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CHAPTER 2: FUNCTIONS, EQUATIONS, AND INEQUALITIES
2.1
LINEAR EQUATIONS, FUNCTIONS, AND MODELS

LINEAR EQUATIONS
o A linear equation in one variable is an equation that is equivalent to one
of the form mx  b  0, where m and b are real numbers and m  0.

EQUATION SOLVING PRINCIPLES
o For any real numbers a, b, and c:
The Addition Principle: If a  b is true, then a  c  b  c is true.
The Multiplication Principle: If a  b is true, then ac  bc is true.

FIVE STEPS FOR PROBLEM SOLVING
1.
Familiarize yourself with the problem situation. If the problem is
presented in words, read the problem carefully! The following
steps can help you to understand the problem.
a) Make a drawing.
b) Make a written list of the known facts, and a list of
what you need to find out.
c) Assign variables to represent unknown quantities.
d) Organize the information in a chart of table.
e) Find further information. Look up a formula, consult
a reference book, ask somebody who is
knowledgeable in the field, or look online.
f) Guess or estimate the answer and check your guess or
estimate.
2.
Translate the problem into one or more equations or inequalities.
3.
Carry out some type of mathematical manipulation.
4.
Check to see whether your solution is reasonable and compare it to
your estimate.
5.
State the answer clearly using a complete sentence.

FORMULAS
o A formula is an equation that can be used to model a situation.
 MOTION
 The distance d traveled by an object moving at rate r in
time t is given by
d  rt.
 Example:
o Question: An airplane that travels 450 mph in
still air encounters a 30-mph headwind. How
long will it take the plane to travel 1050 mi into
the wind?
o Solution: The airplane’s rate is its rate in still
air minus the rate of the headwind, or, 450 mph
minus 30 mph. This gives the plane a rate of
420 mph. Now we know
d
1050mi
d  rt  t  
 2.5hr.
r 420mi / hr


SIMPLE-INTEREST
 The simple interest I on a principal of P dollars at interest
rate r for t years is given by
I  Prt.
 The formula A  P  Prt or A  P(1  rt ) , gives
the amount A to which a principal of P dollars
will grow when invested at simple interest rate r
for t years.
ZEROS OF FUNCTIONS
o If a function’s y-value, or output, is equal to zero, the input is called a zero
of the function.
 An input c of a function f is called a zero of the function, if the
output for c is 0. That is, c is a zero of f if f (c)  0.
x  int ercept
Zero is -5
( 5, 0)
f ( x) 
1
3
x2
2.2
THE COMPLEX NUMBERS

THE COMPLEX NUMBER SYSTEM
o If a graph does not ever cross the x-axis, there are no real number
solutions for the zeros of the function
 Example: f ( x)  x 2  6
o
1 is not a real number since there is no real number x such that
x  1.
2
o THE NUMBER i is defined such that
i  1 and i 2  1.
o COMPLEX NUMBERS
 A complex number is a number of the form a  bi, where a and b
are real numbers. The number a is said to be the real part of
a  bi and the number b (or the number bi) is said to be the
imaginary part of a  bi.
 BE CAREFUL!!! The imaginary term i must not be in the
denominator when you give your final answer.
3
o Example: 2  is not OK as a final answer. You
i
need to multiply by “one” as follows:
3 i 
3i
3i
2    2 2  2
 2  3i
i i
i
1
The complex numbers:
a  bi, b  0
Imaginary numbers:
a  bi , b  0, a  0
Imaginary numbers:
Real numbers:
a  bi , b  0
a  bi, b=0
Pure imaginary numbers:
a  bi , b  0, a  0
Irrational numbers:
Radicals, e, 
Rational numbers:
-1, 1/2 , 0.85

OPERATIONS WITH COMPLEX NUMBERS
o The complex numbers obey the commutative, associative, and distributive
laws.
o We treat them just like we treat polynomials.
 BE CAREFUL!!! (i)(i) = -1

2.3
CONJUGATES AND DIVISION
o The conjugate of a complex number a  bi is a  bi. The numbers a  bi
and a  bi are complex conjugates.
QUADRATIC EQUATIONS, FUNCTIONS, AND MODELS

QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS
o A quadratic equation is an equation equivalent to
ax 2  bx  c  0, a  0,

where a, b, c are real numbers.
The form represented above is called the standard form of a
quadratic equation.
o A quadratic function f is a function that can be written in the form
f ( x)  ax 2  bx  c, a  0,


where a, b, c are real numbers.
EQUATION-SOLVING PRINCIPLES
o The principle of zero products: If ab  0 is true, then a  0 or b  0 ,
and if a  0 or b  0 , then ab  0 .
o The principle of square roots: If x 2  k , then x  k or x   k .
COMPLETING THE SQUARE
o Often used when we are looking for the factored form of a parabola
o We want a coefficient of one in front of the x 2 term
b
2
o Recall that  a  b   a 2  2ab  b 2 . If we switch x for a and
for b we
2
2
2
b
b 
have x 2  bx      x   . This is very useful if we have an
2
2 
equation which is not a perfect square and we want to find exact answers
for its zeros.
 Example: Solve by completing the square to obtain exact solutions
2 x2  5x  3  0
 Solution: First we need to divide by 2 so there is a 1 in front of
the squared term.
2 x2  5x  3 0

2
2
5
3
x2  x   0
2
2
Then we need to add the constant term to both sides
5
3 3
3
x   0
2
2 3
2
5
3
x2  x 
2
2
x2 
Now we need to calculate half of the numerical coefficient of the x term
And add the square of the result to both sides of the equation. Then we’ll
square root both sides and solve.
2
5
3  5
 5
x  x     
2
2  4
 4
2
2
2
5
3  25 

x    
4
2  16 

2
5
49

x  
4
16

5
7
x 
4
4
1
x   ,3
2
o STEPS FOR COMPLETING THE SQUARE
1.
Isolate the terms with variables on one side of the equation and
arrange them in descending order.
2.
Divide by the coefficient of the squared if that coefficient is not 1.
3.
Complete the square by taking half the coefficient of the firstdegree term and adding its square to both sides of the equation.
4.
Express one side of the equation as the square of a binomial.
5.
Use the principal of square roots.
6.
Solve for the variable.

USING THE QUADRATIC FORMULA
o Deriving the quadratic formula
ax 2  bx  c  0, a  0, a  0.
ax 2  bx  c 0

a
a
b
c
x2  x   0
a
a
b
c c
c
x2  x    0 
a
a a
a
b
c
x2  x  
a
a
2
2
2
2
b
c  b 
 b 
x  x     
a
a  2a 
 2a 
2
b
c  b 
 b 
x  x     
a
a  2a 
 2a 
2
b
b2
c  4a  b 2
x  x 2     2
a
4a
a  4a  4a
2
b  b 2  4ac

x


 
2a 
4a 2

2
b 
b 2  4ac

x




2a 
4a 2

2
b 
b 2  4ac

x





2a 
2a

b
b 2  4ac
x 
2a
2a
x
b  b 2  4ac
2a
o The solutions of ax 2  bx  c  0, a  0, are given by
x
b  b2  4ac
2a
o The Discriminant: The expression b 2  4ac shows the nature of the
solutions.
DISCRIMINANT
For ax 2  bx  c  0 :
b 2  4ac  0  one real number solution;
b 2  4ac  0  Two different real number solutions;
b 2  4ac  0  Two different imaginary-number
solutions, complex conjugates.

Equations reducible to quadratic
x 6  x3  8  0 can be written as
o Example:

2.4
x 
3 2
 x3  8  0. Substituting u for x3 ,
u 2  u  8  0, which is quadratic.
Don’t forget to back substitute when you have solved for u.
ANALYZING GRAPHS OF QUADRATIC FUNCTIONS

GRAPHING QUADRATIC FUNCTIONS OF THE TYPE
2
f ( x)  a  x  h   k
o The graph of f ( x)  a  x  h   k is the graph of f ( x)  x 2
 Shifted horizontally h units (to the left if h < 0, to the right if h > 0)
 Shifted vertically k units (down if k < 0, up if k > 0)
 Vertical
 Stretching if a  1
2

Shrinking if 0  a  1
o If a < 0 the graph reflects across the x-axis
2
o Consider f ( x)  a  x  h   k

The point ( h, k ) at which the graph turns is called the vertex. The
vertex will be the maximum of f ( x ) if a < 0. The vertex will be
the minimum of f ( x ) if a > 0.
 Each graph has a line x  h that is called the axis of symmetry.
o Example: f ( x)  3x 2  3x  1

We need to get this function in the form f ( x)  a  x  h   k .
One way to do this is by factoring out -3 and then completing the
square. So we have,
2
f ( x)  3x 2  3x  1
1

 3  x 2  x  
3

2
2
 2
1 1 1
 3  x  x        

 2   2  3 

1 1 1

 3  x 2  x    
4 4 3

2

1
7
 3   x    
2  12 

2
1 7

 3  x   
2 4

2
  1  7
 3  x      
  2  4
7
1
 1 7
The vertex is at the point   ,  . There is a maximum of
at x   . The
4
2
 2 4
1
axis of symmetry is the line x   . Below is the graph of the function:
2

THE VERTEX OF A PARABOLA
 b
 b 
o The vertex of the graph of f ( x)  ax 2  bx  c is   , f     .
 2a  2a  
 First you calculate the x-coordinate, then substitute that value into
the function to find the y-coordinate.

APPLICATIONS
o Maximizing Area Example
 The sum of the base and the height of a parallelogram is 69cm.
Find the dimensions for which the area is a maximum.
A=(base)(height)
b  h  69  b  69  h
A  bh
A(h)  (69  h)(h)
A(h)  69h  h 2
A(h)  h 2  69h
 69  
 b
 b    69
  2a , A   2a      2(1) , A   2(1)  

 



  34.5, A  34.5 
 (34.5,1190.25)
b
The height we found evaluating  = 34.5cm and the base was
2a
69 – 34.5 = 34.5cm.
2.5
MORE EQUATION SOLVING

RATIONAL EQUATIONS
o Equations containing rational expressions are called rational equations.
 You can solve these equations by multiplying both sides of the
equation by the least common denominator (LCD) or, if you have
one fraction equal to another you can cross-multiply.


BE CAREFUL!!! If you have a rational expression that
you need to simplify, you need to get a common
denominator, and then combine the numerators.
Example:
Solve
1 1 1
  5
t 2t 3t
1 1 1 
6t       5  6t
 t 2t 3t 
1
1
1
6t    6t    6t    30t
t 
 2t 
 3t 
6(1)  3(1)  2(1)  30t
11  30t
30t  11
11
t
30

RADICAL EQUATIONS
o A radical equation is an equation in which variables appear in one or
more radicands.

The Principle of Powers
For any positive integer n:
If a  b is true, then a n  bn is true.

Example:
Solve 5 3x  4  2

5
3x  4

5
 (2)5
3x  4  32
3x  28
x

28
3
EQUATIONS WITH ABSOLUTE VALUE
For a  0 and an algebraic expression X :
X  a is equivalent to X  a or X  a.
Example:
Solve 9  x  2  7
9 x2  7
97  x2
2 x2
x2  2
 x  2  2
x4
or
 x  2   2
x0
2.6
SOLVING LINEAR INEQUALITIES

LINEAR INEQUALITIES
o PRINCIPLES FOR SOLVING INEQUALITIES
For any real numbers a, b, c:


The Addition Principle for Inequalities: If a < b is true, then
a  c  b  c is true.

The Multiplication Principle for Inequalities: If a < b and c > 0
are true, then ac  bc is true. If a < b and c < 0 are true, then
ac  bc is true.
COMPOUND INEQUALITIES
o When two inequalities are joined by the word “and” or the word “or”, a
compound inequality is formed.


A compound inequality like
3  2x  5 and 2 x  5  7 is called an conjunction, because it
uses the word “and”. We can write this as one compound
inequality as follows: 3  2x  5  7 .

A compound inequality like 2 x  5  7 or 2x  5  1 is called a
disjunction, because it contains the word “or”. This type of
inequality cannot be abbreviated like the one above.
INEQUALITIES WITH ABSOLUTE VALUE
o For a > 0 and an algebraic expression X:
X  a is equivalent to  a  X  a.
X  a is equivalent to X  a or X  a.