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College Algebra Final Exam Review Sheet
Wednesday 8/04/04
Page 1 of 13
Section 2.1: The Coordinate Plane
Distance Formula:
d  A, B 
x2  x1 2   y2  y1 2
Midpoint Formula:
 x  x 2 y1  y 2 
M  1
,

2 
 2
Section 2.2: Graphs of Equations
1. Know how to graph an equation by calculating and plotting points.
2. Know that the following types of graphs come from the following types of equations:
a. Line: y  mx  b
b. Vertical Parabola: y  ax 2  bx  c
c. Horizontal Parabola: y  ax  b  c
d. Absolute Value: y  ax  b  c
e. Circle: x  h    y  k   r 2
x intercepts are where a graph crosses the x-axis.
a. They can be calculated by setting y = 0 and solving for x.
y intercepts are where a graph crosses the y-axis.
a. They can be calculated by setting x = 0 and solving for y.
If a graph has symmetry with respect to the x-axis, the equation is unchanged when y is replaced by –y.
If a graph has symmetry with respect to the y-axis, the equation is unchanged when x is replaced by –x.
If a graph has symmetry with respect to the origin, the equation is unchanged when x is replaced by –x
and y is replaced by –y.
2
3.
4.
5.
6.
7.
2
Section 2.3: Graphing Calculators
If you have a graphing calculator, know how to enter an equation into it, how to change the window size, and
how to find x intercepts with it.
Section 2.4: Lines
Slope:
Point-slope form of the equation of a line:
Slope-intercept form of the equation of a line:
Two-intercept form of the equation of a line:
Horizontal line:
Vertical line:
Slope relation of parallel lines:
Slope relation of perpendicular lines:
y y 2  y1

x x2  x1
y  y1  mx  x1 
y  mx  b
x y
 1
a b
yb
xa
m1  m2
1
m1  
m2
m
College Algebra Final Exam Review Sheet
Wednesday 8/04/04
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Section 3.1: Algebraic and Graphical Solutions of Equations
1. To solve an equation algebraically, use the rules of algebra to isolate the unknown on one side.
2. To solve an equation graphically, move all terms to one side, set that quantity equal to y, graph y, then
read off the graph to see where y = 0 (i.e., find the x-intercepts).
Section 3.2: Modeling with Equations
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
1
d. Pyramid: V pyr  Bh
3
1
1
e. Cone: Vcone  Bh  r 2 h
3
3
distance
8. Speed problems: speed 
time
College Algebra Final Exam Review Sheet
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Section 3.3: Quadratic Equations
1. The quadratic equation: ax 2  bx  c  0
2. Procedure for Solving a Quadratic Equation by Factoring:
a. Collect all terms on the left side and simplify to the form ax 2  bx  c  0 .
b. Factor the quadratic expression.
c. Set each factor equal to zero. (because if p·q = 0, then either p = 0, or q = 0)
d. Solve the remaining linear equations.
e. Check the solutions in the original equation.
3. Procedure for Solving a Quadratic equation by Completing the Square:
2
a. Remember the product formula: x  p   x 2  2 px  p 2
b. Divide each side by a, the coefficient of the x2 term. (i.e., ax 2  bx  c  0  x 2 
b
c
x 0)
a
a
7
0
3
c. Complete the square by adding and subtracting the square of one-half the coefficient of x.
1 1 7
Example: x 2  x     0
4 4 3
d. Write the equation as a square plus a constant
Example: 3 x 2  3 x  7  0  x 2  x 
2
1
31

 0.
Example:  x   
2  12

e. Solve for x.
2
1
31

x   
2
12

1
31
1 31
1 93
1




93 .
2
12
2 3
2 9
6
1 1
x 
93
2 6
Example: x 
 b  b 2  4ac
4. The Quadratic Formula: x 
2a
5. b2 – 4ac is called the discriminant.
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.
Section 3.4: Complex Numbers
1. i   1
2. i 2  1
3. Know how to add and subtract complex numbers.
4. Know how to multiply complex numbers.
5. Know how to divide complex numbers.
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Section 3.5: Other Equations
1. Know how to solve higher-order polynomial equations by factoring.
2. Know how to solve higher-order polynomial equations by grouping.
3. Know how to solve equations with fractional expressions (multiply both sides by lowest common
denominator).
4. Know how to solve equations with radicals (isolate radical and raise both sides to appropriate power).
5. Know how to solve “quadratic type” equations (make a substitution that results in a quadratic equation).
Section 3.6: Linear Inequalities
1. A linear inequality: 4x  7  19
2. Solving a linear inequality is just like solving a linear equation, except:
a. Remember that multiplying each side of an inequality by the same negative number reverses the
direction of the inequality.
b. Remember that taking the reciprocal of each side of an inequality involving positive numbers
reverses the direction of the inequality.
Section 3.7: Nonlinear Inequalities
1. A nonlinear inequality: x  2x  3  0
2. To solve a nonlinear inequality, get all terms on the left hand side and factor them. Then use the fact
that:
a. The product or quotient of an even number of negative factors is positive.
b. The product or quotient of an odd number of negative factors is negative.
Section 3.8: Absolute Value
1. Absolute value equations:
a. x  c is equivalent to x   c
2. Absolute value inequalities:
a. x  c is equivalent to  c  x  c
b.
x  c is equivalent to  c  x  c
c.
x  c is equivalent to x  c or x  c
d.
x  c is equivalent to x  c or x  c
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Section 4.1: What is a Function?
1. Vocabulary: domain, range, image
2. Know how to evaluate a function at a number and at an expression
3. Know how to find the domain of a function:
a. x values where a denominator is equal to zero can not be in the domain
b. x values where a square root has a negative argument can not be in the domain
Section 4.2: Graphs of Functions
8. Know how to graph a function by calculating and plotting points.
9. Know the graphs of some common functions:
a. Linear function: f ( x)  mx  b
b. Power functions: f ( x)  x n for n = 2, 3, 4, 5, …
c. Root functions: f ( x)  n x for n = 2, 3, 4, 5, …
d. Reciprocal functions: f ( x)  1 n for n = 1, 2, 3, 4, …
x
e. Absolute value function: f ( x)  x
10. The vertical line test: A curve in the coordinate plane is the graph of a function if and only if no
vertical line intersects the curve more than once.
Section 4.3: Applied Functions: Variation
1. Direct Variation (Proportionality): y = kx.
2. Inverse Variation (Inverse Proportionality): y 
3. Joint Variation: z  kxy or z 
k
.
x
k
kx
or z 
xy
y
Section 4.4: Average Rate of Change: Increasing and Decreasing Functions
1. Average Rate of Change: The average rate of change of a function y = f(x) between x = a and x = b is
the slope of the secant line connecting the points (a, f(a)) and (b, f(b)):
y f (b)  f (a)

average rate of change =
.
x
ba
2. Increasing and Decreasing Functions:
a. A function f is increasing on an interval I if f(x1) < f(x2) whenever x1 < x2 in I.
b. A function f is decreasing on an interval I if f(x1) > f(x2) whenever x1 < x2 in I.
Section 4.5: Transformations of Functions
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.
5. Reflecting graphs:
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a. y = –f(x) reflects the graph of y = f(x) in the x-axis.
b. y = f(–x) reflects the graph of y = f(x) in the y-axis.
6. Vertical Stretching and Shrinking of Graphs:
a. y = af(x) (a > 1) stretches the graph of y = f(x) vertically by a factor of a.
b. y = af(x) (0 < a < 1) shrinks the graph of y = f(x) vertically by a factor of a.
7. Even and Odd Functions:
a. f is even if f(–x) = f(x) for all x in the domain of f. (The graph of f is symmetric with respect to
the y-axis.)
b. f is odd if f(–x) = –f(x) for all x in the domain of f. (The graph of f is symmetric with respect to
the origin.)
Section 4.6: Extreme Values of Functions
9. If you have a TI calculator, know how to find the extrema of a graph with it.
10. 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 extremum is a maximum if a < 0.
e. Its extremum is a minimum if a > 0.
b
b 2  4ac

f. It can be re-arranged into the form: y  a x 
 
2a 
4a

  b b 2  4ac 

g. Thus, (h, k) = 
,
4a 
 2a
2
Section 4.7: Combining Functions
6. Addition: (f + g)(x) = f(x) + g(x)
7. Subtraction: (f – g)(x) = f(x) – g(x)
8. Multiplication: (fg)(x) = f(x)g(x)
f
f ( x)
9. Division:  ( x) 
g ( x)
g
10. Composition: (f  g)(x) = f(g(x))
11. Be able to identify the domain of the resulting combined function…
Section 4.8: One-to-One Functions and Their Inverses
6. One-to-One Function: A function with domain A is called a one-to-one function if no two elements of
A have the same image, that is, f(x1)  f(x2) whenever x1 x2.
7. Horizontal Line Test: A function is one-to-one function if and only if no horizontal line intersects its
graph more than once.
8. 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.
9. Properties of Inverse Functions:
a. f-1(f(x)) = x
b. f(f-1 (x)) = x
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10. 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.
Section 5.1: Polynomial Functions and Their Graphs
6. Polynomial Functions: A polynomial of degree n is a function of the form
Px   a n x n  a n 1 x n 1  a n 2 x n 2    a 2 x 2  a1 x  a0
7. Know how to determine the end behavior of a polynomial (it’s determined by the anxn term…).
8. Know how to use zeros of a polynomial as a graphing aid (there are at most n of them).
9. Intermediate Value Theorem for Polynomials: If P is a polynomial function and P(a) and P(b) have
opposite signs, then there exists at least one value c between a and b for which P(c) = 0.
10. Local Extrema of Polynomials: If P is a polynomial of degree n, then the graph of P has at most n – 1
local extrema.
Section 5.2: Dividing Polynomials
3. Division Algorithm: Px  D( x)  Q( x)  R( x) . P is the dividend, D is the divisor, Q is the quotient,
and R is the remainder.
4. Know how to do long division of polynomials.
5. Know how to do synthetic division of polynomials.
6. Remainder Theorem: If the polynomial P(x) is divided by x – c, then the remainder is the value P(c).
7. Factor Theorem: c is a zero of P if and only if x – c is a factor of P.
Section 5.3: Real Zeros of Polynomials
3. All rational zeros of a polynomial P are of the form
p
, where p is a factor of the constant coefficient a0,
q
and q is a factor of the leading coefficient an.
4. Know how to use Descartes’ Rule of Signs.
5. Know how to use the upper and lower bounds theorem.
Section 5.4: The Fundamental Theorem of Algebra
1. The Fundamental Theorem of Algebra…
2. The Complete Factorization Theorem…
3. Zeros Theorem means that you have to keep factoring until you find n zeros…
4. Conjugate Zeros Theorem…
Section 5.5: Rational Functions
1. To graph rational functions:
a. Factor numerator and denominator
b. Find x and y intercepts
c. Find vertical and horizontal (or slant) asymptotes
d. Graph
College Algebra Final Exam Review Sheet
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Section 6.1: Exponential Functions
For a > 0, the exponential function with base a is defined by: f(x) = ax.
The domain is all reals.
The range (for a ≠ 1) is (0, ∞)
The graph has one of the two following shapes:
This is for a > 1
This is for 0 < a < 1
Section 6.2: The Natural Exponential Function
11. The natural exponential function is the exponential function f(x) = ex, where e ≈ 2.7182818.
nt
 r
12. Compound interest is calculated by the formula A(t )  P 1   .
 n
13. Continuously compounded interest is calculated by the formula A(t )  Pert .
14. Exponential growth is modeled by the formula n(t )  n0e rt .
Section 6.3: Logarithmic Functions
4. If a is a positive number and a ≠ 1, then the logarithmic function with base a, denoted as loga, is
defined by loga x = y  ay = x. (i.e., loga x is the exponent to which a must be raised to give x)
College Algebra Final Exam Review Sheet
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5.
6. Properties of Logarithms:
a. log a1  0
b. log a a  1
c. log a a x  x
d. a log a x  x
7. Common Logarithm is the logarithm with base 10. It’s denoted by: log10 x = log x.
8. Natural Logarithm is the logarithm with base e. It’s denoted by: loge x = ln x.
Section 6.4: Laws of Logarithms
3. log a  AB   log a A  log a B
 A
4. log    log a A  log a B
a B
5. log Ac  c log a A
a
 
6. log b x 
log a x
log a b
College Algebra Final Exam Review Sheet
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Section 6.5: Exponential and Logarithmic Equations
8. 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.
9. 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 6.6: Applications of Exponential and Logarithmic Functions
 r
11. Interest compounded n times per year: A(t )  P 1  
 n
rt
12. Interest compounded continuously: A(t )  Pe
13. Exponential growth: n(t )  n0e rt
nt
14. Radioactive decay: m(t )  m0e  rt . The half-life h and the relative rate of growth r are related as:
ln 2
r
h
15. Newton’s Law of Cooling: T (t )  Ts  D0e  kt
16. pH scale: pH = -log[H+]
I
17. Richter scale: M  log , S = 10-4 cm
S
I
18. Decibel scale:   log , I0 =10-12 W/m2
I0
Section 7.1: Systems of Equations
12. Substitution Method:
a. Solve for one variable
b. Substitute
c. Back-Substitute
13. Elimination Method:
a. Adjust coefficients so one variable has equal but opposite coefficients in both equations
b. Add the equations
c. Back-Substitute
14. Graphical Method:
a. Graph each equation
b. Find intersection points
Section 7.2: Pairs of Lines
11. A system of linear equations can have 0, 1, or infinitely many solutions.
 a x  b1 y  c1
b c bc
ac a c
12. 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
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13. Know how to model systems with a system of linear equations.
Section 7.3: Systems of Linear Equations
11. Linear equation in n variables converted to matrix form:
 x 3 y 3z  4
1 1 3 4 



 x 2 y 2 z  10  1 2 2 10 
3x  y 5 z  14

3 1 5 14 
12. Elementary row operations:
a. Add a multiple of a row to another
b. Multiply a row by a constant
c. Interchange two rows
13. 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.
14. Reduced echelon form:
a. Matrix is in echelon form.
b. Every number above and below each leading entry is a 0.
15. Inconsistent systems have no solutions. Dependent systems have infinitely many solutions.
Section 7.4: The Algebra of Matrices
8. Two matrices are equal if and only if they have the same dimension and each corresponding element is
equal.
9. To add two matrices, add each corresponding element.
10. To subtract two matrices, subtract each corresponding element.
11. To multiply a matrix by a scalar, multiply each element by the scalar.
 b1 
b 
2
12. If  a1 a2 ... an  is a row of matrix A, and   is a column of matrix B, then the inner product is the
 
 
bn 
number a1b1 + a2b2 + … + anbn.
13. The product of two matrices A and B is the matrix whose elements are formed by taking the inner
product of each row of A with each column of B.
14. The product of two matrices is not commutative.
15. Properties of matrix multiplication:
a. A(BC) = (AB)C (associative property)
b. A(B + C) = AB + CA (distributive property)
Section 7.5: Inverses of Matrices and Matrix Equations
6. The identity matrix In is the n x n matrix that has all zero entries except for 1 on the main diagonal.
7. The inverse of a square n x n matrix A is another square matrix A –1 that obeys the property
AA –1 = A –1A = In.
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a b 
1  d b 
8. Inverse of a 2 x 2 matrix: If A  
, then A1 

ad  bc  c a 
c d 
9. Inverse of an N x N matrix:
a. Form an augmented matrix by writing the identity matrix next to the given matrix.
b. Do row operations to transform the left hand matrix into the identity matrix.
c. The resulting matrix on the right is the inverse of the original matrix.
10. Solving a matrix equation: If A is a square n x n matrix (representing the matrix of coefficients of a
linear system), and X is a column matrix with n rows (representing the matrix of variables), and B is
another column matrix with n rows (representing the matrix of constant terms), then the solution of
AX = B is given by X = A –1B.
Section 7.6: Determinants and Cramer’s Rule
a b
1. Determinant of a 2 x 2 matrix:
 ad  bc
c d
2. Determinant of a 3 x 3 matrix:
a1
a2
a3
a1
a2
a3 a1 a2
b1
b2
b3  b1
b2
b3 b1 b2
c1
c2
c3
c2
c3 c1 c2
c1
 a1b2 c3  a2b3c1  a3b1c2  c1b2 a3  c2b3a1  c3b1a2
3. The minor Mij of the element aij of a square n x n matrix A is the determinant of the matrix obtained by
deleting the ith row and jth column of A.
4. The cofactor Aij of the element aij of a square n x n matrix A is (-1)i+jMij.
5. The determinant of a square matrix is calculated by multiplying each element in any row or column by
its cofactor, and then adding the results.
6. Invertibility criterion: detA ≠ 0.
7. Row and column transformations of a matrix do not change its determinant.
Dx
8. Cramer’s Rule: xi  i
D
Section 7.7: Systems of Equations and Inequalitites
2. Graphing Inequalities:
a. Graph equation
b. Test Points
c. Find intersection points for a system of inequalities
Section 7.8: Partial Fractions
The partial decomposition of a rational function is obtained by writing the rational function as the sum of
smaller rational functions, each of which has a single factor of the original function in its denominator.
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Section 9.1: Sequences and Summation Notation
A sequence is a function f whose domain is the set of natural numbers. The values f(1), f(2), … are called the
terms of the function.
A partial sum of a sequence is Sn = a1 + a2 + … + an.
n
In sigma notation, Sn =
a
k 1
k
.
Properties of sums:
n
n
n
k 1
k 1
  ak  bk    ak   bk
k 1
n
 ca
k 1
k


 c   ak 
 k 1 
n
Section 9.2: Arithmetic Sequences
An arithmetic sequence is of the form a, a + d, a + 2d, … or an = a + (n – 1)d.
n
n
An arithmetic sequence’s partial sum is given by S n   2a  (n  1)d    a  an  .
2
2
Section 9.3: Geometric Sequences
An arithmetic sequence is of the form a, ar, ar2, … or an = arn-1.
1 rn
An arithmetic sequence’s partial sum is given by Sn  a
.
1 r
Section 9.6: The Binomial Theorem
 n
 n
 n
 n
(a  b)n    a n    a n 1b    a n 2b 2  ...    b n .
 0
 1
 2
 n
 n
n!
 
 r  r !(n  r )!
Section 10.1: Counting
Know the Fundamental Theorem of Counting
Section 10.2: Permutations and Combinations
Know how to calculate permutations, permutations of n objects taken r at a time, distinguishable permutations,
and combinations.