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Intermediate Algebra Final Exam Review Sheet
Friday 12/10/04
Section 1.1: Basic Definitions: The Real Numbers and the Real Number Line
1. blah, blah, blah
Section 1.2: Operation with Real Numbers
1. blah, blah, blah
Section 1.3: Algebraic Expressions
1. blah, blah, blah
Section 1.4: Translating Phrases and Sentences into Algebraic Form
English
Math
is
=
of (with a fraction or %)

sum of
+
difference
-
x more than y
x+y
x less than y
y-x
increase
+
decrease
-
twice x or 2 times x
2x
n times x
nx
Two consecutive integers
1. Area of a rectangle: A = Width  Length
2. Perimeter = sum of the lengths of the sides of a shape
3. Rate  Time = Amount
n, n+1
Section 1.5: First degree Equations and Inequalities
1. To solve linear equations, isolate the variable on one side.
1

4  y    y  6 5  y 
2

4 y  2  y  30  6 y
Example:
9 y  32
y
32
9
2. To solve a linear inequality, isolate the variable using the Rules for Inequalities:
a. a  b  a  c  b  c
b. a  b  a  c  b  c
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Intermediate Algebra Final Exam Review Sheet
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c. a  b  ac  bc when c is positive
d. a  b  ac  bc when c is negative
a b
e. a  b   when c is positive
c c
a b
f. a  b   when c is negative
c c
4 x  7  19
Example:
4 x  12
x3
Section 2.1: Equations as Mathematical Models
1. Draw a picture.
2. Label the picture with given quantities and the unknown.
3. Write down the equation that relates the unknown to the given information.
4. Solve the equation.
5. Check your answer.
6. Some geometric formulas for areas:
a. Rectangle: Arect  lw
b. Circle: Acirc  r 2
1
c. Triangle: Atri  bh
2
7. Some geometric formulas for volumes:
a. Box: Vbox  lwh
b. Cylinder: Vcyl  r 2 h
4
c. Sphere: Vsphere  r 3
3
distance
amount
8. Speed problems: speed 
, or in general, rate 
time
time
9. The total is equal to the sum of its parts!
Section 2.2: First Degree Equations and Applications
1. Combine Sections 1.4 and 1.5…
Section 2.3: First Degree Inequalities and Applications
1. Combine Sections 1.4 and 1.5…
2. shorthand notation for an interval (or “betweenness”): a < x < b.
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Section 2.4: Absolute Value Equations and Inequalities
1. To solve absolute value equations, isolate the absolute value expression, then set that expression to  the
other side.
2 x  6  6  12
2 x  6  12
Example:
x6  6
x  6  6
x  12 or x  0
2. To solve absolute value inequalities, keep in mind that:
a. x  c is equivalent to  c  x  c
b.
x  c is equivalent to x  c or x  c
2 x  6  12
Example:
x6  6
x  6  6 or x  6  6
x  12 or x  0
Section 3.1: The Rectangular Coordinate System and Graphing Straight Lines
1. Know how to plot points on coordinate axes.
2. Know how to generate points from the equation of a line.
3. General Form of a line: Ax + By = C.
4. x intercepts are where a graph crosses the x-axis; calculate by setting y = 0 and solve for x.
5. y intercepts are where a graph crosses the y-axis; calculate by setting x = 0 and solve for y.
Section 3.2: Graphs and Equations
1. Be able to read the coordinates of a point off a graph.
Section 3.3: Relations and Functions
1. A relation is a set of ordered pairs (x, y). The set of all x values is called the domain, and the set of all y
values is called the range.
2. A function is a correspondence between two sets such that to each element of the first set (the domain)
there is assigned exactly one element of the second set (the range).
3. A function is a relation in which no two distinct ordered pairs have the same first coordinate.
4. You can tell that a graph is a function if it passes the “vertical line test.”
5. Know how to find the domain of a function:
a. Start with the domain being all real numbers.
b. Eliminate x values from the domain where a denominator is equal to zero.
c. Eliminate x values from the domain where a square root has a negative argument.
Section 3.4: Function Notation
1. Example: f ( x)  3x 2  2 x  5
2. f(a) is the value of f(x) when a is substituted in for x.
Intermediate Algebra Final Exam Review Sheet
3. Example: if g ( x ) 
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x
3
3
2a  5
2a  5
 , and g (2a  5) 

, then g (3) 
x5
35 8
2a  5  5
2a
Section 3.5: Interpreting Graphs
1. blah, blah, blah…
Section 4.1: Straight Lines and Slopes
y y 2  y1
1. slope: m 

x x2  x1
2. A line with positive slope rises.
3. A line with negative slope falls.
4. A horizontal line has zero slope.
5. Two lines are parallel if they have the same slope: m1  m2
1
6. Two lines are perpendicular if: m1  
m2
Section 4.2: Equations of a Line and Linear Functions as Mathematical Models
1. Point-slope form of the equation of a line: y  y1  mx  x1 
2. Slope-intercept form of the equation of a line: y  mx  b A line with positive slope rises.
Section 4.3: Linear Systems in Two Variables
1. Two or more equations considered together are called a system of equations.
2. The points of intersection of the graphs of the equations are solutions of both equations.
3. A system of linear equations can have 0 solutions (inconsistent), 1 solution, or infinitely many solutions
(dependent).
4. Substitution Method of solving a system of equations:
a. Solve for one variable
b. Substitute
c. Back-Substitute
5. Elimination Method of solving a system of equations:
a. Adjust coefficients so one variable has equal but opposite coefficients in both equations
b. Add the equations
c. Back-Substitute
 a x  b1 y  c1
b c bc
ac a c
6. For the linear system  1
, the solution is: x  2 1 1 2 and y  1 2 2 1 .
a1b2  a2b1
a1b2  a2b1
a2 x  b2 y  c2
7. Know how to model systems with a system of linear equations.
Section 4.4: Graphing Linear Inequalities in Two Variables
1. Graph the line, then pick a test point to decide which side of the line to shade in.
Section 5.1: Polynomial Functions as Mathematical Models
1. Definition: A monomial is an algebraic expression that is either a constant or a product of constants and
one or more variables with whole number exponents.
2. Definition: A polynomial is a finite sum of monomials.
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3. Definition: The degree of a monomial is the sum of the exponents of its variables. The degree of a
nonzero constant is zero.
4. Definition: The degree of a polynomial is the highest degree of any monomial in it.
5. Definition: A polynomial in one variable is a function of the form
P(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0. n is the degree of the polynomial.
Section 5.2: Polynomials: Sums, Differences, and Products
1. To add or subtract polynomials, combine like terms.
2. Distributive property: a(b + c) = ab + ac.
3. To multiply polynomials, use distribution, or multiply each term in the first polynomial by each term in
the second polynomial.
a. Example: (x2 + 3x + 9)(x2 – 2x +3) = x4 –2x3 +3x2
+3x3 -6x2 +9x
+9x2 -18x +27
4
3
=x
+x
+6x2 -9x
+27
Section 5.3: General Forms and Special Products
1. General Form #1: (x + a)(x + b) = x2 + (a + b)x + ab
2. General Form #2: (ax + b)(cx + d) = acx2 + (ad + bc)x + bd
3. FOIL Method: First, Outer, Inner, Last
a. Example: (2x + 3)(5x-9) = 10x2 – 18x + 15x – 27 = 10x2 – 3x – 27
4. Difference of Two Squares: (a + b)(a – b) = a2 – b2
5. Square of Sum: (a + b)2 = a2 + 2ab + b2
6. Square of Difference: (a – b)2 = a2 – 2ab + b2
Section 5.4: Factoring Out the Greatest Common Factor
1. Common Monomial Factoring: Factor out the greatest monomial (i.e., constants and variables of the
highest power possible) common to all terms in a polynomial.
a. Example: 3x2 – 15x4 + 81x6 = 3x2(1 – 5x2 + 27x4)
b. Example: 5a(x – 2y) – 3b(x – 2y) = (5a – 3b)(x – 2y)
2. Factoring by Grouping: Grouping and factoring parts of a polynomial to factor the polynomial itself.
a. Example: 3xb – 2b – 15x + 10 = 3xb – 2b – 15x + 10 = b(3x – 2) – 5(3x – 2) = (b – 5)(3x – 2)
Section 5.5: Factoring Trinomials
1. Factoring x2 + qx + p:
a. Use the fact that (x + a)(x + b) = x2 + (a + b)x + ab
b.  look for factors of p that sum to q
c. Example: x2 – 4x – 12 =
i. factors of 12 are: 1, 2, 3, 4, 6, 12
ii. the factors that add up to –4 are –6 and +2
iii.  x2 – 4x – 12 = (x – 6)(x + 2)
2. Factoring Ax2 + Bx + C:
a. Use the fact that (ax + b)(cx + d) = acx2 + (ad + bc)x + bd
b.  list factors of A and B, then try out all the possibilities until you get it right
c. Example: 2x2 – 9x – 18 =
i. factors of 2 are: 1, 2
Intermediate Algebra Final Exam Review Sheet
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ii. factors of 18 are: 1, 2, 3, 6, 9, 18
iii. (2x – 2)(x + 9) = 2x2 + 18x – 2x – 18 = 2x2 – 16x – 18
iv. (2x + 3)(x – 6) = 2x2 – 12x + 3x – 18 = 2x2 – 9x – 18 
3. Factoring Trinomials by Grouping:
a. You’re on your own…
4. Factoring Trinomials by the AC Method:
a. You’re on your own…
5. Factoring Using Special Products:
a. Difference of Two Squares: a2 – b2 = (a + b)(a – b)
i. Example: 9x2 – 81 = (3x + 3)(3x – 3)
b. Square of Sum: a2 + 2ab + b2 = (a + b)2
i. Example: x2 + 12x + 36 = (x + 6)2
c. Square of Difference: a2 – 2ab + b2 = (a – b)2
i. Example: x2 - 2x + 1 = (x – 1)2
d. Difference of Cubes: a3 – b3 = (a – b)(a2 + ab + b2)
i. Example: x3 – 8 = (x – 2)(x2 + 2x + 4)
e. Sum of Cubes: a3 + b3 = (a + b)(a2 – ab + b2)
i. Example: 27x3 + 64 = (3x + 4)(9x2 – 12x + 16)
Section 5.6: Solving Polynomial Equations by Factoring
1. Zero-Product Rule: If a  b = 0, then either a = 0 or b = 0.
2. To solve a (polynomial) nonlinear equation, get all terms on one side, then factor and apply the zero
product rule.
a. Example:
x3 – 4x2
= 12x
3
2
x – 4x – 12x = 0
x(x3 – 4x2 – 12)= 0
x(x – 6)(x + 2) = 0
x = 0 or
x–6=0
or
x+2=0
x=0
x=6
x = -2
Section 5.7: Polynomial Division
1. Example:
2 x3  x  18
?
x3
2 x 2  6 x  17
x  3 2 x 3  0 x 2  x  18

 2 x3  6 x 2

6x2  x

 6 x 2  18 x

 17 x  18
  17 x  51
33
Intermediate Algebra Final Exam Review Sheet
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2 x3  x  18
33
 2 x 2  6 x  17 
x3
x3
Section 6.1: Rational Functions
1. A rational function is a function of the form f ( x) 
p( x)
, where p(x) and q(x) are polynomials and
q( x)
q(x)  0.
Section 6.2: Equivalent Fractions
ak a
 (b, k  0)
bk b
x 2  y 2 ( x  y)( x  y) ( x  y)
a. Example:


( x  y)2 ( x  y)( x  y) ( x  y)
1. The Fundamental Principle of Fractions:
Section 6.3: Multiplication and Division of Rational Expressions
a c ac
1. Multiplication of Fractions:  
(b, d  0)
b d bd
a c a d
2. Division of Fractions:    (b, c, d  0)
b d b c
x 3
( x  3)
4 2 x  1 4( x  3)(2 x  1)
a. Example:
 (8 x  4) 
 

4
2
2 x  5x  3
(2 x  1)( x  3) 1
1
( x  3)(2 x  1)
Section 6.4: Sums and Differences of Rational Expressions
1. Addition and Subtraction of Rational Expressions:
a b ab
 
(Note the need for a common
c c
c
denominator)
3x  1 2 x  1  3x  1 x  1  2 x  1 x  1



x 1 x 1
 x  1 x  1  x  1 x  1
2. Example: 

(3x 2  2 x  1)  (2 x 2  3x  1)
 x  1 x  1
x2  5x  2
 x  1 x  1
Section 6.5: Mixed Operations and Complex Fractions
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1. To simplify a compound fraction, multiply numerator and denominator by the L.C.D. of all fractions,
then simplify.
1 1 1 1
xy xy



yx
x y x y xy
x
y

 

Example:
1 1 1 1 xy xy xy y  x



x y x y
x
y
Section 6.6: Fractional Equations and Inequalities
1. To solve equations with fractional expressions, multiply both sides of the equation by the L.C.D.
1
1
5


x 1 x  2 4
1 
5
 1


  4  x  1 x  2    4  x  1 x  2 
4
 x 1 x  2 
4  x  2   4  x  1  5  x 2  x  2 
Example:
5 x 2  3 x  14  0
 5 x  7  x  2   0
x
7
5
or
x2
Section 6.7: Literal Equations
Key: Solve these types of equations the same as you would any other type of linear or nonlinear equation.
Section 6.8: Applications
See previous chapters’ notes on solving word problems.
Section 7.1: Natural Number and Integer Exponents
1. Rules of Exponents:
a. a m  a n  a m n
b.
c.
a 
m n
abn
 a mn
 a nb n
am
d.
 a mn
n
a
n
an
a
e.    n
b
b
0
f. a  1 for a  0
1
g. a  n  n for a  0 
a
Section 7.2: Scientific Notation
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Section 7.3: Rational Exponents and Radical Notation
1. a
2. a
3.
n
1
n
m
n a
 n am 
n
an 
 a
n
n
 a
n
m
a
Section 7.4: Simplifying Radical Expressions
1. n a n b  n ab
  
n
a
n
b
b
3. Simplest Radical Form:
a. Factors of radicand have exponents less than index.
b. No fractions under the radical.
c. No radicals in the denominator of a fraction.
d. Greatest common factor of the index and exponents of radicand is 1.
4. To rationalize the denominator, multiply numerator and denominator by the conjugate of the
denominator.
1
1
2 3
2 3
2 3
1
1
3
3
Example:
Example:




 2 3



2
3
43
3
3 3
2  3 2  3 2  3 22  3
2.
a
n
 
Section 7.5: Adding and Subtracting Radical Expressions
1. You can only combine terms with identical radicals.
2. You may be able to simplify the radical in some terms to get like terms.
Section 7.6: multiplying and Dividing Radical Expressions
1. Multiply out radical expressions like you would any other factors…
Section 7.7: Radical Functions and Equations
1. To solve equations with a radical, isolate the radical, then raise both sides to the appropriate power to
eliminate the radical.
Example:
5  x 1  x  2
5 x  x 3

5 x

2
  x  3
2
5  x  x2  6x  9
x2  5x  4  0
 x  1 x  4   0
x  1 or x  4
Section 7.8: Complex Numbers
1. i   1
Intermediate Algebra Final Exam Review Sheet
2.
3.
4.
5.
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i 2  1
Know how to add and subtract complex numbers.
Know how to multiply complex numbers.
Know how to divide complex numbers (multiply numerator and denominator by the conjugate of the
denominator).
Section 8.1: Quadratic Functions as Mathematical Models
Section 8.2: Solving Quadratic Equations: The Factoring and Square Root Methods
1. See section 5.5 for tips on factoring…
2. Example:
5 x 2  26 x  5  0
 5 x  1 x  5  0
x
1
or x  5
5
Section 8.3: Solving Quadratic Equations: Completing the Square
1. To solving a quadratic equation by completing the square:
b
c
x  0)
a
a
b. Complete the square by adding and subtracting the square of one-half the coefficient of x.
c. Write the equation as a square plus a constant
d. Solve for x.
3x 2  3x  7  0
a. Divide each side by a, the coefficient of the x2 term. (i.e., ax 2  bx  c  0  x 2 
7
0
3
1 1 7
 2
x  x    0
4 4 3

x2  x 
2
Example:
1  31

x   0
2  12

2
1  31

x  
2  12

x
1
31
1 31
1 93
1




93
2
12
2 3
2 9
6
1 1
x 
93
2 6
Section 8.4: Solving Quadratic Equations: The Quadratic Formula
b  b2  4ac
1. The solution to the quadratic equation ax + bx + c = 0 is x 
.
2a
2. b2 – 4ac is called the discriminant
2
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a. If b2 – 4ac > 0, then the equation has two distinct real solutions.
b. If b2 – 4ac = 0, then the equation has exactly one real solution.
c. If b2 – 4ac < 0, then the equation has two distinct complex solutions.
Example:
x2  x 1  0
x
x
1  12  4 1 1
2 1
1  5
 0.618 or
2
x
1  5
 1.618
2
Section 8.5: Equations Reducible to Quadratic Form
1. To solve “quadratic-like” equations, make a variable substitution where u = the middle term, and u2 =
the other term.
Example:
x4  5x2  4  0
Let u  x 2
u 2  5u  4  0
 u  1 u  4   0
u  1 or u  4
x 2  1 or
x 2  4
x  1 or
x  2i
Section 8.6: Graphing Quadratic Functions
3. Some facts about the quadratic function f(x) = ax2 + bx + c:
a. Its graph is a parabola.
b. Its standard form is f(x) = a(x – h)2 + k
c. The vertex, or extremum of its graph is at the point (h, k).
d. Its vertex is a maximum if a < 0.
e. Its vertex is a minimum if a > 0.
b
2a
g. The x intercepts (if there are any) are the solutions to the quadratic equation ax2 + bx + c = 0.
f. The x coordinate of the vertex (and the axis of symmetry) is given by h  
Example:
y intercept is (0, 2):
y  f (0)  0  0  2  2
1
Let f ( x)   x 2  3x  2
2
x intercepts are about (-.606, 0) and (6.606, 0):
Intermediate Algebra Final Exam Review Sheet
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y  f ( x)  0
1
 x 2  3x  2  0
2
 1
3  32  4     2 
3  13
 2
x

3
1
 1
2  
 2
Vertex is at (3, 6.5):
b
3
x

3
2a
 1
2  
 2
1
y   32  3(3)  2  6.5
2
13
Two extra points are (4, 6) and (2, 6):
1
x  4  y   42  3(4)  2  6
2
Section 8.7: Quadratic and Rational Inequalities
1. To solve a nonlinear inequality, get all terms on the left hand side and factor them. Find the intervals of
sign changes, then make a table or diagram to show the signs of each factor and the overall quantity.
x 2  5 x  6
x2  5x  6  0
Example:  x  2  x  3  0
Interval:
 x  2 :
 x  3 :
 x  2  x  3
Solution: 2  x  3
x2 2 x3 x3









Intermediate Algebra Final Exam Review Sheet
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Section 8.8: The Distance Formula: Circles
1. Distance Formula: d 
 x2  x1    y2  y1 
2
2
 x  x 2 y1  y 2 
2. Midpoint Formula: M   1
,

2 
 2
2
2
3. The equation x  h    y  k   r 2 is the standard form of the equation of a circle with center (h, k)
and radius r.
Section 9.1: More on Function Notation: Split Functions
 x  4 if x  2
1. Example of a split function: f ( x)  
 x  1 if x  2
Section 9.1: Composition and the Algebra of Functions
1. Addition: (f + g)(x) = f(x) + g(x)
2. Subtraction: (f – g)(x) = f(x) – g(x)
3. Multiplication: (fg)(x) = f(x)g(x)
f
f ( x)
4. Division:  ( x) 
provided g(x)  0
g ( x)
g
5. Composition: (f  g)(x) = f(g(x))
Example:
Let f ( x)  x  1 and g ( x)  x 2  2.
f
g  x   x 2  2  1  x 2  1
g
f  x  


2
x 1  2  x 1 2  x 1
Section 9.3: Types of Functions
1. Know how to graph a function by calculating and plotting points.
2. Know the graphs of some common functions:
a. Linear function: f ( x)  mx  b
b. Quadratic function: f ( x)  ax 2  bx  c
c. Polynomial functions: f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0. n is the degree of the
polynomial; a polynomial has at most n zeros and at most n – 1 local extrema.
d. Square root function: f ( x)  x
e. Absolute value function: f ( x)  x
3. Vertical shifts of graphs:
a. y = f(x) + c (c > 0) shifts the graph of y = f(x) upward by c units.
b. y = f(x) – c (c > 0) shifts the graph of y = f(x) downward by c units.
4. Horizontal shifts of graphs:
a. y = f(x – c) (c > 0) shifts the graph of y = f(x) to the right by c units.
b. y = f(x + c) (c > 0) shifts the graph of y = f(x) to the left by c units.
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Section 9.4: Inverse Functions
1. A one-to-one function is a function in which at most one x value is associated with a y value.
2. Horizontal Line Test: A function is one-to-one function if and only if no horizontal line intersects its
graph more than once.
3. Definition of the Inverse of a Function: Let f be a function with domain A and range B. Then its
inverse function f-1 has domain B and range A and is defined by f-1(y) = x  f(x) = y for any y in B.
4. Properties of Inverse Functions:
a. f-1(f(x)) = x
b. f(f-1 (x)) = x
5. Know how to find the inverse of a function:
a. Set y = f(x)
b. Solve for x
c. Switch variables x  y
d. y = f-1(x) now.
Example:
f ( x)  x  1
y  f ( x)  x  1
x
y 1
x2  y  1
y  f 1 ( x)  x 2  1
6. The graph of f-1 is obtained by reflecting the graph of f in the line y = x.
7. A point with coordinates (a, b) on the graph of f corresponds to a point with coordinates (b, a) on the
graph of f-1.
Section 9.5: Variation
1. Direct Variation: y = kx
k
x
3. Joint Variation: z  kxy
2. Inverse Variation: y 
Section 10.1: Exponential Functions
1. For a > 0, the exponential function with base a is defined by: f(x) = ax.
2. The domain is all reals.
3. The range (for a ≠ 1) is (0, ∞)
4. Common bases are a = 2, 10, and e.
5. The graph has one of the two following shapes (for a  1):
Intermediate Algebra Final Exam Review Sheet
This is for a > 1
Friday 12/10/04
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This is for 0 < a < 1
Section 10.2: Logarithms and Logarithmic Functions
1. y = loga x and ay = x are alternative ways of expressing the same relationship. A logarithm is just an
exponent.
Section 10.3: Properties of Logarithms
1. log a1  0
2. log a a  1
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3. log a a x  x
4. a log a x  x
5. log a  AB   log a A  log a B
 A
6. log    log a A  log a B
a B
7. log Ac  c log a A
a
 
Section 10.4: Common Logarithms, Natural Logarithms, and Change of Base
log a x
1. log b x 
log a b
2. Common Logarithm is the logarithm with base 10. It’s denoted by: log10 x = log x.
3. Natural Logarithm is the logarithm with base e. It’s denoted by: loge x = ln x.
Section 10.5: Exponential and Logarithmic Equations
1. To solve exponential equations (i.e., variable is in the exponent):
a. Isolate the exponential expression on one side of the equation.
b. Take the logarithm of both sides.
c. Solve for the variable.
2. To solve logarithmic equations:
a. Combine logarithmic terms into one and isolate that term on one side of the equation.
b. Write the equation in exponential form.
c. Solve for the variable.
Section 10.6: Applications: Exponential and Logarithmic Functions as Mathematical Models
1. pH: pH = -log[H3O+]
nt
 r
2. Compound interest: A  P  1  
 n
3. Continuously compounded interest: A  Pe rt
4. Exponential growth: n(t )  n0e rt
Section 11.1: 3 x 3 Linear Systems
1. Substitution still works…
2. Elimination still works…
Section 11.2: Solving Linear Systems Using Augmented Matrices
1. Convert the linear system to an augmented matrix as follows:
 x 3 y 3z  4
1 1 3 4 

2.  x 2 y 2 z  10  1 2 2 10 
3x  y 5 z  14

3 1 5 14
3. Perform elementary row operations to get the matrix into echelon form:
a. Add a multiple of a row to another
b. Multiply a row by a constant
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c. Interchange two rows
4. Echelon form:
a. The leading entry (first nonzero number) in each row is 1.
b. The leading entry in each row is to the right of the leading entry in the row immediately above.
c. Rows that are all zeros are at the bottom.
5. Use back-substitution to solve for remaining variables.