Transcript PreCalulus

PreCalculus
Section 1-3
Graphs of Functions
Objectives
• Find the domain and range of functions
from a graph.
• Use the vertical line test for functions.
• Determine intervals on which functions are
increasing, decreasing, or constant.
• Determine relative maximum and relative
minimum values of functions.
• Identify and graph step functions and other
piecewise defined functions.
• Identify even and odd functions.
The Graph of a Function
f(x)
• The graph of a function f is the collection
of ordered pairs (x, f(x)) such that x is in
the domain of f.
• The geometric interpretations of x and f(x)
are;
– x: the directed distance from the y-axis
– f(x): the directed distance from the x-axis
x
The domain is {x| -2≤x<5}
The range is {y| -4≤y≤4}.
Domain & Range
The domain of the function y = f (x) is the set of
values of x for which a corresponding value of y exists.
The range of the function y = f (x) is the set of values of y
which correspond to the values of x in the domain.
y
Example: Find the domain and
range of f (x) = x2 – 2x – 2 by
investigating its graph.
Range
x
The domain is the real numbers.
The range is { y: y ≥ -3}
or [-3, + ∞).
4
-4
Domain = all real numbers
Domain & Range
Example:
Find the domain and range.
y
4
3
2
1
x
4 3 2 1
1
2
3
4
2
3
4
Domain: {-3, -2, -1, 0, 1, 2}
Range: {-2, 0, 1, 2, 4}
Examples: Domain and Range from a Graph
Calculate the Max. & Min.
∴ R = -6≤y≤6 or [-6, 6] Calculate the Min.
∴ R = y≥-3 or [-3, ∞)
Range
Domain
Calculate the zeros
∴ D = -3≤x≤3 or [-3, 3]
Range
Domain
The parabola continues to ∞ and
covers the entire x-axis ∴ D = ℝ or (-∞,∞)
y =|x|
Your Turn
Range
Domain
Domain: ℝ or (-∞, +∞)
Range: y≥0 or [0, +∞)
f(x) = (x +
2)2
Your Turn
-4
Range
Domain: ℝ or (-∞, +∞)
Range: y≥-4 or [-4, +∞)
Domain
f ( x) 
x  3  4 , x≠ -3
Your Turn
Domain: x>-3 or (-3, +∞)
Range
Range: y>-4 or (-4, +∞)
Domain
f(x) = -(x +
2)3
-1
Your Turn
Range
Domain: ℝ or (-∞, +∞)
Range: ℝ or (-∞, +∞)
Domain
f(x) =
2x+2
-3
Your Turn
Range
Domain: ℝ or (-∞, +∞)
Range: y>-3 or (-3, +∞)
Domain
Your Turn
D = -5≤x≤9 or [-5, 9]
R = -5≤y≤5 or [-5, 5]
Determine the Domain and
Range for Each Function From
Their Graph
D = -4≤x<2 or [-4, 2)
R = {-2, -1, 0}
Relation
A relation is a correspondence that associates
values of x with values of y.
The graph of a relation is the set of ordered pairs
(x, y) for which the relation holds.
Example: The following equations define relations:
y = x2
x2 + y2 = 4
y2 = x
y
y
y
(4, 2)
(0, 2)
x
(4, -2)
x
x
(0, -2)
Functions - Vertical Line Test
• To determine whether or not a relation is in fact a
function, we can draw a vertical line through the graph of
the relation.
• If the vertical line intersects the graph more than once,
then that means the graph of the relation is not a
function.
• If the vertical line intersects the graph once then the
graph shows that the relation is a function.
Examples: Vertical Line Test
Vertical Line Test
A set of points in the xy-plane is the graph of a function
if and only if every vertical line intersects the graph in at
most one point.
y
y
x
Function
One point of
intersection
y
x
Not a Function
Two points of
intersection
x
Not a Function
Two points of
intersection
Your Turn
Vertical Line Test: Apply the vertical line test to
determine which of the relations are functions.
y
y
x
The graph does not pass
the vertical line test.
It is not a function.
x
The graph passes the
vertical line test.
It is a function.
Evaluating a Function Graphically
Example: Function g’s graph is shown.
Use the graph of g to
evaluate g(–1).
Evaluating a Function Graphically
Solution
(a) To evaluate g(-1) on the graph.
(b) Find x = –1 on the x-axis. Move upward
to the graph of g.
(c) Move across to the yaxis. Read the y-value:
g(–1) = 3.
Your Turn
y
Find f(2)






x















f(2) = 3










Your Turn
y
Find f(3)






x



















f(3) = -3







Your Turn
y

Find f(7), f(3), and f(1)





x















f(7) = 2




f(3) = 0
f(1) = DNE





Increasing and Decreasing Functions
1.
Increasing function
–
–
–
2.
Decreasing function
–
–
–
•
The range values increase from
left to right
The graph rises from left to right
Positive slope
The range values decrease from
left to right
The graph falls from left to right
Negative slope
To decide whether a function is increasing, decreasing, or
constant on an interval, ask yourself “What does the graph do
as x goes from left to right?”
Graphs of Increasing and Decreasing
Functions
If could walk from left to right along the graph of an
increasing function, it would be uphill.
For a decreasing function,
we would walk downhill.
Definition:
Increasing, Decreasing, and Constant Functions
Suppose that a function f is defined over an interval I.
a. f increases on I if, whenever x1  x 2 , f ( x1 )  f ( x 2 )
b. f decreases on I if, whenever x1  x 2 , f ( x1 )  f ( x 2 )
c. f is constant on I if, for every x and x , f ( x )  f ( x )
1
2
1
2
Figure 7, pg. 2-4
Increasing and Decreasing Functions
• Increasing functions- rise from left to right
• Decreasing functions- falls from left to right
• Constant functions- flat (horizontal line)
**A function that is increasing, decreasing, or constant, is
described within its domain (in terms of x-values).**
Ask yourself: What does y do as x goes from left to right?
Intervals of Increase or Decrease
We need to identify where the function is
increasing or decreasing
y
-3
Increasing:
x<1 or (-, 1)
1
5
x
Decreasing:
x>1 or (1, +)
Increasing, Decreasing, and Endpoints
The concepts of increasing and decreasing apply only to
intervals of the real number line and NOT to individual
points.
Decreasing: x<0 or (–∞, 0)
Increasing: x>0 or (0, ∞)
Do NOT say that the
function f both increases
and decreases at the point
(0, 0). The point (0, 0) is
the ‘turning point’.
Example
Find the Intervals
on the Domain in
which the Function
is Increasing,
Decreasing, and/or
Constant.
Increasing: 3<x<5
Decreasing: x<-1 and x>5
Constant: -1<x<3
Your Turn
• Determine the intervals over which the function is
increasing, decreasing, or constant.
Solution: Ask “What is happening to the y-values as x
is getting larger?” decreasing : x  1 or (  ,1)
increasing : 1  x  3 or (1, 3)
constant : x>3 or (3,  )
Relative Minimum and Maximum Values
of a Function
A function f has a relative (local) maximum at x = c
if there exists an open interval (r, s) containing c
such that f ( x )  f ( c ) for all x between r and s.
A function f has a relative (local) minimum at x = c
if there exists an open interval (r, s) containing c
such that f ( x )  f ( c ) for all x between r and s.
Relative
Maximums
Relative
Minimums
Relative Maximum Values of a Function
Assuming the graph is continuous (no break) at the point
where the function changes from increasing to
decreasing, that point is called a relative maximum point.
Relative Minimum Values of a Function
In the same manner, a relative minimum point occurs
when the graph changes from decreasing to
increasing.
Relative Minimum and Maximum Values
of a Function
**Maximum and Minimum values are based
on y-values (or the output f(x)) of the
function**
• Relative Maximum (Peak)
– Highest value in some open interval
• Relative Minimum (Valley)
– Lowest value in some open interval
Example:
d. Relative Minimum
Indicate and label each maximum
or minimum point on the function
graph, from left to right.
e. Relative Maximum
b. Relative Maximum
s. Relative Minimum
Your Turn: Indicate and label each
maximum or minimum
point on the function
graph, from left to right.
b, Local
c, & d
Minimums
Minimums
e. Maximum
r. Minimum
s. Maximum
Your Turn:
Find each maximum or minimum value
and label as a maximum or minimum
from left to right.
f(6)=3
Relative
Maximum
f(4)=4
Relative
Maximum
f(2)=1
Relative
Minimum
f(5)=2
Relative
Minimum
Use Graphing Calculator to Find
Relative Minimum and Maximum
Values of a Function
• Example:
– f(x) = -4x2 – 7x + 3
– Answer: Max. (-0.875, 6.063)
• Example:
– f(x) = -x3 + x
– Answer: Min. (-0.58, -0.38), Max. (0.58, 0.38)
• Your Turn:
– F(x) = x3 – 2x2
– Answer: Max. (0, 0), Min. (1.3, -1.2)
Graphing Step Functions and
Piecewise – Defined Functions
• A step function is a specific kind of piecewise function in
which the different parts of the equation look like steps
when graphed.
Greatest Integer Function – Step Function
• The greatest integer function, denoted by ⟦x⟧ and
defined as the largest integer less than or equal to x, has
an infinite number of breaks or steps – one at each
integer value in its domain.
• Also called the rounding-down or the floor function.
• Basic characteristics;
–
–
–
–
–
Domain: (-∞, +∞)
Range: {y: y = n, n є Z}
x-intercepts: in the interval [0, 1)
y-intercept: (0, 0)
Is a piecewise function.
– Constant between each pair of consecutive integers.
– Jumps vertically one unit at each integer value.
GREATEST INTEGER FUNCTION
When greatest integer acts on a number, the value that
represents the result is the greatest integer that is less
than or equal to the given number. There are several
descriptors in that expression. First of all you are looking
only for an integer. Secondly, that integer must be less
than or equal to the given number and finally, of all of the
integers that satisfy the first two criteria, you want the
greatest one. The brackets which indicate that this
operation is to be performed is as shown: ‘[ ]’.
Example: [1.97] = 1
Example: [-1.97] = -2
There are many integers less than 1.97; {1,
0, -1, -2, -3, -4, …} Of all of them, ‘1’ is the
greatest.
There are many integers less than -1.97;
{-2, -3, -4, -5, -6, …} Of all of them, ‘-2’ is
the greatest.
It may be helpful to visualize this function a little more clearly by
2
using a number line.  14
-6.31
-7
-6
5
-5
-4
-3
6.31
3
-2
Example: [6.31] = 6
-1
0
1
2
3
4
5
6
7
8
Example: [-6.31] = -7
When you use this function, the answer is the integer on the immediate
left on the number line. There is one exception. When the function
acts on a number that is itself an integer. The answer is itself.
Example: [5] = 5
Example:
Example: [-5] = -5
 
2
 3   0.6  0
 
Example:
  14 
 5    2.8  3


If there is an operation inside the greatest integer brackets, it must
be performed before applying the function.
Example: [5.5–3.6] = [1.9] = 1
Example: [3.6–5.5] = [-1.9] = -2
Example: [5.5+3.6] = [9.1] = 9
Example: [5.53.6] = [19.8] = 19
Calculating the Greatest Integer Function
⟦x⟧ is defined as the largest integer less than or
equal to x.
Example: ⟦2.5⟧ = 2
• Your Turn
1)
2)
3)
4)
5)
6)
⟦33.4⟧
⟦29/10⟧
⟦-5.2⟧
⟦2π⟧
⟦-√7⟧
⟦16⟧
• Answers
1)
2)
3)
4)
5)
6)
33
2
-6
6
-3
16
Graph of the Greatest Integer
Function
The greatest integer function can be used to construct a Cartesian
graph. The simplest of which is demonstrated below.
f(x) = [x]
To see what the graph looks like, it is necessary to determine some
ordered pairs which can be determined with a table of values.
x
0
1
2
3
-1
-2
f(x) = [x]
f(0) = [0] = 0
f(1) = [1] = 1
f(2) = [2] = 2
f(3) = [3] = 3
f(-1) = [-1] = -1
f(-2) = [-2] = -2
If we only choose integer
values for x then we will not
really see the function
manifest itself. To do this
we need to choose noninteger values.
x
f(x) = [x]
0 f(0) = [0] = 0
0.5 f(0.5) = [0.5] = 0
0.7 f(0.7) = [0.7] = 0
0.8 f(0.8) = [0.8] = 0
0.9 f(0.9) = [0.9] = 0
1
f(1) = [1] = 1
1.5 f(1.5) = [1.5] = 1
1.6 f(1.6) = [1.6] = 1
1.7 f(1.7) = [1.7] = 1
1.8 f(1.8) = [1.8] = 1
1.9 f(1.9) = [1.9] = 1
2
f(2) = [2] = 2
-0.5 f(-0.5) =[-0.5]=-1
-0.9 f(-0.9) =[-0.9]=-1
-1
f(-1) = [-1] = -1
When all these
points are strung
together the graph
looks something like
this – a series of
steps.
For this reason it is
sometimes called the
‘STEP FUNCTION’.
Notice that the left
of each step begins
with a closed
(inclusive) point but
the right of each step
ends with an open
(excluding point)
We can’t really
state the last (most
right) x-value on
each step because
there is always
another to the right
of the last one you
may name. So
instead we describe
the first x-value
that is NOT on a
given step.
Example: (1,0)
Rather than place a
long series of points
on the graph, a line
segment can be
drawn for each step
as shown to the
right.
The graphs shown
thus far have been
magnified to make a
point. However,
these graphs are
usually shown at a
normal scale as you
can see on the next
slide.
f(x) = [x]
This is a rather tedious way to
construct a graph and for this reason
there is a more efficient way to
construct it. Basically the greatest
integer function can be presented with
4 parameters, as shown below.
f(x) = a[bx - h] + k
By observing the impact of these
parameters, we can use them to
predict the shape of the graph.
f(x) = [x]
In these 3
examples,
parameter ‘a’ is
changed. As a
increases, the
distance between
the steps
increases.
a=1
a=3
f(x) = 3[x]
f(x) = 2[x]
a=2
f(x) = -[x]
a = -1
f(x) = -2[x]
a = -2
When ‘a’ is negative, notice that the slope of the steps is changed.
Downstairs instead of upstairs. But as ‘a’ changes from –1 to –2,
the distance between steps increases. The further that ‘a’ is
from 0, the greater the separation between steps. This can be
described with a formula.
Vertical distance between Steps = |a|
f(x) = [x]
b=1
f(x) = [2x]
b=2
1

f (x )   x 
2 
b
1
2
As ‘b’ is increased
from 1 to 2, each step
gets shorter. Then
as it is decreased to
0.5, the steps get
longer.
1
Length of Step 
b
f(x) = [-x]
b = -1
When ‘b’ is negative, notice that the
slope of the steps and the orientation of
each step changes. It now is open on the
left but closed on the right – opposite to
the way it is when ‘b’ is positive.
b > 0
b < 0
f(x) = -[-x]
a = -1
Notice that when both ‘a’ and ‘b’ are
negative the slope of the steps
becomes positive again. Both
parameters affect the slope.
Slope through closed
points of each step
= ab
If ab > 0, steps are increasing.
If ab < 0, steps are decreasing.
b = -1
Parameters h and k do not have an impact in defining the shape of the
graph. These parameters simply translate the graph. This translation
can be taken into account by determining a starting point. The
previous four formulae can be used to construct the greatest integer
function provided that there is an ordered pair to start with – the
starting point. For this we can use the y-intercept, f(0).
Example: g ( x )  2  1 x  3   1 a  2 ; b  1 ; h   3 ; k   1
2

2. Orientation of each step:
1. Starting point: (0,5)
b > 0
1

g( 0 )  2  ( 0 )  3   1
2

4. Vertical distance between Steps = |a|
 2 3   1  2 ( 3 )  1  5
3 . Length

of Step 
1
1
2
2
2
=|2| = 2
1
b
5. Slope through closed
=
points of each step
ab
1
 (2 )   1
2

1
1
or
1
1
Example:
1

g( x )  2  x  3   1
2

1. Starting point: (0,5)
1

g( 0 )  2  ( 0 )  3   1
2

 2 3   1  2 ( 3 )  1  5
2. Orientation of each step:
b > 0
3 . Length

of Step 
1
1
1
b
2
2
After placing the first step, the
closed point on the next step must be
vertically aligned with the open point
on the previous one, keeping in mind
the slope of the steps (up or down).
4. Vertical distance between Steps = |a|
=|2| = 2
5. Slope through closed
=
points of each step
ab
1
 (2 )   1
2

1
1
or
1
1
Your Turn: f(x) = 3[-x – 3] + 5
a = 3; b = -1; h = 3; k = 5
1. Starting point: (0,-4)
f(0) = 3[-(0)-3]+5
=3(-3)+5 = -4
2. Orientation of each step:
b < 0
3 . Length

of Step 
1
1
1
b
1
4. Vertical distance
between Steps =
|a|
=|3| = 3
5. Slope through closed
points of each step =
ab
 ( 3 )   1   3

 3
1
or
 3
1
Five Steps to Construct Greatest Integer Graph
f(x) = a[bx - h] + k
1. Starting point: f(0)
2. Orientation of each step:
b > 0
b < 0
3 . Length
of Step 
1
b
4. Vertical distance
between Steps =
|a|
5. Slope through closed
points of each step
= ab
Greatest Integer Function as a
Piecewise Function


f (x)  


1, if 0  x  1
2, if 1  x  2
3, if 2  x  3
4, if 3  x  4


f ( x)  


1, if 0  x  1
2, if 1  x  2
3, if 2  x  3
4, if 3  x  4
Graph of
Greatest Integer
Function as a
Piecewise
Function
Example: Application of a Step Function
Downtown Parking charges a $5 base fee for parking through 1 hour, and
$1 for each additional hour or fraction thereof. The maximum fee for 24
hours is $15. Sketch a graph of the function that describes this pricing
scheme.
Solution
Sample of ordered pairs (hours, price): (.25,5), (.75,5), (1,5), (1.5,6),
(1.75,6).
During the 1st hour: price = $5
During the 2nd hour: price = $6
During the 3rd hour: price = $7

During the 11th hour: price = $15
It remains at $15 for the rest of
the 24-hour period.
Plot the graph on the interval (0,24].
The TI-89 command for the Greatest Integer function is
floor (x).
Y=
flo o r  x 
CATALOG
Graphing
the
Greatest
Integer
Function
with a
Graphing
Calculator
F
floor(
Graph a Piecewise Function
by Hand
1) Write all of the equations of the piecewise
functions in y = mx + b form
2) Look at your first equation and its domain. Graph
your first equation over that domain.
3) Put a closed circle on the beginning or ending
points if that point is included in the domain
(greater than or equal to, less than or equal to). If
the point is not included in the domain, use an
open circle (less than or greater than).
4) Repeat for the other equations of the function.
Example: Graph the piecewise
function f(x).
 x  , if x  1
f ( x)  
  x  3, if x  1
1
2
3
2
•For all x’s < 1, use the top graph (to the left
of 1)
•For all x’s ≥ 1, use the bottom graph (to the
right of 1)
 12 x  32 , if x  1
f ( x)  
  x  3, if x  1
x=1 is the breaking
point of the graph.
To the left is the top
equation.
To the right is the
bottom equation.
Solution
Your Turn: Graph the piecewise
function f(x).
2
2
x  , if x  2
3
3
f ( x)  
  x  1, if x  2

Example: Graphing a Piecewise-Defined
Function with a Graphing Calculator
• Graph each equation and its domain separately.
– Example:
 x  2, x  1
f (x)  
2
x

1
, x 1



– Input: y1=x+2|x<1
y2=(x-1)2|x≥1
• Your Turn:
1.
 | x  3 |, x   3

f ( x )   4,  3  x  2
 5  x, x  2

2.
3,
x  1


2
f ( x )    x  1   2,  1  x  1

x  4, x  1

Function Symmetry with Respect to the y-Axis
Even Function
If we were to “fold” the graph
of f(x) = x2 along the y-axis,
the two halves would coincide
exactly. We refer to this
property as symmetry.
Symmetry with Respect to the y-Axis – Even Function
If a function f is defined so that
f ( x)  f ( x)
for all x in its domain, then the graph of f is symmetric with respect to
the y-axis.
2
For example, if f ( x )  x , then
f (  4 )  f ( 4 )  16
For any real number x , f (  x )  (  x )
f ( 3)  f (3)  9.
2
 x
2
 f ( x ).
Even Functions have y-axis Symmetry
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
So for an even function, for every point (x, y) on
the graph, the point (-x, y) is also on the graph.
Function Symmetry with Respect to the Origin
ODD Function
• If we “rotate” the graph of f (x) = x3
180˚ about the origin, the two parts
would coincide. We say that the
graph is symmetric with respect to the origin.
Symmetry with Respect to the Origin - Odd Function
If a function f is defined so that
f ( x)   f ( x)
for all x in its domain, then the graph of f is symmetric with
respect to the origin.
•
For example, Given f ( x )  x 3 , we have
f ( 2 )   f ( 2 )  8
f (  1 )   f (1 )   1 ,
or for any real number x , f (  x )  (  x )
3
 x
3
  f ( x ).
Odd Functions have Origin Symmetry
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
So for an odd function, for every point (x, y) on the
graph, the point (-x, -y) is also on the graph.
Symmetry with Respect to the x-Axis
• If we “fold” the graph of x = y2 along the x-axis, the two halves
of the parabola coincide. This graph exhibits symmetry with
respect to the x-axis. (Note, this relation is not a function. Use
the vertical line test on its graph below.)
Symmetry with Respect to the x-Axis
If replacing y with –y in an equation results in the same equation,
then the graph is symmetric with respect to the x-axis.
e.g.
x  y
2
x  ( y )
2

x  (  1) y
2
2

x  y
2
x-axis Symmetry
We wouldn’t talk about a function with x-axis symmetry
because it wouldn’t BE a function.
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
Even and Odd Functions
A function f is called an even function if f (  x )  f ( x ) for all x in the
domain of f. (Its graph is symmetric with respect to the y-axis.)
A function f is called an odd function if f (  x )   f ( x ) for all x in the
domain of f. (Its graph is symmetric with respect to the origin.)
To determine odd or even function, calculate f(-x). If f(-x)=f(x) even, if f(-x)=-f(x) odd,
and if f(-x)≠f(x) & f(-x)≠-f(x) neither.
Example
Decide if the functions are even, odd, or neither.
1.
f (x)  6 x  9 x
3
f ( x)  6( x)  9( x)
3
 6 x  9 x   f ( x )
3
The function is odd.
2. f ( x )  3 x  5 x
2
f (  x )  3(  x )  5(  x )
2
 3x  5x
2
Since f (  x )   f ( x ) and f (  x )  f ( x ) ,
f is neither even nor odd.
A function is even if f( -x) = f(x) for every number x in
the domain.
So if you substitute a –x into the function and you get
the original function back again it is even.
f x   5 x  2 x  1
4
2
Is this function even?
YES
f  x   5 (  x )  2 (  x )  1  5 x  2 x  1
4
f x   2 x  x
3
2
4
Is this function even?
2
NO
3
3
f  x   2 (  x )  (  x )   2 x  x
A function is odd if f( -x) = - f(x) for every number x in
the domain.
So if you substitute a –x into the function and you get the
negative of the function back again (all terms change signs)
it is odd.
f x   5 x  2 x  1
4
2
Is this function odd?
NO
f  x   5 (  x )  2 (  x )  1  5 x  2 x  1
4
2
f x   2 x  x
3
4
2
Is this function odd?
YES
f  x   2 (  x )  (  x )   2 x  x
3
3
If a function is not even or odd we just say neither
(meaning neither even nor odd)
Determine if the following functions are even, odd or
neither.
Not the original and all
3
terms didn’t change
f x  5x 1
signs, so NEITHER.
 
f  x   5  x   1   5 x  1
3
3
f x   3 x  x  2
4
2
Got f(x) back so
EVEN.
f  x    3 (  x )  (  x )  2   3 x  x  2
4
2
4
2
Your Turn
Functions may be described as even, odd, or neither.
Even functions have graphs symmetric with the y-axis.
Odd functions have graphs symmetric with the origin.
To test algebraically whether a function is even or odd:
Substitute -x for x in the function.
The function is even if f(-x) = f(x). ( function is unchanged)
The function is odd if f(-x) = -f(x). ( all signs change)
1) Determine whether f(x) = 5 – 3x is even,
odd, or neither.
– Answer: Neither
2) Determine whether f(x) = - x2 + 3 is even,
odd, or neither.
–
Answer : Even
Homework
• Section 1.3, pg. 38 – 41:
Vocabulary Check #1 – 6 all
Exercises: #1-9 odd, 15-35 odd, 43, 47,
51, 53, 59 -69 odd, 95-100 all
• Read Section 1.4, pg. 42 – 47