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CHAPTER 1 LECTURE NOTES
1.1

INTRODUCTION TO GRAPHING
GRAPHS
 Points and Ordered Pairs
o The Coordinate Plane: Many ideas in mathematics are best
explained with a graph in a plane. Each point in the plane
corresponds to an ordered pair. Graphs are set up with two axes,
one for each number in an ordered pair. The horizontal axis is
often called the x-axis, and the vertical axis is often called the yaxis. We call x the first coordinate, x-coordinate, or abscissa. We
call y the second coordinate, y-coordinate, or ordinate. This is
called the Cartesian coordinate system, after Rene Descartes, a
French mathematician and philosopher.

(-11.75, 8)
QUADRANT II
(-, +)
QUADRANT III
(-, -)
 (-7, -6.5)

QUADRANT I
(+, +)
 (3.25,5)

(4, -2.75)
QUADRANT IV
(+, -)
SOLUTIONS OF EQUATIONS
o If an equation has two variables, its solutions are pairs of numbers.
 Example: Determine whether the pair (2, 7) is a solution of the
equation 5 x  y  3 . To find out, we just plug in 2 where the x is and
7 where the y is in the given equation. This gives us
5(2)  (7)  10  7  3 . Since the left-hand side equals the right-hand
side, the ordered pair (2, 7) is a solution of the given equation.

GRAPHS OF EQUATIONS
o If an equation’s degree is 1, like 3 x  5 y  2 , then you have an equation for a
line. All you need to graph a line is two points. One way to graph a line is to
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plot two points and then draw a line through the points. If you have an
equation, you can substitute any value of x or y (which fall in the domain or
range) and then solve for the other variable.

For Example: Suppose you have the line 5 x  8 y  10 . To find two
points, we could choose to substitute 0 in for y and then solve for x
5 x  8(0)  10
5 x  0  10
.
5 x  10
5 x 10

5
5
x2
This would give us the ordered pair (aka point) (2, 0). Then we could
choose to in 10 for x and solving like we did above, we would have the
ordered pair (10, -5). These two solutions give you two points on the line.
You simply plot the points and connect the dots and you have graphed the
line!
o Nonlinear Equations

To graph nonlinear equations, you’ll need more than two points. I
would substitute at least 5 values in for x to get an idea of what the
graph will look like.

Examples of Some Nonlinear Equations that You Should Know

y  x2
2

y  x3

y | x |
3

y
1
x
 GRAPHING USING INTERCEPTS
o To Determine Intercepts: Consider the ordered pair (a, b).

The x-intercept is (a, 0). To find a, let y = 0 and solve the
original equation for x.

The y-intercept is (0, b). To find b, let x = 0 and solve the
original equation for y.
o Example: Graph the equation 5 x  8 y  10 . To find the x-intercept, we
would substitute 0 in for y and then solve for x.
5 x  8(0)  10
5 x  0  10
5 x  10
.
5 x 10

5
5
x2
4
This gives us the ordered pair (2, 0). Now you find the y-intercept and
graph the line using the coordinate plane provided below.

THE DISTANCE FORMULA
o DERIVED USING THE PYTHAGOREAN THEOREM
 If triangle ABC is a right triangle, with sides a and b representing
the legs and side c representing the hypotenuse, we have the
relationship a 2  b 2  c 2 .
 This gives us the distance between two points,  x1 , y1  and
 x2 , y2  as

d
 x2  x1    y2  y1 
2
2
MIDPOINTS OF SEGMENTS
o If the endpoints of a segment are  x1 , y1  and  x2 , y2  , then the
 x  x   y  y  
coordinates of the midpoint are  1 2 , 1 2 
2
2



THE EQUATION OF A CIRCLE
o The equation of a circle with center  h, k  and radius r, in standard form,
is  x  h    y  k   r 2
2
1.2
2
FUNCTIONS AND GRAPHS
 Functions
o Relation: A relation is a correspondence between a first set, called
the domain, and a second set, called the range, such that each
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member of the domain corresponds to at least one member of the
range.
o Function: A function is a rule that assigns to each element from
one set exactly one element from another set.
o Domain and Range: The set of all possible values of the
independent variable in a function is called the domain of the
function, and the resulting set of possible values of the dependent
variable is called the range.

Example: Consider the points (2, 3), (1, -5), (8, 4), and
(3, 3). If we grouped these ordered pairs in terms of
domain and range we’d have the following
correspondence.
23
3
1  5
84

This correspondence is a function since each member of
the domain (the x values) is assigned to exactly one
member of the range (the y values).
We can express the previous function as sets of ordered
pairs. It would be written as {(2, 3), (1, -5), (8, 4), (3, 3)}.
We can also graph this set of ordered pairs as follows:
(8, 4)
(2, 3) (3, 3)
(1, -5)
The domain is {1, 2, 3, 8} and the range is {-5, 3, 4}. We
use the roster notation since we are dealing with specific
points only.
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
Now you try! Find the domain and range of the function
graphed below:
Domain:
Range:
What y value corresponds to the value x = 0?
o Vertical Line Test: If a vertical line intersects a graph in more than
one point, the graph is not the graph of a function.
 Function Notation and Equations
o The symbol f ( x ) is read “f of x”, “f at x”, or “the value of f at x”.
It DOES NOT mean f times x!!! The expression f ( x)  2 x  5 is
equivalent to y  2 x  5 . The first expression is in function
notation and the second expression is in equation notation.
 Example: Find the indicated function value.
 f (2) , for f ( x)  15 x  4
f (2)  15(2)  4
 30  4
 34
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1.3
LINEAR FUNCTIONS: SLOPE, AND APPLICATIONS
 SLOPE OF A LINE: The slope of a line is defined as the vertical change
(the “rise”) over the horizontal change (the “run”) as one travels along the
line. In symbols, taking two different points (x1, y1) and (x2, y2) on the
line, the slope is
m
rise Change in y y2  y1


,
run Change in x x2  x1
where x1  x2
o Example: Find the slope of the lines drawn in the graph below.
l
k
The slope for line l:
The slope for line k:
 SLOPE-INTERCEPT FORM: If a line has a slope m and y-intercept b,
then the equation of the line in slope-intercept form is
y  mx  b
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o Example: Write the equation 12 y  2 x  14 in the slope-intercept
form of the line.
We simply solve for y:
12 y  2 x  14
12 y  2 x  2 x  2 x  14
12 y  0  2 x  14
12 y  2 x  14
12 y 2 x 14


12
12 12
1 7
y 
6 6
 Zero Slope and Lines with Undefined Slope:
o Slope of a horizontal line: Every horizontal line has a slope of 0.
Since the rise in the slope formula m 
rise y2  y1

, is always 0 for
run x2  x1
a horizontal line, the entire ratio is 0.
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o Slope of a Vertical Line: Every vertical line has an undefined
slope.
rise y2  y1

, is always 0 for a
run x2  x1
vertical line, the denominator of the ratio is 0, so the slope is
undefined.
Since the run in the slope formula m 
 SOLVING EQUATIONS GRAPHICALLY
o Consider the line 2 x  4  8 . To find the x value for which 2x  4
will equal 8, we graph f ( x)  2 x  4 and g ( x)  8 on the same set
of axes.
The solution or intersection appears to be at the point  2,8 .
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 RECOGNIZING LINEAR EQUATIONS
o The Standard Form of a Linear Equation: Any equation in the
form of Ax  By  C , where A, B, and C are real numbers and A
and B are not both zero, is linear. Any equation of the form
Ax  By  C is said to be a linear equation in standard form.
1.4
EQUATIONS OF LINES AND MODELING
 POINT-SLOPE FORM: If a line has slope m and passes through the point
(x1, y1), then an equation of the line is given by
y  y1  m( x  x1 )
the point-slope form of the equation of a line.
 PARALLEL LINES: Two lines are parallel if and only if they have the
same slope, or if they are both vertical.
 PERPENDICULAR LINES: Two lines are perpendicular if and only if
the product of their slopes is -1, or if one is vertical and the other is
horizontal.
 CURVE FITTING
o SCATTERPLOTS
o LINEAR REGRESSION
 THE CORRELATION COEFFICIENT
1.5


MORE ON FUNCTIONS
INCREASING, DECREASING, AND CONSTANT FUNCTIONS
o A function f is said to be increasing on an open interval I, if for all a and b
in that interval, a<b implies f (a)  f (b) .
o A function f is said to be decreasing on an open interval I, if for all a and
b in that interval, a<b implies f (a )  f (b) .
o A function f is said to be constant on an open interval I, if for all a and b
in that interval, f (a )  f (b) .
RELATIVE MAXIMUM AND MINIMUM VALUES
o Suppose that f is a function for which f (c ) exists for some c in the
domain of f. Then:
f (c ) is a relative maximum if there exists an open interval I
containing c such that f (c)  f ( x) , for all x in I where x  c ; and
f (c ) is a relative minimum if there exists an open interval I
containing c such that f (c)  f ( x) , for all x in I where x  c ;
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
FUNCTIONS DEFINED PIECEWISE
o Piecewise functions use different output formulas for different parts of the
domain
 Example: Graph the piecewise function
for x  0
1,
f ( x)   2
for x  0
x ,

GREATEST INTEGER FUNCTION
o f ( x)  x  the greatest integer function less than or equal to x
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1.6
THE ALGEBRA OF FUNCTIONS
 THE SUM, DIFFERENCE, PRODUCT, OR QUOTIENT OF TWO
FUNCTIONS
o  f  g  x   f  x   g  x 
o
o
o
 f  g  x   f  x   g  x 
 f  g  x   f  x   g  x 
f  x
, provided
 f / g  x  
g  x

g ( x ) is not zero.
Example: Let f ( x)  2 x 2 and g ( x)  5  3x 2

 f  g  x   f  x   g  x 
  2 x 2    5  3x 2 

 5x2  5
 f  g  x   f  x   g  x 
  2 x 2    5  3x 2 

  x2  5
 f  g  x   f  x   g  x 
  2 x 2  5  3x 2 

 10 x 2  6 x 4
 f / g  x  
f  x
g  x
2x2

5  3x 2
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 DOMAINS AND GRAPHS
o The domain of f  g , f  g , or f  g is the set of all values common to the
domains of f and g.
f
g
o The domain of f/g is the set of all values common to the domains of f and
g, excluding any values for which g(x) is 0.
f
g


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 DIFFERENCE QUOTIENTS
o The slope of the line, called a secant line, is
f  x  h   f ( x) f  x  h   f ( x)

h
 x  h  x
y
 x  h, f  x  h  
f ( x  h)
secant line
 x, f  x  
f ( x)
x
xh
h
 THE COMPOSITION OF FUNCTIONS
o We often come across situations where the output of functions depend on
inputs which are functions, as opposed to single values.
o The composite function f g , the composition of f and g, is defined as
g  ( x)  f  g  x   ,
where x is in the domain of g and g ( x ) is in the domain of f.
f
1.7 SYMMETRY AND TRANSFORMATIONS
 SYMMETRY
o If, for any point ( x, y ) on a graph, the point ( x,  y ) is also on the graph,
then the graph is said to symmetric with respect to the x-axis.
o If, for any point ( x, y ) on a graph, the point ( x, y ) is also on the graph,
then the graph is said to symmetric with respect to the y-axis.
o If, for any point ( x, y ) on a graph, the point ( x,  y ) is also on the graph,
then the graph is said to symmetric with respect to the origin.
 ALGEBRAIC TESTS OF SYMMETRY
o X-AXIS: If replacing y with –y produces an equivalent equation, then the
graph is symmetric with respect to the x-axis.
o Y-AXIS: If replacing x with –x produces an equivalent equation, then the
graph is symmetric with respect to the y-axis.
o ORIGIN: If replacing x with –x and y with –y produces an equivalent
equation, then the graph is symmetric with respect to the origin.
15
 EVEN AND ODD FUNCTIONS
o If the graph of a function f is symmetric with respect to the y-axis, we say
that it is an even function. That is, for each x in the domain of f,
f ( x)  f ( x).
o If the graph of a function f is symmetric with respect to the origin, we say
that it is an odd function. That is, for each x in the domain of f,
f ( x)   f ( x).
 VERTICAL TRANSLATION
For b > 0,
The graph of y 
units;
The graph of y 
units
For d > 0,
The graph of y 
units;
The graph of y 
units
f ( x )  b is the graph of y  f ( x) shifted up b
f ( x )  b is the graph of y  f ( x) shifted down b
f ( x  d ) is the graph of y  f ( x) shifted right d
f ( x  d ) is the graph of y  f ( x) shifted left d
 REFLECTIONS
o The graph of y   f ( x) is the reflection of the graph y  f ( x) across the
x-axis
o The graph of y  f ( x ) is the reflection of the graph y  f ( x) across the
y-axis
o If a point ( x, y ) is on the graph of y  f ( x) , then ( x,  y ) is on the graph
of y   f ( x) , and ( x, y ) is on the graph of y  f ( x )
 VERTICAL STRETCHING OR SHRINKING
o The graph of af ( x) can be obtained from the graph of y  f ( x) by
Stretching vertically for a  1, or
Shrinking vertically for 0  a  1.
o For a < 0, the graph is also reflected across the x-axis.
 HORIZONTAL STRETCHING OR SHRINKING
o The graph of f (cx) can be obtained from the graph of y  f ( x) by
Shrinking horizontally for c  1, or
Stretching horizontally for 0  c  1.
o For c < 0, the graph is also reflected across the y-axis
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