
–
In other words, for each input there is exactly one
In
each
exactly
corresponding output.
output 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
No
least 30 semester hours.
least • No student may be classified as a junior until after earning at least
No
60 hours.
60 • No student may be classified as a senior until after earning at least
No
90 hours.
90 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), f(20) and f(61):
Evaluate
),
), ),
and
):
–
–
–
– •
• f(20) = Freshman
Freshman
f(30) = Sophomore
Sophomore
f(0) = Freshman
Freshman
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 One More Example (Cont.)
Domain of f is the set of nonnegative integers
Alternatively, some individuals say the domain is the set of
positive rational numbers, since technically one could
positive
since
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
nonnegative
numbers
, but this set includes irrational numbers.
numbers
It is impossible to earn an irrational number of credit
hours. For example, one could not earn
hours.
hours.
Range of f is {Fr, Soph, Jr, Sr}
Range Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 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 = y2, iis y a function of x? Why or why not?
s
Given
•
Given x = y2, iis x a function of y? Why or why not?
s
Given
•
Given y = x2 +7, iis y a function of x? Why, why not?
s
Given Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 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
value of H?
value
–
No. One example is
•
if C = Freshman, then H could be 3 or 10 (or lots of
other values for that matter)
other Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 24 Identifying Functions (Cont.)
• Is classification a function of years spent at SHU? That is, is
Is
C = f(Y)? • That is, for each value of Y is there just one corresponding
That
value of C?
value
–
No. One example is
• 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.
taken. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 25 Identifying Functions (Cont.)
•
• Given x = y2, iis y a function of x?
s
Given
That is, given a value of x, iis there just one corresponding
s
That
value of y?
–
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 = y2, iis x a function of y?
s
Given
That is, given a value of y, is there just one corresponding
That
value of x?
–
Yes, given a value of y, there is just one corresponding
Yes,
value of x, namely y2.
namely Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 27 Identifying Functions (Cont.)
•
• Given y = x2 +7, iis y a function of x?
s
Given
That is, given a value of x, is there just one corresponding
That
value of y?
–
Yes, given a value of x, there is just one corresponding
Yes,
there
value of y, namely x2 +7.
namely Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 28 Five Ways to Represent
a Function
•
•
•
•
• Rev.S08 Verbally
Numerically
Diagrammaticly
Symbolically
Graphically http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 29 Verbal Representation
• Rev.S08 Referring to the previous example:
–
If you have less than 30 hours, you are a freshman.
If
–
If you have 30 or more hours, but less than 60 hours,
If
you are a sophomore.
–
If you have 60 or more hours, but less than 90 hours,
If
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 Numeric
Representation Rev.S08 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. 31 Symbolic Representation Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 32 iag
Re r a
p re mm
s en a t
ta t ic
io
n H Rev.S08 C 0
1
2
•
•
•
29
30
31
•
•
•
59
60
61
•
•
•
89
90
91
•
•
• Freshman Sophomore Junior Senior http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 33 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
horizontal
outputs
graphed on the vertical axis.
vertical Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 34 Vertical Line Test
• Another way to determine if a graph represents a function,
Another
simply visualize vertical lines in the xyplane. If each vertical
xyplane.
line intersects a graph at no more than one point, then it is
line
then
the graph of a function. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 35 What is a Constant Function?
A function f represented by f(x) = b,
function
where b is a constant (fixed number), is a
constant function.
constant
f(x) = 2 Examples:
Examples: Note: Graph of a constant function is a horizontal line.
constant
horizontal
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 36 What is a Linear Function?
A function f represented by f(x) = ax + b,
function
ax
where a and b are constants, iis a linear function.
constants s linear
(It will be an identity function, if constant a = 1 and constant b = 0.)
(It
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
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 37 Rate of Change
of a Linear Function xy
−2 −1
−1 1
03
15
27
39
Rev.S08 Table of values for f(x) = 2x + 3.
• Note throughout the table, as x
Note
increases by 1 unit, y increases by 2
units. In other words, the RATE OF
CHANGE of y with respect to x is
CHANGE
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
change of a linear function is SLOPE.
change http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 38 The Slope of a Line
• The slope m of the line passing through the points (x1, y1) and
The
of
and
(x2, y2) is Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 39 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)
(3, (2, 1) • The slope being 2 means that for each unit x increases, the
The slope
corresponding increase in y is 2. The rate of change of y with
rate
respect to x is 2/1 or 2.
respect Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 40 How to Write the PointSlope Form of the
Equation of a Line?
The line with slope m passing through the point (x1, y1) has equation
The
point
y = m(x − x 1) + y 1
or
or
y − y 1 = m(x − x 1) Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 41 How to Write the Equation of the Line Passing
How
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
the slope m and a point (x1, y1) are needed.
the slope
point
Let (x1, y1) = (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 SlopeIntercept Form
The line with slope 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?
½? Since the point (0, − 2) has an xcoordinate of 0, the point is a ySince
2)
coordinate
iintercept. Thus b = − 2
ntercept.
Using slopeintercept form
Using slopeintercept
y=mx+b
the equation is
the
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
The
y− axis at (0,3) so the
axis
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 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
for
a line.
line. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 46 How to Find xIntercept and
yintercept?
• • To find the xintercept, let
intercept,
y = 0 and solve for x.
– 2x – 3(0) = 6
– 2x = 6
– x=3
To find the yintercept, let
intercept,
x = 0 and solve for y.
– 2(0) – 3y = 6
– –3y = 6
– y = –2 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. (0, 2)
(3, 0) 47 What are the Characteristics of
Horizontal Lines?
Slope is 0, since Δy = 0 and m = Δy / Δx
since
Equation is: y = mx + b
mx y = (0)x + b
y = b where 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. http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 48 What are the Characteristics of
Vertical Lines?
•
• Slope is undefined, since Δx = 0 and m = Δy /Δx
since
Example:
•
•
• Rev.S08 Note that regardless of the value
Note
of y, the value of x is always 3.
the
Equation is x = 3 (or x + 0y = 3)
Equation
Equation of a vertical line is x = k
Equation
where k is the xintercept. 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
Parallel
same slant, thus they have
same
thus
the same slopes.
same Rev.S08 Perpendicular lines have slopes
which are negative reciprocals
which
(unless one line is vertical!)
(unless 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 slope of any line perpendicular to y = − 4x – 2 is ¼
The
(− 4 and ¼ are negative reciprocals)
and
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:
pointslope
y = m(x − x1) + y1
y = (1/4)(x − 3) + (− 1) y = − 4x – 2 In slopeintercept 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 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
25
5
75
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.
rate
This means that for an increase of 1 ton of rock the price
increases by $25.
increases
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 52 Example of a Nonlinear Function x
0
1
2 y
0
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
increases from 1 to 2, y increases by 3 units. 1 does not equal 3.
This function does NOT have a CONSTANT RATE OF CHANGE of
does
CONSTANT
y with respect ...
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