Precalculus%20Review%20I

1
1
Preliminaries

• Precalculus Review I
• Precalculus Review II
• The Cartesian Coordinate System
• Straight Lines

2
1.1

• Precalculus Review I

3
The Real Number Line

• We can represent real numbers geometrically by
  points on a real number, or coordinate, line
• This line includes all real numbers.
• Exactly one point on the line is associated with
  each real number, and vice-versa.

Origin
Positive Direction
Negative Direction
4 3 2 1 0 1 2 3 4
p
4
Finite Intervals

• Open Intervals
• The set of real numbers that lie strictly between
  two fixed numbers a and b is called an open
  interval (a, b).
• It consists of all the real numbers that satisfy
  the inequalities a lt x lt b.
• It is called open because neither of its
  endpoints is included in the interval.

5
Finite Intervals

• Closed Intervals
• The set of real numbers that lie between two
  fixed numbers a and b, that includes a and b,
  is called a closed interval a, b.
• It consists of all the real numbers that satisfy
  the inequalities a ? x ? b.
• It is called closed because both of its
  endpoints are included in the interval.

6
Finite Intervals

• Half-Open Intervals
• The set of real numbers that between two fixed
  numbers a and b, that contains only one of its
  endpoints a or b, is called a half-open interval
  (a, b or a, b).
• It consists of all the real numbers that satisfy
  the inequalities a lt x ? b or a ? x lt b.

7
Infinite Intervals

• Examples of infinite intervals include
• (a, ?), a, ?), (?, a), and (?, a.
• The above are defined, respectively, by the set
  of real numbers that satisfy x gt a, x ? a, x lt a,
  x ? a.

8
Exponents and Radicals

• If b is any real number and n is a positive
  integer, then the expression bn is defined as the
  number
• bn b b b b
• The number b is called the base, and the
  superscript n is called the power of the
  exponential expression bn.
• For example
• If b ? 0, we define b0 1.
• For example
• 20 1 and (p)0 1, but 00 is undefined.

n factors
9
Exponents and Radicals

• If n is a positive integer, then the expression
  b1/n is defined to be the number that, when
  raised to the nth power, is equal to b, thus
• Such a number is called the nth root of b, also
  written as
• Similarly, the expression bp/q is defined as the
  number
• (b1/q)p or
• Examples

(b1/n)n b
10
Laws of Exponents

• Law Example
• 1. am an am n x2 x3 x2 3 x5
• 2.
• 3. (am)n am n (x4)3 x4 3 x12
• 4. (ab)n an bn (2x)4 24 x 4 16x4
• 5.

11
Examples

• Simplify the expressions

Example 1, page 6
12
Examples

• Simplify the expressions

(assume x, y, m, and n are positive)
Example 2, page 7
13
Examples

• Rationalize the denominator of the expression

Example 3, page 7
14
Examples

• Rationalize the numerator of the expression

Example 4, page 7
15
Operations With Algebraic Expressions

• An algebraic expression of the form axn, where
  the coefficient a is a real number and n is a
  nonnegative integer, is called a monomial,
  meaning it consists of one term.
• Examples
• 7x2 2xy 12x3y4
• A polynomial is a monomial or the sum of two or
  more monomials.
• Examples
• x2 4x 4 x4 3x2 3 x2y xy y

16
Operations With Algebraic Expressions

• Constant terms, or terms containing the same
  variable factors are called like, or similar,
  terms.
• Like terms may be combined by adding or
  subtracting their numerical coefficients.
• Examples
• 3x 7x 10x 12xy 7xy 5xy

17
Examples

• Simplify the expression

Example 5, page 8
18
Examples

• Simplify the expression

Example 5, page 8
19
Examples

• Perform the operation and simplify the expression

Example 6, page 8
20
Examples

• Perform the operation and simplify the expression

Example 6, page 9
21
Factoring

• Factoring is the process of expressing an
  algebraic expression as a product of other
  algebraic expressions.
• Example

22
Factoring

• To factor an algebraic expression, first check to
  see if it contains any common terms.
• If so, factor out the greatest common term.
• For example, the greatest common factor for the
  expression
• is 2a, because

23
Examples

• Factor out the greatest common factor in each
  expression

Example 7, page 9
24
Examples

• Factor out the greatest common factor in each
  expression

Example 8, page 10
25
Factoring Second Degree Polynomials

• The factors of the second-degree polynomial with
  integral coefficients
• px2 qx r
• are (ax b)(cx d), where ac p, ad bc q,
  and bd r.
• Since only a limited number of choices are
  possible, we use a trial-and-error method to
  factor polynomials having this form.

26
Examples

• Find the correct factorization for x2 2x 3
• Solution
• The x2 coefficient is 1, so the only possible
  first degree terms are
• (x )(x )
• The product of the constant term is 3, which
  gives us the following possible factors
• (x 1)(x 3)
• (x 1)(x 3)
• We check to see which set of factors yields 2
  for the x coefficient
• (1)(1) (1)(3) 2 or (1)(1)
  (1)(3) 2
• and conclude that the correct factorization is
• x2 2x 3 (x 1)(x 3)

27
Examples

• Find the correct factorization for the expressions

Example 9, page 11
28
Roots of Polynomial Expressions

• A polynomial equation of degree n in the variable
  x is an equation of the form
• where n is a nonnegative integer and a0, a1, ,
  an are real numbers with an ? 0.
• For example, the equation
• is a polynomial equation of degree 5.

29
Roots of Polynomial Expressions

• The roots of a polynomial equation are the values
  of x that satisfy the equation.
• One way to factor the roots of a polynomial
  equation is to factor the polynomial and then
  solve the equation.
• For example, the polynomial equation
• may be written in the form
• For the product to be zero, at least one of the
  factors must be zero, therefore, we have
• x 0 x 1 0 x 2 0
• So, the roots of the equation are x 0, 1, and 2.

30
The Quadratic Formula

• The solutions of the equation
• ax2 bx c 0 (a ? 0)
• are given by

31
Examples

• Solve the equation using the quadratic formula
• Solution
• For this equation, a 2, b 5, and c 12, so

Example 10, page 12
32
Examples

• Solve the equation using the quadratic formula
• Solution
• First, rewrite the equation in the standard form
• For this equation, a 1, b 3, and c 8, so

Example 10, page 12
33
1.2

• Precalculus Review II

34
Rational Expressions

• Quotients of polynomials are called rational
  expressions.
• For example

35
Rational Expressions

• The properties of real numbers apply to rational
  expressions.
• Examples
• Using the properties of number we may write
• where a, b, and c are any real numbers and b and
  c are not zero.
• Similarly, we may write

36
Examples

• Simplify the expression

Example 1, page 16
37
Examples

• Simplify the expression

Example 1, page 16
38
Rules of Multiplication and Division

• If P, Q, R, and S are polynomials, then
• Multiplication
• Division

39
Example

• Perform the indicated operation and simplify

Example 2, page 16
40
Rules of Addition and Subtraction

• If P, Q, R, and S are polynomials, then
• Addition
• Subtraction

41
Example

• Perform the indicated operation and simplify

Example 3b, page 17
42
Other Algebraic Fractions

• The techniques used to simplify rational
  expressions may also be used to simplify
  algebraic fractions in which the numerator and
  denominator are not polynomials.

43
Examples

• Simplify

Example 4a, page 18
44
Examples

• Simplify

Example 4b, page 18
45
Rationalizing Algebraic Fractions

• When the denominator of an algebraic fraction
  contains sums or differences involving radicals,
  we may rationalize the denominator.
• To do so we make use of the fact that

46
Examples

• Rationalize the denominator

Example 6, page 19
47
Examples

• Rationalize the numerator

Example 7, page 19
48
Properties of Inequalities

• If a, b, and c, are any real numbers, then
• Property 1 If a lt b and b lt c, then a lt c.
• Property 2 If a lt b, then a c lt b c.
• Property 3 If a lt b and c gt 0, then ac lt bc.
• Property 4 If a lt b and c lt 0, then ac gt bc.

49
Examples

• Find the set of real numbers that satisfy
• 1 ? 2x 5 lt 7
• Solution
• Add 5 to each member of the given double
  inequality
• 4 ? 2x lt 12
• Multiply each member of the inequality by ½
• 2 ? x lt 6
• So, the solution is the set of all values of x
  lying in the interval 2, 6).

Example 8, page 20
50
Examples

• Solve the inequality x2 2x 8 lt 0.
• Solution
• Factorizing we get (x 4)(x 2) lt 0.
• For the product to be negative, the factors must
  have opposite signs, so we have two possibilities
  to consider
• The inequality holds if (x 4) lt 0 and (x 2) gt
  0, which means x lt 4, and x gt 2, but this
  is impossible x cannot meet these two
  conditions simultaneously.
• The inequality also holds if (x 4) gt 0 and (x
  2) lt 0, which means x gt 4, and x lt
  2, or 4 lt x lt 2.
• So, the solution is the set of all values of x
  lying in the interval ( 4, 2).

Example 9, page 20
51
Examples

• Solve the inequality
• Solution
• For the quotient to be positive, the numerator
  and denominator must have the same sign, so we
  have two possibilities to consider
• The inequality holds if (x 1) ? 0 and (x 1) lt
  0, which means x ? 1, and x lt 1, both of
  these conditions are met only when x ? 1.
• The inequality also holds if (x 1) ? 0 and (x
  1) gt 0, which means x ? 1, and x gt 1, both
  of these conditions are met only when x gt 1.
• So, the solution is the set of all values of x
  lying in the intervals ( ?, 1 and (1, ?).

Example 10, page 21
52
Absolute Value

• The absolute value of a number a is denoted a
  and is defined by

53
Absolute Value Properties

• If a, b, and c, are any real numbers, then
• Property 5 a a
• Property 6 ab a b
• Property 7 (b ? 0)
• Property 8 a b a b

54
Examples

• Evaluate the expression
• p 5 3
• Solution
• Since p 5 lt 0, we see that
• p 5 (p 5).
• Therefore
• p 5 3 (p 5) 3
• 8 p
• 4.8584

Example 12a, page 22
55
Examples

• Evaluate the expression
• Solution
• Since , we see that
• Similarly, , so
• Therefore,

Example 12b, page 22
56
Examples

• Evaluate the inequality x ? 5.
• Solution
• If x ? 0, then x x, so x ? 5 implies x ? 5.
• If x lt 0, then x x , so x ? 5 implies x
  ? 5 or x ? 5.
• So, x ? 5 means 5 ? x 5, and the solution
  is 5, 5.

Example 13, page 22
57
Examples

• Evaluate the inequality 2x 3 ? 1.
• Solution
• From our last example, we know that 2x 3 ? 1
  is equivalent to 1 ? 2x 3 ? 1.
• Adding 3 throughout we get 2 ? 2x ? 4.
• Dividing by 2 throughout we get 1 ? x ? 2, so the
  solution is 1, 2.

Example 14, page 22
58
1.3

• The Cartesian Coordinate System

59
The Cartesian Coordinate System

• At the beginning of the chapter we saw a
  one-to-one correspondence between the set of real
  numbers and the points on a straight line (one
  dimensional space).

60
The Cartesian Coordinate System

• The Cartesian coordinate system extends this
  concept to a plane (two dimensional space) by
  adding a vertical axis.

4 3 2 1 1 2 3 4
61
The Cartesian Coordinate System

• The horizontal line is called the x-axis, and the
  vertical line is called the y-axis.

y
4 3 2 1 1 2 3 4
x
62
The Cartesian Coordinate System

• The point where these two lines intersect is
  called the origin.

y
4 3 2 1 1 2 3 4
Origin
x
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The Cartesian Coordinate System

• In the x-axis, positive numbers are to the right
  and negative numbers are to the left of the
  origin.

y
4 3 2 1 1 2 3 4
Positive Direction
Negative Direction
x
64
The Cartesian Coordinate System

• In the y-axis, positive numbers are above and
  negative numbers are below the origin.

y
4 3 2 1 1 2 3 4
Positive Direction
x
Negative Direction
65
The Cartesian Coordinate System

• A point in the plane can now be represented
  uniquely in this coordinate system by an ordered
  pair of numbers (x, y).

y
( 2, 4)
4 3 2 1 1 2 3 4
(4, 3)
x
(3,1)
( 1, 2)
66
The Cartesian Coordinate System

• The axes divide the plane into four quadrants as
  shown below.

y
4 3 2 1 1 2 3 4
Quadrant I (, )
Quadrant II (, )
x
Quadrant IV (, )
Quadrant III (, )
67
The Distance Formula

• The distance between any two points in the plane
  may be expressed in terms of their coordinates.

• Distance formula
• The distance d between two points P1(x1, y1) and
  P2(x2, y2) in the plane is given by

68
Examples

• Find the distance between the points ( 4, 3) and
  (2, 6).
• Solution
• Let P1( 4, 3) and P2(2, 6) be points in the
  plane.
• We have
• x1 4 y1 3 x2 2 y2 6
• Using the distance formula, we have

Example 1, page 26
69
Examples

• Let P(x, y) denote a point lying on the circle
  with radius r and center C(h, k). Find a
  relationship between x and y.
• Solution
• By definition in a circle, the distance between
  P(x, y) and C(h, k) is r.
• With distance formula we get
• Squaring both sides gives

y
P(x, y)
C(h, k)
r
k
x
h
Example 2, page 27
70
Equation of a Circle

• An equation of a circle with center C(h, k) and
  radius r is given by

71
Examples

• Find an equation of the circle with radius 2 and
  center (1, 3).
• Solution
• We use the circle formula with r 2, h 1,
  and k 3

y
(1, 3)
3
2
x
1
Example 3, page 27
72
Examples

• Find an equation of the circle with radius 3 and
  center located at the origin.
• Solution
• We use the circle formula with r 3, h 0, and
  k 0

y
3
x
Example 3, page 27
73
1.4

• Straight Lines

74
Slope of a Vertical Line

• Let L denote the unique straight line that passes
  through the two distinct points (x1, y1) and (x2,
  y2).
• If x1 x2, then L is a vertical line, and the
  slope is undefined.

y
L
(x1, y1)
(x2, y2)
x
75
Slope of a Nonvertical Line

• If (x1, y1) and (x2, y2) are two distinct points
  on a nonvertical line L, then the slope m of L is
  given by

y
L
(x2, y2)
y2 y1 ?y
(x1, y1)
x2 x1 ?x
x
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Slope of a Nonvertical Line

• If m gt 0, the line slants upward from left to
  right.

y
L
m 2
?y 2
?x 1
x
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Slope of a Nonvertical Line

• If m lt 0, the line slants downward from left to
  right.

y
m 1
?x 1
?y 1
x
L
78
Examples

• Sketch the straight line that passes through the
  point (2, 5) and has slope 4/3.

• Solution
• Plot the point (2, 5).
• A slope of 4/3 means that if x increases by 3,
  y decreases by 4.
• Plot the point (5, 1).
• Draw a line across the two points.

y
6 5 4 3 2 1
?x 3
(2, 5)
?y 4
(5, 1)
x
1 2 3 4 5 6
L
Example 1, page 34
79
Examples

• Find the slope m of the line that goes through
  the points (1, 1) and (5, 3).
• Solution
• Choose (x1, y1) to be (1, 1) and (x2, y2) to be
  (5, 3).
• With x1 1, y1 1, x2 5, y2 3, we find

Example 2, page 35
80
Equations of Lines

• Let L be a straight line parallel to the y-axis.
• Then L crosses the x-axis at some point (a, 0) ,
  with the x-coordinate given by x a, where a is
  a real number.
• Any other point on L has the form (a, ), where
  is an appropriate number.
• The vertical line L can therefore be described as
• x a

y
L
(a, )
(a, 0)
x
81
Equations of Lines

• Let L be a nonvertical line with a slope m.
• Let (x1, y1) be a fixed point lying on L and (x,
  y) be variable point on L distinct from (x1, y1).
• Using the slope formula by letting (x, y) (x1,
  y1) we get
• Multiplying both sides by x x2 we get

82
Point-Slope Form

• An equation of the line that has slope m and
  passes through point (x1, y1) is given by

83
Examples

• Find an equation of the line that passes through
  the point (1, 3) and has slope 2.
• Solution
• Use the point-slope form
• Substituting for point (1, 3) and slope m 2, we
  obtain
• Simplifying we get

Example 5, page 36
84
Examples

• Find an equation of the line that passes through
  the points (3, 2) and (4, 1).
• Solution
• The slope is given by
• Substituting in the point-slope form for point
  (4, 1) and slope m 3/7, we obtain

Example 6, page 36
85
Perpendicular Lines

• If L1 and L2 are two distinct nonvertical lines
  that have slopes m1 and m2, respectively, then L1
  is perpendicular to L2 (written L1 - L2) if and
  only if

86
Example

• Find the equation of the line L1 that passes
  through the point (3, 1) and is perpendicular to
  the line L2 described by
• Solution
• L2 is described in point-slope form, so its slope
  is m2 2.
• Since the lines are perpendicular, the slope of
  L1 must be
• m1 1/2
• Using the point-slope form of the equation for L1
  we obtain

Example 7, page 37
87
Crossing the Axis

• A straight line L that is neither horizontal nor
  vertical cuts the x-axis and the y-axis at , say,
  points (a, 0) and (0, b), respectively.
• The numbers a and b are called the x-intercept
  and y-intercept, respectively, of L.

y
y-intercept
(0, b)
x-intercept
x
(a, 0)
L
88
Slope Intercept Form

• An equation of the line that has slope m and
  intersects the y-axis at the point (0, b) is
  given by
• y mx b

89
Examples

• Find the equation of the line that has slope 3
  and y-intercept of 4.
• Solution
• We substitute m 3 and b 4 into y mx b,
  and get
• y 3x 4

Example 8, page 38
90
Examples

• Determine the slope and y-intercept of the line
  whose equation is 3x 4y 8.
• Solution
• Rewrite the given equation in the slope-intercept
  form.
• Thus,
• Comparing to y mx b we find that m ¾ , and
  b 2.
• So, the slope is ¾ and the y-intercept is 2.

Example 9, page 38
91
Applied Example

• An art object purchased for 50,000 is expected
  to appreciate in value at a constant rate of
  5000 per year for the next 5 years.
• Write an equation predicting the value of the art
  object for any given year.
• What will be its value 3 years after the
  purchase?
• Solution
• Let x time (in years) since the object was
  purchased
• y value of object (in dollars)
• Then, y 50,000 when x 0, so the y-intercept
  is b 50,000.
• Every year the value rises by 5000, so the slope
  is m 5000.
• Thus, the equation must be y 5000x 50,000.
• After 3 years the value of the object will be
  65,000
• y 5000(3) 50,000 65,000

Applied Example 11, page 39
92
General Form of an Linear Equation

• The equation
• Ax By C 0
• where A, B and C are constants and A and B are
  not both zero, is called the general form of a
  linear equation in the variables x and y.

93
Theorem 1

• An equation of a straight line is a linear
  equation conversely, every linear equation
  represents a straight line.

94
Example

• Sketch the straight line represented by the
  equation
• 3x 4y 12 0
• Solution
• Since every straight line is uniquely determined
  by two distinct points, we need find only two
  such points through which the line passes in
  order to sketch it.
• For convenience, lets compute the x- and
  y-intercepts
• Setting y 0, we find x 4 so the x-intercept
  is 4.
• Setting x 0, we find y 3 so the y-intercept
  is 3.
• Thus, the line goes through the points (4, 0) and
  (0, 3).

Example 12, page 40
95
Example

• Sketch the straight line represented by the
  equation
• 3x 4y 12 0
• Solution
• Graph the line going through the points (4, 0)
  and (0, 3).

y
L
1 1 2 3 4
(4, 0)
x
1 2 3 4 5 6
(0, 3)
Example 12, page 40
96
End of Chapter