Solving Polynomial and Rational Inequalities
1.
2.
3.
4.
Make one side of the inequality equal to 0.
Put into factored form with common denominator (if more than one rational
term).
Find the boundary numbers
Make a sign graph.
Note: Be able to use a variety of notations for your answer.
x7

[7,)
{x x  7}
_____________________


Example: Solve 3x(x  6)(x 2  9)(x 2  25)  0

Example: 2 
12 11x
x

6x
6

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Absolute Value
 x, x  0
x  
x, x  0

3  5x 

(2  7x) 2 

Example: Solve
x 1
2
3x  2

Example: 5  3x  6

Example: Find all values of x for which the radical represents a real number.
x 2  7x 12
x3  8
3
x 1
x3

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P.1
Graphs and Models
There are four ways to represent a function:




Verbally (by a description in words)
Numerically (by a table of values)
Algebraically (by an explicit formula)
Graphically (by a graph)
You can sketch a graph by point-plotting (can be tedious or difficult to obtain accuracy) or by
using a graphing utility (if you set the window appropriately). There is information that can
help us draw a graph more accurately.
Intercepts of a Graph:
Find the x- and y-intercepts of the graph of y  x  4 x .
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
Even Functions and Odd Functions – Symmetry:
A function y = f(x) is an
even function of x if:
odd function of x if:
for every x in the domain.
A graph can have symmetry to the x-axis (meaning we do not call it a function of x) if:
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Example: Find any intercepts, test for symmetry, and graph each function.
y  2x 3  x
x  y2 1

Points of Intersection:
You can find the point(s) of intersection of two or more graphs by
(1)
(2)
Example: Find all points of intersection of the graphs of
x 2  y  3 and
x  y  1. Sketch each graph to see if your
answer is reasonable.

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P.2 Linear Models and Rates of Change
Which of the following table of values represents a line? Why?
What does “slope” tell us about a graph? How do you calculate it?
Sketch a line that represents each of the following:
m>0
m<0
m=0
m is undefined
Give an equation whose graph would be a vertical line and one that would produce a
horizontal line.
There are 3 common forms of the Equation of a Line:
(1)
Slope-Intercept Form:
(2)
Point-Slope Form:
(3)
General Form:
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Slope can also be called an “Average Rate of Change” – in this case the units of
measurement for the vertical and horizontal change are different, like in velocity. When the
units of measurement are the same, the unit cancels, and slope is considered a “ratio”.
Example: (a) A soccer player kicked the ball. After 1 second, the height of the ball was
30 feet. After 2 seconds, the height was 58 feet. Calculate the average velocity of the
ball in feet per second.
(b) In a water-ski jumping tournament, the ramp rises to a height of 6 feet on top of a
raft that is 21 feet long. What is the slope of the ski ramp?
What do you know about slopes of parallel lines? perpendicular lines?
Example: Find the general forms of the equation of the line that passes through the
point (2, -1) and is (a) parallel to 2x – 3y = 5 (b) perpendicular to 2x -3y = 5.
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P. 3 Functions and Their Graphs
What is a relation?
When is y a function of x?
What is the domain of a function?
What is the range of a function?
Functions are not always in terms of x and y:
Function notation is a convenient way to write a function:
Example: For the function f defined by f(x) where f(x) = x2+7, evaluate the following:
(a)
f(3a)
(b)
f(b-1)
(c)
f (x  x)  f (x)
x
*Be certain you understand this!!

Finding the Domain and Range of a Function:
Restrictions in the domain of a function result from the following:
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Example: Find the domain and range of the function:
(a)
f (x)  x 1
(b)
g(x)  x 2  x  6


Functions Defined by More than One Equation (Piecewise Functions):
x, x  0

Graph: f (x)  x 2 , 0  x  1
1,
x 1


See if you can write the piecewise function for the following graph:
(1,1)
(0,0)
(2,1)
(1,0)
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Example: Find the domain of each function:
(a)
f (x)  6
3x(2x  1)
2x  1
(b)
f (x)  7
x 2 1
x 2


Graphs of a Function from a Parent Function
Remember that you can often sketch the graph of a function using the key points of the
known parent function (see page 22).
Use transformations to graph each of the following:
(a)

y  2 x 1
(b)
y  (x  3) 2  2

Classification and Combinations of Functions
The modern notion of a function was derived from many 17th and 18th century
mathematicians. Of particular note was ______________________. By the end of the 18th
century, mathematicians and scientists had concluded that most real-world situations could
be represented by mathematical models taken from a basic collection of functions called
elementary functions. Elementary functions are divided into 3 categories:
1) ______________________________________
2) ______________________________________
3) ______________________________________
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The most common type of algebraic function is a _____________________ function, which
can be written in this form:
We can predict the end behavior by looking at the degree and the leading coefficient:
Even degree:
Odd degree:
Another type of algebraic function is the _____________________ function, written in the
form f (x) 
p(x)
q(x)
where
q(x)  0.
Composite Functions are formed when one functions replaces the independent variable of
another function. Common notations for composite functions are:

The domain of the composite function must be a subset of the domain of the “inner”
function.
Example: Given f (x)  2x  3 and g(x)  x 2  1
(a) f(g(x)) and (b) g(f(x)). Give the domain of each.
, find

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EXPLORATION of Parametric Equations
A relation is a set of ordered pairs (x,y) of real numbers. The graph of a relation is the set
of points in the plane that correspond to the ordered pairs of the relation. If x and y are
functions of a third variable t, called a parameter, then we can use the parametric mode of a
graphing utility to obtain a graph of the relation.
Example: Graph the relation determined by x  t,
y  t, t  0
.

We can confirm this algebraically:
Example: Graph the relation determined by x  2cos(t),
y  2sin( t), 0  t  2

Again, we can confirm this algebraically:
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Activity: Match the parametric equations with their graph.
State the approximate dimensions of the viewing window. Give
a parameter interval that traces the curve exactly once.
Assignment: Graph the curve. Find a Cartesian equation for a curve that contains the
parametrized curve.
1. x  4  t, y  t,
t0
2. x  cos(t), y  sin( t), 0  t  
3.

x  t 2, y  4  t 2 , 0  t  2
4.
Graph the curve x  2cot(t), y  2sin 2 (t), 0  t  
. This curve
is called the “witch of Agnesi”. Maria Agnesi was a gifted
scholar and linguist whose Latin essay written in the 1740s
defending higher education for women was published when she

was only nine years
old. She was a well-published scientist
by age 20. Today, Agnesi is chiefly remembered for this bellshaped curve. The name is a result of mistranslation. She
named it “versiera” or “turning curve”, but the term was
probably confused with “avversiera” which means “wife of the
devil” which was translated into “witch”.
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