Download College Algebra Notes Contents Joseph Lee Metropolitan Community College

College Algebra Notes
Joseph Lee
Metropolitan Community College
Contents
Introduction
2
Unit 1
Rational Expressions . . . . . . . . . . . . . . . . .
Quadratic Equations . . . . . . . . . . . . . . . . .
Polynomial, Radical, Rational, and Absolute Value
Linear and Absolute Value Inequalities . . . . . . .
Polynomial and Rational Inequalities . . . . . . . .
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3
3
9
12
18
24
Unit 2
Functions . . . . . . . . . . . . . . . . . . .
Extremum, Symmetry, Piecewise Functions,
Graphing Functions . . . . . . . . . . . . .
Operations on Functions . . . . . . . . . . .
Inverse Functions . . . . . . . . . . . . . . .
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and the Difference Quotient
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27
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30
37
44
47
Unit 3
Quadratic Functions . . . . .
Polynomial Functions . . . .
The Division Algorithm . . .
The Fundamental Theorem of
Rational Functions . . . . . .
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Equations
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51
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57
60
63
68
Unit 4
Exponential Functions . . . . . . . . . .
Logarithmic Functions . . . . . . . . . .
Properties of Logarithms . . . . . . . . .
Exponential and Logarithmic Equations
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72
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88
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94
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101
105
Unit 5
Gaussian Elimination
Matrices . . . . . . . .
Circles . . . . . . . . .
Ellipses . . . . . . . .
Hyperbolas . . . . . .
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Algebra
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Joseph Lee
Introduction
The question I receive most often, regardless of the course, is, “When am I ever going to use this?” I think
the question misses the point entirely. While I do not determine which classes students need to get their
degree, I do think it is a good policy that students are required to take my course – for more reasons than
just my continued employment, which I support as well.
If a student asked an English instructor why he or she had to read Willa Cather’s My Ántonia, the
instructor would not argue that understanding nineteenth century prairie life was essential to becoming a
competent tax specialist or licensed nurse. The instructor would not argue that reading My Ántonia would
benefit the student directly through a future application. Instead, the benefit of reading this beautiful piece
of American literature is entirely intrinsic. The mere enjoyment and appreciation is enough to justify its
place in a post-secondary education. Moreover, the results arrived to througout the course are as beautiful as any prose or poetry a student will encounter in his or her studies here at Metro or any other college.
2
College Algebra Notes
Joseph Lee
Unit 1
Rational Expressions
Domain of a Rational Expression
A rational expression will be defined as long as the denominator does not equal zero.
Example 1. State the domain of the rational expression.
x
x+3
Example 2. State the domain of the rational expression.
2x + 3
3x − 2
Example 3. State the domain of the rational expression.
x2
x−4
+ 5x + 6
Example 4. State the domain of the rational expression.
x2 + 8x + 7
x2 + 1
3
College Algebra Notes
Joseph Lee
Example 5. Simplify. State any domain restrictions.
4x − 8
x−2
Example 6. Simplify. State any domain restrictions.
x2
x−3
− 5x + 6
Example 7. Simplify. State any domain restrictions.
x2 − 14x + 49
x2 − 6x − 7
4
College Algebra Notes
Joseph Lee
Example 8. Multiply. State any domain restrictions.
3x2 − 5x − 2
2x
·
3x + 1
4x2 + 8x
Example 9. Multiply. State any domain restrictions.
x2 + x − 6 x2 + 4x + 4
· 2
4 − x2
x + 4x + 3
Example 10. Divide. State any domain restrictions.
2x2 − 9x − 5
4x2 − 1
÷
2x2 − 13x + 15 4x2 − 8x + 3
5
College Algebra Notes
Joseph Lee
Example 11. Add. State any domain restrictions.
3x − 3
x2 − 5x
+ 2
2
x − 7x + 12 x − 7x + 12
Example 12. Add. State any domain restrictions.
x2
x+3
x−1
+ 2
+ x − 20 x − 5x + 4
6
College Algebra Notes
Joseph Lee
Example 13. Subtract. State any domain restrictions.
4
x − 26
− 2
x + 2 x − 3x − 10
Example 14. Simplify the complex rational expression. State any domain restrictions.
2
x
4
1− 2
x
1−
7
College Algebra Notes
Joseph Lee
Example 15. Simplify the complex rational expression. State any domain restrictions.
x
+2
x+3
2
x+
x+3
Example 16. Simplify the complex rational expression. State any domain restrictions.
9
6
+
x x2
1 12
1− − 2
x x
1+
8
College Algebra Notes
Joseph Lee
Quadratic Equations
Definition: Quadratic Equation
A quadratic equation is an equation that can be written as
ax2 + bx + c = 0
where a, b, and c are real numbers and a 6= 0.
Zero Factor Property
If a · b = 0, then a = 0 or b = 0.
Example 1. Solve.
x2 − 5x + 6 = 0
Example 2. Solve.
3x(x − 2) = 4(x + 1) + 4
Square Root Property
√
If x2 = a, then x = ± a.
9
College Algebra Notes
Joseph Lee
Example 3. Solve.
3x2 + 4 = 58
Example 4. Solve.
(x − 3)2 = 4
Example 5. Solve.
(2x − 1)2 = −5
10
College Algebra Notes
Joseph Lee
Quadratic Formula
For any quadratic equation ax2 + bx + c = 0,
x=
−b ±
√
b2 − 4ac
2a
Example 6. Solve.
3x2 − 5x − 2 = 0
Example 7. Solve.
x2 − 3x − 7 = 0
11
College Algebra Notes
Joseph Lee
Polynomial, Radical, Rational, and Absolute Value Equations
Example 1. Solve.
x3 − 16x = 0
Example 2. Solve.
8x3 + 6x = 12x2 + 9
Example 3. Solve.
x+1=
√
12
x + 13
College Algebra Notes
Joseph Lee
Example 4. Solve.
p
x2 − x + 3 − 1 = 2x
Example 5. Solve.
√
x−1=
√
13
2x + 2
College Algebra Notes
Joseph Lee
Example 6. Solve.
(x − 1)2/3 = 4
Example 7. Solve.
x6 − 6x3 + 9 = 0
14
College Algebra Notes
Joseph Lee
Example 8. Solve.
x−2 + 2x−1 − 15 = 0
Example 9. Solve.
6
8
−
+1=0
(x − 4)2 x − 4
15
College Algebra Notes
Joseph Lee
Definition. Absolute Value.
The absolute value of a real number x is the distance between 0 and x on the real number line.
The absolute value of x is denoted by |x|.
Example 10. Solve.
|x| = 7
Observation 1
For any nonnegative value k, if |x| = k, then x = k or x = −k.
Example 11. Solve.
|x − 3| = 2
Example 12. Solve.
|3x + 5| = 8
16
College Algebra Notes
Joseph Lee
Example 13. Solve.
|x + 2| = −3
Observation 2
For any negative value k, the equation |x| = k has no solution.
Example 14. Solve.
|x + 9| − 3 = 1
Example 15. Solve.
−2|3x + 2| + 1 = 0
17
College Algebra Notes
Joseph Lee
Linear and Absolute Value Inequalities
Definition: Union and Intersection
Let A and B be sets.
The union of A and B, denoted A ∪ B is the set of all elements that are members of A, or B, or
both.
The intersection of A and B, denoted A ∩ B is the set of all elements that are members of both
A and B.
Example 1. Let A = {1, 2, 3} and B = {2, 4, 6}. Determine both A ∪ B and A ∩ B.
A∪B =
A∩B =
Example 2. Let B = {2, 4, 6} and C = {1, 3, 5}. Determine both B ∪ C and B ∩ C.
B∪C =
B∩C =
Example 3. Let D = {x | 0 < x < 4} and E = {x | 2 < x < 6}. Determine both D ∪ E and D ∩ E.
D∪E =
D∩E =
Interval Notation
For any real numbers a and b, the following are sets written in interval notation.
(a, b) = {x | a < x < b}
(a, b] = {x | a < x ≤ b}
[a, b) = {x | a ≤ x < b}
[a, b] = {x | a ≤ x ≤ b}
18
College Algebra Notes
Joseph Lee
Example 4. Write the following sets in interval notation.
{x | − 3 ≤ x < 5} =
{x | 7 < x ≤ 10} =
Example 5. Write the following sets in set-builder notation.
(3, 8) =
[−2, 5] =
Unbounded Intervals
(a, ∞) = {x | x > a}
[ a, ∞) = {x | x ≥ a}
(−∞, b) = {x | x < b}
(−∞, b ] = {x | x ≤ b}
Example 6. Write the following sets in interval notation.
{x | x ≥ −2} =
{x | x < −2} =
Example 7. Let A = (1, 4) and B = (2, 5). Determine both A ∪ B and A ∩ B.
A∪B =
A∩B =
Example 8. Let B = (2, 5) and C = [3, 6]. Determine both B ∪ C and B ∩ C.
B∪C =
B∩C =
19
College Algebra Notes
Joseph Lee
Example 9. Let D = (0, 4] and E = [5, 9). Determine both D ∪ E and D ∩ E.
D∪E =
D∩E =
Example 10. Solve.
3x − 7 < 5
Solution:
Example 11. Solve.
−2x − 7 ≤ 19
Solution:
Example 12. Solve.
1 < 4x − 3 ≤ 11
Solution:
20
College Algebra Notes
Joseph Lee
Example 13. Solve.
−2 ≤
1 − 2x
≤3
3
Solution:
Example 14. Solve.
|x| < 4
Solution:
Observation 3
For any nonnegative value k, the inequality |x| < k may be expressed as
−k < x < k.
Similarly, for |x| ≤ k, we have −k ≤ x ≤ k.
Example 15. Solve.
|x| > 4
Solution:
Observation 4
For any nonnegative value k, the inequality |x| > k may be satisfied by either
x>k
or x < −k.
Similarly, for |x| ≥ k, we know x ≥ k or x ≤ −k.
21
College Algebra Notes
Joseph Lee
Example 16. Solve.
|x + 8| ≤ 2
Solution:
Example 17. Solve.
|6x + 2| ≥ 2
Solution:
Example 18. Solve.
|4 − x| < 8
Solution:
22
College Algebra Notes
Joseph Lee
Example 19. Solve.
|1 − 7x| > 13
Solution:
Example 20. Solve.
|x − 3| > −2
Solution:
Observation 5
For any negative value k, the inequality |x| > k holds for any value of x.
Example 21. Solve.
|3x + 2| < −5
Solution:
Observation 6
For any negative value k, the inequality |x| < k has no solution.
23
College Algebra Notes
Joseph Lee
Polynomial and Rational Inequalities
Example 1. Solve.
x2 − 7x + 12 = 0
Solution:
Example 2. Solve.
x2 − 7x + 12 > 0
Solution:
Example 3. Solve.
x2 + x ≤ 20
Solution:
24
College Algebra Notes
Joseph Lee
Example 4. Solve.
4x2 ≥ 4x + 3
Solution:
Example 5. Solve.
x−3
≥0
x+4
Solution:
25
College Algebra Notes
Joseph Lee
Example 6. Solve.
2x2 − 5x + 3
≥0
2−x
Solution:
Example 7. Solve.
x
≥2
x+4
Solution:
26
College Algebra Notes
Joseph Lee
Unit 2
Functions
Definition: Relation
A relation is a correspondence between two sets.
Elements of the second set are called the range.
Elements of the first set are called the domain.
Definition: Function
A function is a specific type of a relation where each element in the domain corresponds to exactly one element in the range.
Example 1. Determine the domain and range of the following relation1 . Does the relation define a
function?
{(Joseph, turkey), (Joseph, roast beef), (Michael, ham)}
Domain:
Range:
Function?
Example 2. Determine the domain and range of the following relation. Does the relation define a
function?
{(1, 3), (2, 4), (−1, 1)}
Domain:
Range:
Function?
Example 3. Determine the domain and range of the following relation. Does the relation define a
function?
{(3, 5), (4, 5), (5, 5)}
Domain:
Range:
Function?
1
This relation relates math instructors and the sandwiches they enjoy.
27
College Algebra Notes
Joseph Lee
Example 4. Determine whether the equation defines y as a function of x.
x2 + y = 1
Example 5. Determine whether the equation defines y as a function of x.
x + y2 = 1
Example 6. Determine whether the equation defines y as a function of x.
x2 + y 2 = 1
Example 7. Determine whether the equation defines y as a function of x.
x3 + y 3 = 1
28
College Algebra Notes
Joseph Lee
Example 8. Evaluate the function for the given values.
f (x) = x2 + 2x + 1
f (4) =
f (−x) =
f (x + h) =
Example 9. Evaluate the function for the given values.
f (x) = x2 − x − 6
f (−3) =
f (−x) =
f (x + h) =
Example 10. Evaluate the function for the given values.
f (x) = x3 − 3x2 + 3x − 1
f (2) =
f (−x) =
29
College Algebra Notes
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Quotient
Increasing Functions, Decreasing Functions, Constant Functions
Let f be a function and (a, b) be some interval in the domain of f . The function is called
• increasing over (a, b) if f (x) < f (y) for every x < y,
• decreasing over (a, b) if f (x) > f (y) for every x < y, and
• constant over (a, b) if f (x) = f (y) for every x and y
(where a < x < y < b).
Example 1. Determine over which intervals the function f is increasing, decreasing, or constant.
Increasing:
Decreasing:
Constant:
Relative Maximum:
Relative Minimum:
Domain:
Range:
Zeros of the function:
30
College Algebra Notes
Joseph Lee
Example 2. Determine over which intervals the function f is increasing, decreasing, or constant.
Increasing:
Decreasing:
Constant:
Relative Maximum:
Relative Minimum:
Domain:
Range:
Example 3. Determine over which intervals the function f is increasing, decreasing, or constant.
Increasing:
Decreasing:
Constant:
31
College Algebra Notes
Joseph Lee
Even and Odd Functions
A function f is called even if
f (−x) = f (x).
A function f is called odd if
f (−x) = −f (x).
Example 4. Determine if f is even, odd, or neither.
f (x) = x2 − 4
Example 5. Determine if g is even, odd, or neither.
g(x) = x3 − 2x
Example 6. Determine if h is even, odd, or neither.
h(x) = (x − 2)2
32
College Algebra Notes
Joseph Lee
Example 7. Evaluate the piecewise function.


2x + 8
f (x) = x2


1
if x ≤ −2
if − 2 < x ≤ 1
if x > 1
f (−3) =
f (−1) =
f (2) =
f (4) =
Example 8. Evaluate the piecewise function.
(
x
f (x) =
−x
if x ≥ 0
if x < 0
f (−2) =
f (−1) =
f (1) =
f (2) =
Example 9. Graph the piecewise function.
(
x+2
f (x) =
1
if x ≤ 0
if x > 0
y
3
2
1
−3
−2
−1
1
−1
33
2
3
x
College Algebra Notes
Joseph Lee
Example 10. Graph the piecewise function.
(
x
f (x) =
−x
if x ≥ 0
if x < 0
y
3
2
1
−3
−2
−1
1
2
3
x
−1
Example 11. Graph the piecewise function.


2x + 8
f (x) = x2


1
if x ≤ −2
if − 2 < x ≤ 1
if x > 1
y
4
3
2
1
−3
−2
−1
1
−1
34
2
3
x
College Algebra Notes
Joseph Lee
Difference Quotient
For a function f (x), the difference quotient is
f (x + h) − f (x)
,
h
h 6= 0.
Example 12. Find the difference quotient of the given function.
f (x) = 2x + 3
Example 13. Find the difference quotient of the given function.
f (x) = 5x − 6
35
College Algebra Notes
Joseph Lee
Example 14. Find the difference quotient of the given function.
f (x) = x2 + 1
Example 15. Find the difference quotient of the given function.
f (x) = x2 − 4x
36
College Algebra Notes
Joseph Lee
Graphing Functions
The function f (x) = x2 is called the square function.
x
−2
−1
0
1
2
f (x)
4
1
0
1
4
The function f (x) = x3 is called the cube function.
x
−2
−1
0
1
2
The function f (x) =
x
0
1
4
9
f (x)
−8
−1
0
1
8
√
x is called the square root function.
f (x)
0
1
2
3
37
College Algebra Notes
The function f (x) =
x
−8
−1
0
1
8
Joseph Lee
√
3
x is called the cube root function.
f (x)
−2
−1
0
1
2
The function f (x) = |x| is called the absolute value function.
x
−2
−1
0
1
2
f (x)
2
1
0
1
2
Transformations of f (x)
f (x) + c
f (x) − c
f (x + c)
f (x − c)
−f (x)
f (−x)
cf (x)
f (cx)
vertical shift up c units
vertical shift down c units
horizontal shift left c units
horizontal shift right c units
reflection over the x-axis
reflection over the y-axis
vertical stretch or compression by a factor of c
horizontal compression or stretch by a factor of c
38
College Algebra Notes
Joseph Lee
Example 1. Graph g(x) = x2 + 1.
Let f (x) = x2 . Note g(x) =f (x) + 1.
x
−2
−1
0
1
2
f (x)
4
1
0
1
4
g(x)
5
2
1
2
5
Example 2. Graph g(x) = (x − 2)2 .
Let f (x) = x2 . Note g(x) =f (x − 2).
x−2
−2
−1
0
1
2
f (x − 2)
4
1
0
1
4
x
0
1
2
3
4
g(x)
4
1
0
1
4
Example 3. Graph h(x) = (x + 4)2 + 1.
Let f (x) = x2 and g(x) = (x + 4)2 .
Note h(x) =g(x) + 1 =f (x + 4) + 1.
x+4
−2
−1
0
1
2
f (x + 4)
4
1
0
1
4
x
−6
−5
−4
−3
−2
g(x)
4
1
0
1
4
x
−6
−5
−4
−3
−2
h(x)
5
2
1
2
5
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College Algebra Notes
Joseph Lee
Example 4. Graph k(x) = −(x − 3)2 − 1.
Let f (x) = x2 , g(x) = (x − 3)2 , and h(x) = −(x − 3)2 .
Note k(x) =h(x) − 1 =−g(x) − 1 =−f (x − 3) − 1.
x−3
−2
−1
0
1
2
Example 5. Graph g(x) =
Let f (x) =
x
0
1
4
9
f (x)
0
1
2
3
√
√
f (x − 3)
4
1
0
1
4
x
1
2
3
4
5
g(x)
4
1
0
1
4
x − 2.
x. Note g(x) =f (x) − 2.
g(x)
−2
−1
0
1
40
x
1
2
3
4
5
h(x)
−4
−1
0
−1
−4
x
1
2
3
4
5
k(x)
−5
−2
−1
−2
−5
College Algebra Notes
Joseph Lee
Example 6. Graph g(x) =
Let f (x) =
x−2
0
1
4
9
√
√
x − 2.
x. Note g(x) =f (x − 2).
f (x)
0
1
2
3
x
2
3
6
11
g(x)
0
1
2
3
√
Example 7. Graph k(x) = − x + 1 + 2.
Let f (x) =
√
x, g(x) =
√
√
x + 1, and h(x) = − x + 1.
Note k(x) =h(x) + 2 =−g(x) + 2 =−f (x + 1) + 2.
x+1
0
1
4
9
f (x + 1)
0
1
2
3
x
−1
0
3
8
g(x)
0
1
2
3
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x
−1
0
3
8
h(x)
0
−1
−2
−3
x
−1
0
3
8
k(x)
2
1
0
−1
College Algebra Notes
Joseph Lee
Example 8. Graph h(x) = (x − 5)3 − 2.
Let f (x) = x3 and g(x) = (x − 5)3 .
Note h(x) =g(x) − 2 =f (x − 5) − 2.
x−5
−2
−1
0
1
2
f (x − 5)
−8
−1
0
1
8
x
3
4
5
6
7
g(x)
−8
−1
0
1
8
x
3
4
5
6
7
h(x)
−10
−3
−2
−1
6
Example 9. Graph k(x) = −|x + 2| + 1.
Let f (x) = |x|, g(x) = |x + 2|, and h(x) = −|x + 2|.
Note k(x) =h(x) + 1 =−g(x) + 1 =−f (x + 2) + 1.
x+2
−2
−1
0
1
2
f (x − 3)
2
1
0
1
2
x
−4
−3
−2
−1
0
g(x)
2
1
0
1
2
42
x
−4
−3
−2
−1
0
h(x)
−2
−1
0
−1
−2
x
−4
−3
−2
−1
0
k(x)
−1
0
1
0
−1
College Algebra Notes
Joseph Lee
Example 10. The graph of the function f is given below.
(a) Graph g(x) = f (x) − 2.
(b) Graph h(x) = f (x + 2).
(c) Graph k(x) = −f (x − 1) + 2.
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Operations on Functions
Basic Operations on Functions
Let f (x) and g(x) be functions. The following basic operations of addition, subtraction, multiplication, and division may be performed on the functions as follows:
• (f + g)(x) = f (x) + g(x)
• (f − g)(x) = f (x) − g(x)
• (f · g)(x) = f (x) · g(x)
f
f (x)
•
(x) =
g
g(x)
If the domain of f (x) is A and the domain of g(x) is B, then the domain of f + g, f − g, and f · g is A ∩ B.
The domain of f /g is A ∩ B restricted for any x values such that g(x) = 0.
Example 1. Let f (x) = 3x − 2 and g(x) = x + 7. Find f + g, f − g, f · g, and f /g. State the domain of
each function.
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Composition of Functions
Let f (x) and g(x) be functions. The composition of f and g, denoted f ◦ g, is given by
(f ◦ g)(x) = f (g(x)).
Example 2. Let f (x) = 3x − 2 and g(x) = x + 7. Find f ◦ g and g ◦ f . State the domain of each function.
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College Algebra Notes
Example 3. Let f (x) =
Joseph Lee
1
x
and g(x) =
. Find f ◦ g and g ◦ f . State the domain of each function.
x+7
x+3
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College Algebra Notes
Joseph Lee
Inverse Functions
Inverse Functions
Two functions f and g are called inverse functions if
(f ◦ g)(x) = (g ◦ f )(x) = x.
Example 1. Verify that f and g are inverse functions. Graph both f and g.
f (x) = 3x + 4
g(x) =
x−4
3
Example 2. Verify that f and g are inverse functions. Graph both f and g.
√
f (x) = x3 + 2
g(x) = 3 x − 2
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Recall the definition of a function:
One-to-one Function
A function f is called one-to-one if each element in the range corresponds to exactly one element
in the domain. If a function is one-to-one, then it has an inverse function.
Example 3. Which of the following functions are one-to-one?
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College Algebra Notes
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Example 4. Determine the inverse of the one-to-one function. State the domain and range of f and f −1 .
1
f (x) = x − 3
2
Example 5. Determine the inverse of the one-to-one function. State the domain and range of f and f −1 .
f (x) = x2 − 3,
49
x≥0
College Algebra Notes
Joseph Lee
Example 6. Determine the inverse of the one-to-one function. State the domain and range of f and f −1 .
√
f (x) = x − 4
Example 7. Determine the inverse of the one-to-one function. State the domain and range of f and f −1 .
√
f (x) = 3 x + 2
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College Algebra Notes
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Unit 3
Quadratic Functions
Vertex
The vertex of a parabola is the point where the parabola achieves its minimum or maximum value.
Example 1. Graph f (x) = x2 .
Vertex:
Example 2. Graph f (x) = (x + 4)2 − 1.
Vertex:
Example 3. Graph f (x) = 2(x − 5)2 + 3.
Vertex:
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College Algebra Notes
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Example 4. Graph f (x) = −3(x − 3)2 − 1.
Vertex:
Standard and General Form of a Parabola
A quadratic function is said to be in standard form if it is written as
f (x) = a(x − h)2 + k.
A quadratic function is said to be in general form if it is written as
f (x) = ax2 + bx + c.
Example 5. Graph f (x) = x2 + 8x + 15.
Vertex:
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Example 6. Graph f (x) = x2 + 6x + 7.
Vertex:
Example 7. Graph f (x) = x2 − 8x + 19.
Vertex:
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Joseph Lee
Example 8. Graph f (x) = 2x2 − 4x − 3.
Vertex:
Example 9. Graph f (x) = x2 − 5x + 1.
Vertex:
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Joseph Lee
Example 10. Write the quadratic function f (x) = ax2 + bx + c in standard form.
Vertex:
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Joseph Lee
b
.
Example 11. Evaluate the quadratic function f (x) = ax2 + bx + c for x = − 2a
Vertex Formula
For any quadratic function f (x) = ax2 + bx + c, the vertex is located at
b
b
.
− ,f −
2a
2a
Example 12. Graph f (x) = 3x2 − 6x + 2.
Example 13. Graph f (x) = −2x2 − 7x + 5.
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College Algebra Notes
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Polynomial Functions
Polynomial Function
A polynomial function is a function of the form
f (x) = an xn + an−1 xn−1 + ... + a2 x2 + a1 x + a0 ,
where n is a nonnegative integer and each ai is a real number. Assuming an 6= 0, the degree of the
polynomial function is n and an is called the leading coefficient.
Example 1. Graph the following power functions.
a. f (x) = x
b. f (x) = x2
c. f (x) = x3
d. f (x) = x4
e. f (x) = x5
f. f (x) = x6
g. f (x) = x7
End Behavior
The end behavior of a function is the value f (x) approaches as x approaches −∞ or as x approaches ∞.
Example 2. Identify the end behavior of each of the power functions in Example 1.
Power Function
f (x) = x
f (x) = x2
f (x) = x3
f (x) = x4
f (x) = x5
f (x) = x6
f (x) = x7
x −→ −∞
f (x) −→
f (x) −→
f (x) −→
f (x) −→
f (x) −→
f (x) −→
f (x) −→
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x −→ ∞
f (x) −→
f (x) −→
f (x) −→
f (x) −→
f (x) −→
f (x) −→
f (x) −→
College Algebra Notes
Joseph Lee
End Behavior of Any Polynomial Function
The end behavior of any polynomial function is the same as the end-behavior of its highest degree
term.
Zero of a Function
If f (c) = 0, then c is called a zero of the function.
If c is a zero of a function, then (c, 0) is an x-intercept on the graph of the function.
Example 3. Sketch a graph of f (x) = (x + 4)(x + 1)(x − 2).
Example 4. Sketch a graph of f (x) = (x + 4)(x + 1)2 (x − 2).
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College Algebra Notes
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Example 5. Sketch a graph of f (x) = (x + 4)2 (x + 1)3 (x − 2).
Multiplicity of a Zero
If (x − c)n is a factor of f (x), but (x − c)n+1 is not a factor of f (x), then c is a zero of multiplicity n.
If c is a zero of multiplicity n, then:
• if n is odd, the graph crosses the x-axis,
• if n is even, the graph touches the x-axis, but does not cross.
The Intermediate Value Theorem
If f (x) is a polynomial function and a and b are real numbers with a < b, then if either
• f (a) < 0 < f (b), or
• f (b) < 0 < f (a),
then there exists a real number c such that a < c < b and f (c) = 0.
Example 6. Use the Intermediate Value Theorem to verify f (x) = x3 + x + 1 has a zero on the closed
interval [−1, 0].
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College Algebra Notes
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The Division Algorithm
The Division Algorithm
Let p(x) be a polynomial of degree m and let d(x) be a nonzero polynomial of degree n where
m ≥ n. Then there exists unique polynomials q(x) and r(x) such that
p(x) = d(x) · q(x) + r(x)
where the degree of q(x) is m − n and the degree of r(x) is less than n. The polynomial d(x) is called the
divisor, q(x) is called the quotient, and r(x) is called the remainder.
Example 1. Use long division to divide
guaranteed by the Division Algorithm.
x2 − 3x − 6
. State the quotient, q(x), and remainder, r(x),
x+4
x4 − 4x3 + 6x2 − 4x + 1
. State the quotient, q(x), and remainder,
x−1
r(x), guaranteed by the Division Algorithm.
Example 2. Use long division to divide
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College Algebra Notes
Example 3. Use long division to divide
anteed by the Division Algorithm.
Joseph Lee
x3 + x + 1
. State the quotient, q(x), and remainder, r(x), guarx+1
x3 − 3x2 − 10x + 24
. State the quotient, q(x), and remainx−2
der, r(x), guaranteed by the Division Algorithm.
Example 4. Use synthetic division to divide
Example 5. Use synthetic division to divide
x2 − 10x + 24
. State the quotient, q(x), and remainder,
x+5
r(x), guaranteed by the Division Algorithm.
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College Algebra Notes
Example 6. Use synthetic division to divide
guaranteed by the Division Algorithm.
Joseph Lee
x3 − 7x + 12
. State the quotient, q(x), and remainder, r(x),
x−3
The Remainder Theorem
Let p(x) be a polynomial. Then p(c) = r(x) where r(x) is the remainder guarenteed from the division algorithm with d(x) = x − c.
Example 7. Evaluate f (x) = x4 − 3x3 + 5x2 − 7x + 8 for f (2) using the remainder theorem.
Example 8. Evaluate f (x) = x5 + 4x3 − 9x + 2 for f (−1) using the remainder theorem.
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College Algebra Notes
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The Fundamental Theorem of Algebra
The Factor Theorem
Let p(x) and d(x) be polynomials. If r(x) = 0 by the division algorithm, then d(x) is a factor of
p(x).
Example 1. Use the remainder theorem to verify that −3 is a zero of f (x) = x3 − 3x2 − 10x + 24. Then
find all other zeros.
Example 2. Use the remainder theorem to verify that 7 is a zero of f (x) = x3 − 5x2 − 13x − 7. Then find
all other zeros.
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College Algebra Notes
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The Rational Zeros Theorem
Let p(x) be a polynomial function with integer coefficients:
p(x) = an xn + an−1 xn−1 + ... + a1 x + a0 .
Then any rational zero of the polynomial will be of the form
±
factor of a0
factor of an
leading coefficient an and constant term a0 .
Example 3. List all possible rational zeros for f (x) = x3 − 3x2 − 10x + 24 given by the Rational Zeros
Theorem.
Example 4. List all possible rational zeros for f (x) = 2x3 + 3x2 − 32x + 15 given by the Rational Zeros
Theorem.
Descartes’ Rule of Signs
Let p(x) be a polynomial function.
The number of positive real zeros is equal to or less than by an even number the number of sign
changes of p(x).
The number of negative real zeros is equal to or less than by an even number the number of sign
changes of p(−x).
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College Algebra Notes
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The Fundamental Theorem of Algebra
Let p(x) be a polynomial function of degree n.
plicities.
Then p(x) has n complex zeros, including multi-
Example 5. Find all zeros of the function f (x) = x3 − 4x2 + x + 6.
Example 6. Find all zeros of the function f (x) = x3 + 7x2 + 16x + 12.
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Example 7. Find all zeros of the function f (x) = x4 − 4x3 − 19x2 + 46x − 24.
Example 8. Find all zeros of the function f (x) = x4 − x3 − 2x2 − 4x − 24.
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Complex Conjugate Theorem
Let p(x) be a polynomial with real coefficients. If a + bi is a zero of the polynomial, then its
complex conjugate a − bi is also a zero of the polynomial.
Example 9. Find a third degree polynomial f (x) with zeros of i and 3 such that f (0) = −3.
Example 10. Find a third degree polynomial f (x) with zeros of 1 + i and −1 such that f (1) = 2.
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Rational Functions
Example 1. State the domain of the rational function.
f (x) =
x2
x−1
−x−6
Example 2. Graph the rational function.
f (x) =
1
x
Example 3. Graph the rational function.
f (x) =
1
+2
x−3
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Vertical Asymptotes
Let r(x) =
n(x)
be a simplified rational function. If c is a zero of d(x), then x = c is a vertical
d(x)
asymptote.
Horizontal Asymptotes
Let r(x) =
n(x)
be a rational function.
d(x)
1. If the degree of the denominator, d(x), is greater than the degree of the numerator, n(x), then the
line y = 0 is the horizontal asymptote.
2. If the degree of the denominator, d(x), is equal to the degree of the numerator, n(x), then the line
y = ab is the horizontal asymptote, where a is the leading coefficient of n(x) and b is the leading
coefficent of d(x).
3. If the degree of the denominator, d(x), is less than the degree of the numerator, n(x), then there is
no horizontal asymptote.
Holes
f (x) · n(x)
Let r(x) =
be a rational function. If c is a zero of f (x), then there is a hole at
f (x) · d(x)
Example 4. Find any vertical or horizontal asymptotes. Identify any holes in the graph.
f (x) =
3x
x2 − 9
69
n(c)
c,
.
d(c)
College Algebra Notes
Joseph Lee
Example 5. Find any vertical or horizontal asymptotes. Identify any holes in the graph.
f (x) =
3x2
x2 − 9
Example 6. Find any vertical or horizontal asymptotes. Identify any holes in the graph.
f (x) =
3x3
x2 − 9
Example 7. Find any vertical or horizontal asymptotes. Identify any holes in the graph.
f (x) =
3x + 9
x2 − 9
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Example 8. Graph the rational function.
f (x) =
x2
1
+x−6
Example 9. Graph the rational function.
f (x) =
2x
x2 − 4
Example 10. Graph the rational function.
f (x) =
x2
71
1
+1
College Algebra Notes
Joseph Lee
Unit 4
Exponential Functions
Exponential Function
The function
f (x) = bx ,
where b > 0 and b 6= 1, is called an exponential function.
Example 1. Graph f (x) = 2x . State its domain and range.
Example 2. Graph f (x) = 2x + 3. State its domain and range.
Example 3. Graph f (x) = 2x+3 . State its domain and range.
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Example 4. Graph f (x) = 2−x . State its domain and range.
Example 5. Graph f (x) = 3x+2 − 4. State its domain and range.
Natural Base
Consider the expression 1 +
1 n
n
for various values of n. See the table below.
n
1+
1
10
100
1,000
10,000
100,000
1 n
n
2
2.59374
2.70481
2.71692
2.71815
2.71827
n
As n gets bigger, the expression 1 + n1 gets bigger as well, but this sequence has an upper bound. This
particular upper bound is called the natural base, e.
e = 2.71828...
Example 6. Graph f (x) = ex . State its domain and range.
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College Algebra Notes
Joseph Lee
Example 7. Graph f (x) = e−x . State its domain and range.
Example 8. Graph f (x) = −3ex + 1. State its domain and range.
Example 9. Graph f (x) = e2x − 3. State its domain and range.
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College Algebra Notes
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Logarithmic Functions
Logarithm
If by = x, then logb x = y. The expression logb x is read “the logarithm base b of x” or “log base
b of x.”
Example 1. Write the following exponential equations as logarithmic equations.
(a) 24 = 16
(b) 53 = 125
1
(c) 81 2 = 9
(d) 4−3 =
1
64
Example 2. Write the following logarithmic equations as exponential equations.
(a) log10 1000 = 3
(b) log3 243 = 5
(c) log27 3 =
1
3
(d) −3 = log2
1
8
Example 3. Evaluate the following logarithms.
(a) log5 25
(a) log10 10000
(a) log3 1
(a) log2 64
(a) log36
1
6
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College Algebra Notes
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Example 4. Find the inverse function of f (x) = 2x .
Example 5. Graph the function f (x) = 2x and g(x) = log2 x.
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College Algebra Notes
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Example 6. Graph the function f (x) = log3 x. State the domain and range.
Natural and Common Logarithms
The natural logaritm of x is denoted ln x and ln x = loge x.
The common logaritm of x is denoted log x and log x = log10 x.
Example 7. Graph the function f (x) = log x. State the domain and range.
Example 8. Graph the function f (x) = ln x. State the domain and range.
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Example 9. Graph the function f (x) = log2 (x − 3). State the domain and range.
Example 10. Graph the function f (x) = log2 x − 3. State the domain and range.
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Properties of Logarithms
Basic Properties of Logarithms
1. logb 1 = 0
2. logb b = 1
3. logb bx = x
4. blogb x = x
Example 1. Evaluate the following expressions.
(a) log8 1
(b) log4 4
(c) log2 27
(d) 3log3 4
Example 2. Evaluate the following expressions.
(a) ln 1
(b) log 10
(c) ln e3
(d) eln(2x)
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College Algebra Notes
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Product Rule for Logarithms
logb (M · N ) = logb M + logb N
Proof.
Quotient Rule for Logarithms
logb
M
N
= logb M − logb N
Proof.
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Power Rule for Logarithms
logb M N = N logb M
Proof.
Example 3. Expand the logarithmic expression.
log3
x2 y
9z 3
Example 4. Condense the logarithmic expression.
ln x + 5 ln y − 3 ln z
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College Algebra Notes
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Change of Base Formula
logb M =
loga M
loga b
Proof.
Example 5. Approximate the logarithm.
log4 50
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Exponential and Logarithmic Equations
One-to-one Property for Exponential Functions
If bx = by , then x = y.
Example 1. Solve the equation.
34x+1 = 81
Example 2. Solve the equation.
43x−1 = 8x+5
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One-to-one Property for Logarithmic Functions
If x = y and x > 0, then logb x = logb y.
Example 3. Solve the equation.
7x−3 = 21
Example 4. Solve the equation.
42x−3 = 53x+4
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Example 5. Solve the equation.
e2x+5 = 18
Example 6. Solve the equation.
e2x − ex − 6 = 0
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Example 7. Solve the equation.
log3 (x + 3) = 2
Example 8. Solve the equation.
ln(x + 7) = 4
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Example 9. Solve the equation.
log2 (x − 5) + log2 (x + 2) = 3
Second One-to-one Property for Logarithmic Functions
If logb x = logb y and x > 0 and y > 0, then x = y.
Example 10. Solve the equation.
ln(x + 3) − ln x = ln 7
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College Algebra Notes
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Unit 5
Gaussian Elimination
Example 1. Consider the following system of equations.
x+y+z =6
2x − 3y − z = 5
3x + 2y − 2z = −1
It may be written as an augmented matrix as follows.


6
1
1
1
 2 −3 −1
5 
3
2 −2 −1
Elementary Row Operations
The following operations may be performed on an augmented matrix and preserve its associated
system of equations:
1. Multiply each element of a row by a nonzero scalar.
2. Add a multiple of one row to another row.
3. Swap two rows.
Example 2. The following are examples of the three elementary row operations.


1
1
1
6
 2 −3 −1
5 
3
2 −2 −1

4R1
=

4
4
4 24
 2 −3 −1
5 
3
2 −2 −1

4
4
4
24
 2 −3 −1
5 
−1
8
0 −11

−2R2 +R3
=

R1 ↔R3
=

−1
8
0 −11
 2 −3 −1
5 
24
4
4
4
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College Algebra Notes
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Gaussian Elimination with Back-Substitution
To perform Gaussian elimination with back-substitution, use elementary row operations to transform the augmented matrix to the following form.


1 a b c
 0 1 d e 
0 0 1 f
Then z = f , y + df = e, and x + ay + df = c.
Example 3. Perform Gaussian elimination with back-substitution to solve the system of equations.
x+y+z =6
2x − 3y − z = 5
3x + 2y − 2z = −1
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Example 4. Perform Gaussian elimination with back-substitution to solve the system of equations.
3x + 4y = −4
5y + 3z = 1
2x − 5z = 7
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Example 5. Perform Gaussian elimination with back-substitution to solve the system of equations.
2x + 3y − z = 7
−3x + 2y − 2z = 7
5x − 4y + 3z = −10
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Example 6. Perform Gaussian elimination with back-substitution to solve the system of equations.
x + 2y − z = 3
2x + 3y − 5z = 3
5x + 8y − 11z = 9
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Example 7. Perform Gaussian elimination with back-substitution to solve the system of equations.
3x − y + 2z = 4
x − 5y + 4z = 3
6x − 2y + 4z = −8
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Matrices
Matrix
An m × n matrix is an array of numbers with m rows and n

a1,1 a1,2 · · ·
 a2,1 a2,2 · · ·

A= .
..
..
 ..
.
.
am,1 am,2 · · ·
columns.

a1,n
a2,n 

.. 
. 
am,n
Each ai,j is called an element, and m × n is called the order of the matrix.
1 2
5 6
Example 1. Let A =
and B =
. Find A + B.
3 4
7 8
1 2
. Find 3A.
Example 2. Let A =
3 4
1 2
5 6
Example 3. Let A =
and B =
. Find 4A − B.
3 4
7 8
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Matrix Multiplication
Let A be an m × n matrix and let B be an n × p matrix. Then AB = X is an m × p matrix
were each element
xi,j = ai,1 · b1,j + ai,2 · b2,j + ... + ai,n · bn,j .
1 2
5 6
Example 4. Let A =
and B =
. Find AB.
3 4
7 8
5 6
1 2
. Find BA.
and B =
Example 5. Let A =
7 8
3 4
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

1 2
1 2
Example 6. Let A =
and C = 3 4. Find CA.
3 4
5 6


1 2
1 2
and C = 3 4. Find AC.
Example 7. Let A =
3 4
5 6
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Circles
Distance Formula
The distance between any two points (x1 , y1 ) and (x2 , y2 ) is given by
p
d = (x2 − x1 )2 + (y2 − y1 )2 .
y2
(x2 , y2 )
d
y1
(x1 , y1 )
y2 − y1
x2 − x1
x1
x2
Example 1. Find the distance between the points (−3, 4) and (3, −1).
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Midpoint Formula
The point halfway between two points (x1 , y1 ) and (x2 , y2 ) is called the midpoint and is given
by
x 1 + x 2 y1 + y2
.
,
2
2
y2
(x2 , y2 )
y1
x1 + x2 y1 + y2
,
2
2
(x1 , y1 )
x1
x2
Example 2. Locate the midpoint between (−3, 4) and (3, −1).
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Example 3. Verify the point given by the midpoint formula
and (x2 , y2 ).
x1 +x2 y1 +y2
,
2
2
is equidistant from (x1 , y1 )
Circles
A circle is the set of all points a fixed distance, called the radius, from a fixed point, called the
center.
Centered at (0, 0) with radius r
x2 + y 2 = r2
(x, y)
r
(0, 0)
Centered at (h, k) with radius r
(x − h)2 + (y − k)2 = r2
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Example 4. Graph x2 + y 2 = 1.
Example 5. Graph (x − 3)2 + (y + 1)2 = 9.
Example 6. Graph x2 + 8x + y 2 − 4y − 16 = 0.
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Ellipses
Ellipses Centered at (0, 0)
An ellipse is the set of all points a fixed distance from two fixed point, called the foci.
For both of the following equations, a > b and c2 = a2 − b2 .
Horizontal Major Axis
x2 y 2
+ 2 =1
a2
b
(0, b)
(−a, 0)
(0, 0)
(a, 0)
(0, −b)
The foci are located at (c, 0) and (−c, 0).
Vertical Major Axis
x2 y 2
+ 2 =1
b2
a
(0, a)
(−b, 0)
(b, 0)
(0, −a)
The foci are located at (0, c) and (0, −c).
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College Algebra Notes
Example 1. Graph
x2 y 2
+
= 1.
25
9
Example 2. Graph
x2
y2
+
= 1.
4
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Joseph Lee
Example 3. Graph 9x2 + 16y 2 = 144.
Example 4. Write the equation of the ellipse that has vertices at (0, 6) and (0, −6) and foci at (0, 5) and
(0, −5).
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Ellipses Centered at (h, k)
Horizontal Major Axis
(x − h)2 (y − k)2
+
=1
a2
b2
The vertices are located at (h + a, k) and (h − a, k).
The covertices are located (h, k + b) and (h, k − b).
The foci are located at (h + c, k) and (h − c, k).
Vertical Major Axis
(x − h)2 (y − k)2
+
=1
b2
a2
The vertices are located at (h, k + a) and (h, k − a).
The covertices are located (h + b, k) and (h − b, k).
The foci are located at (h, k + c) and (h, k − c).
Example 5. Graph
(x + 4)2 (y − 2)2
+
= 1.
20
36
Example 6. Graph x2 + 9y 2 = 9.
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Example 7. Graph 4x2 − 8x + 9y 2 + 90y + 193 = 0.
Example 8. Graph 4x2 + 24x + y 2 − 10y + 57 = 0.
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Hyperbolas
Hyperbolas Centered at (0, 0)
A hyberbola is the set of all points a fixed distance, when you subtract, from two fixed point, called the foci.
For both of the following equations, c2 = a2 + b2 .
Horizontal Transverse Axis
x2 y 2
− 2 =1
a2
b
(0, b)
(−c, 0)
(c, 0)
(−a, 0)
(a, 0)
(0, −b)
The vertices are located at (a, 0) and (−a, 0).
The foci are located at (c, 0) and (−c, 0).
Vertical Transverse Axis
y 2 x2
− 2 =1
a2
b
(0, c)
(0, a)
(−b, 0)
(b, 0)
(0, −a)
(0, −c)
The vertices are located at (0, a) and (0, −a).
The foci are located at (0, c) and (0, −c).
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