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Final Exam
Review Slides
Example: Absolute Value
Simplify the following expressions using yourknowledge of absolute values:
a.
b.
c.
21 7 14
3 3
The Order of Mathematical Operations
Order of Operations1. If the expression is a fraction, simplify the numerator and
denominator individually, according to the guidelines in the following steps.
2. Parentheses, braces and brackets are all used as grouping symbols. Simplify expressions within each set of grouping symbols, if any are present, working from the innermost outward.
3. Simplify all powers (exponents) and roots.4. Perform all multiplications and divisions in the expression in
the order they occur, working from left to right.5. Perform all additions and subtractions in the expression in the
order they occur, working from left to right.
Properties of Exponents and Their Use
Properties of ExponentsThroughout this table, a and b may be taken to represent constants, variables or more complicated algebraic expressions. The letters n and m represent integers.
Property1. Product Rule
2. Quotient Rule
3. Zero Exponent Rule
4. Negative Exponent Rule
n m n ma a a
nn m
m
aa
a
1nnaa
0 1.a
Properties of Exponents and Their Use
Power Rules 5.
6.
7.
mn nma a
n n nab a b
n n
n
a a
b b
Example: Scientific Notation
The distance from Earth to the sun is approximately 93,000,000 miles. Scientific notation takes advantage of the observation that multiplication of a number by 10 moves the decimal point one place to the right, and we can repeat this process as many times as necessary. Thus, in scientific notation,
93,000,000.. 7.9.3 10
Example 2: Scientific Notation
a. The mass of an electron, in kilograms, is approximately 0.000000000000000000000000000000911. Scientific notation takes advantage of the observation that multiplication of a number by moves the decimal point one place to the left, and we can repeat this process as many times as necessary. Thus, in scientific notation,
110
0.000000000000000000000000000000911 31=9.11 1 .0
Simplifying Radical Expressions
In the following properties, a and b may be taken to represent constant variable, or more complicated algebraic expressions. The letters n and m represent natural numbers.
Property1. Product Rule
2. Quotient Rule
n n nab a b
n
nn
a a
b b
Simplifying Radical Expressions
Rationalizing DenominatorsDenominator consists of two terms, one or both of which are square roots.Once again, remember that we cannot multiply the denominator by A – B unless we multiply the numerator by this same factor.Thus, multiply the fraction by
The factor A – B is called the conjugate radical expression of A + B.
–
–.
A B
A B
Common Factoring Methods
• Method 1: Greatest common factor.
• Method 2: Factoring by grouping.
• Method 3: Factoring special binomials.
• Method 4: Factoring trinomials.
– Case 1: Leading coefficient is 1.
– Case 2: Leading coefficient is not 1.
Simplify the following complex number expressions.
Example: The Algebra of Complex Numbers
2 1 2i i
4 3i
The product of two complex numbers leads to four products via the distributive property.
Roots and Complex Numbers
We earlier said that if are real numbers, then
If are not real
numbers, then these properties do not necessarily hold. For instance:
In order to apply either of these two properties, first simplify any square roots of negative numbers by rewriting them as pure imaginary numbers.
and a b
and .a a
a b abbb
9 4
3 2i i26i
6
9 4
366
and a b
Types of Equations
There are three types of equations:
1. A conditional equation has a countable number of solutions. For example, x + 7 = 12
has one solution, 5. The solution set is {5}.
2. An identity is true for all real numbers and has an infinite number of solutions. For example,
is true for all real number values of . The solution set is R.
3. A contradiction is never true and has no solution. For example, is false for any value of . The solution set is Ø.
x6x x
x
2( 1)x x x x
Solving Linear Equations
To solve a linear equation (in x): 1. Simplify each side of the equation separately by removing any
grouping symbols and combining like terms. 2. Add or subtract the same expression(s) on both sides of the
equation in order to get the variable term(s) on one side and the constant term(s) on the other side of the equation and simplify.
3. Multiply or divide by the same nonzero quantity on both sides of the equation in order to get the numerical coefficient of the variable term to be one.
4. Check your answer by substitution in the original equation.
Example: Solving Absolute Value Equations
Step 1: Rewrite the absolute value equation without absolute values.
3 2 5x
3 2 5x
3 7x
7
3x 1x
Step 2: Solve the two equations
or 3x – 2 = -5
or 3x = -3
or
Solve:
Example: Absolute Value Equations
4 3 2 0x Solve:
|4x + 3| = -2 False, absolute value is never negative. No solution; the solution set is Ø.
Solve: |6x – 2| = 06x – 2 = 06x = 2x = ⅓
If |ax + b| = 0, then ax + b = 0.
Example: Linear Inequalities
Solve the following linear inequality.
10 4( 2) (2 )x x Step 1: Distribute. 10 4 8 2x x
4 18 2x x
5 20x Step 3: Divide by . Note the reversal of the inequality sign.
5
4x
Solution is 4,
Step 2: Combine like terms.
Example: Solving Compound Inequalities
Solve the compound inequality. 12 5 2 15x
12 10 5 15x
2 5 25x 2
55
x
25
5x
2Solution is 5,
5
Note: each inequality is reversed since we are dividing by a negative number!
Example: Solving Absolute Value Inequalities
Solve the following absolute value inequality.5 3 2 9y
5 3 7y Step 1: Subtract . 2
7 5 3 7y Step 2: Rewrite the inequality without absolute values.
4 5 10y Step 3: Solve as compound inequality. 4
25
y
4
Solution is ,25
Example: Solving Quadratic Equations by Factoring
Solve the quadratic equation by factoring.
2 3 2
5 5
xx
Step 1: Multiply both sides by .5
Step 2: Subtract from both sides so is on one side.
25 3 2x x 25 3 2 0x x 2
0
Step 3: Factor and solve the two linear equations.
(5 2)( 1) 0x x 5 2 0x 1 0x
2
5x
1x
or
or
Example: The Quadratic Formula
Solve using the quadratic formula.
24 7 15x x
2 15 074x x a b c
24
2
7 7 4 1
4
5x
2 4
2x
cb b a
a
7 289
8x
7 17
8x
53,
4x
Operations with Rational Expressions
• To add or subtract two rational expressions, a common denominator must first be found.
• To multiply two rational expressions, the two numerators are multiplied and the two denominators are multiplied.
• To divide one rational expression by another, the first is multiplied by the reciprocal of the second.
• No matter which operation is being considered, it is generally best to factor all the numerators and denominators before combining rational expressions.
Example: Add/Subtract Rational Expressions
Subtract the rational expression.
2 2
5 2 2
5 6 4
x x
x x x
5 2 2
2 3 2 2
x x
x x x x
2 35 2 2
2 2 3 3 2 2
x xx x
x x x x x x
2 25 8 4 2 6
2 2 3 2 2 3
x x x x
x x x x x x
23 14 4
, 2, 2, 32 2 3
x xx
x x x
Step 1: Factor both denominators.
Step 2: Multiply to obtain the least common denominator (LCD) and simplify.
Example: Divide Rational Expressions
Divide the rational expression.2 2
3
20 10 25
2 6
x x x x
x x
3
2
4 5 6
2 5
x x x
x x
3
2
6 4 5
2 5
x x x
x x
3 2
23 4, 0,5
5
x xx
x
Example: Complex Rational Expressions
Simplify the complex rational expression. 1 1x z xz
1 1
1
x z xx z xz x z x
x x z
z x z x
z
z x z x
1
, 0, 0, 0x z x zx x z
Step 1: Multiply the numerator and the denominator by the LCD . x z x
Step 2: Cancel the common factor of . z
Example: Work-Rate Problem
One hose can fill a swimming pool in 10 hours. The owner buys a second hose that can fill the pool in half the time of the first one. If both hoses are used together, how long does it take to fill the pool?The work rate of the first hose is
1
10
The work rate of the second hose is 1
5 1 1 1
10 5 x Step 1: Set up the problem.
Step 2: Multiply both sides by the LCD , and solve. 10x
2 10x x
3 10x 1
33
x hours
Example: Solving Radical Equations
Solve the radical equation.
1 1x x 1 1x x
2 2
1 1x x 21 2 1x x x 20 3x x 0 3x x
0, 3x 0x
Note that , so -3 is an extraneous solution.
1 ( 3) ( 3) 1
Recognizing Linear Equations in Two Variables
Linear Equations in Two VariablesA linear equation in two variables, say the variables and , is an equation that can be written in the form
where , , and are constants and and are not both zero. This form of such an equation is called the standard form.
xy
ax by c
a b c a b
Example: Linear Equations
Determine if the equation is a linear equation.
2 3 5 2 1x y x x y
2 6 5 2 1x y x x y
3 6 2 1x y x y
4 4 1x y
The equation is linear.
3 6 2 1x y x y
Example: Intercepts
Find the - and -intercepts of the equation and graph.x y3 4 12x y
3 0 4 12y 3y
3 4 0 12x 4x
-intercept: 0, 3y
-intercept: 4,0x
Example: Finding Slope Using Two Points
Determine the slopes of the line passing through the following points.
8,1 and 2,33 1
2 8m
2
10m
1
5m
2 1
2 1
y ym
x x
Slope-Intercept Form of a Line
If the equation of a non-vertical line in and is solved for , the result is an equation of the form
The constant is the slope of the line, and the line crosses the -axis at ; that is, the -intercept of the line is . If the variable does not appear in the equation, the slope is 0 and the equation is simply of the form .
xy
y
.y mx b
my b y 0,b x
y b
Point-Slope Form of a Line
Given an ordered pair and a real number an equation for the line passing through the point with slope is
Note that , , and are all constants, and that and are variables. Note also that since the line, by definition, has slope , vertical lines cannot be described in this form.
1 1,x y m,
1 1,x y m
m 1x 1y xy
m
1 1 .y y m x x
The Slopes of Parallel and Perpendicular lines
ImportantParallel lines have the same slope. For example:
Perpendicular lines have slopes that are negative reciprocals of each other. For example:
1 2 and1 1
2
2.m m
1 2 and .2 3
3 2m m
Example: Slopes of Parallel Lines
Find the equation, in slope-intercept form, for the line which is parallel to the line and which passes through the point .
8 2 10x y 1,5 8 2 10x y
Step 1: Write equation in slope-intercept form.
2 10 8y x 4 5y x
5 4 1y x Step 2: Use point-slope form.
5 4 4y x 4 1y x
Step 3: Solve for to obtain slope-intercept form.
y
Use slope to write a new equation that passes through the point .
4m 1,5
Functions and the Vertical Line Test
FunctionsA function is a relation in which every element of the domain is paired with exactly one element of the range. Equivalently, a function is a relation in which no two distinct ordered pairs have the same first coordinate.
Implied Domain of a Function
The domain of the function is implied by the formula used in defining the function. It is assumed that the domain of the function consists of all real numbers at which the function can be evaluated to obtain a real number: any values for the argument that result in division by zero or an even root of a negative number must be excluded from the domain.
Quadratic Functions and Their Graphs
Vertex Form of a Quadratic FunctionThe graph of the function where a, h and k are real numbers and is a parabola whose vertex is (h,k). The parabola is narrower than if and is broader than . The parabola opens upward if a is positive and downward if a is negative.
2g x a x h k
0a 2f x x
1a 2 if 0 1f x x a
Commonly Occurring Functions
Piecewise-Defined FunctionA piecewise-defined function is a function defined in terms of two or more formulas, each valid for its own unique portion of the real number line. In evaluating a piecewise-defined function f at a certain value for x, it is important to correctly identify which formula is valid for that particular value.
The function f is a linear function on the interval and a quadratic function on the interval To graph f, we graph each portion separately, making sure that each formula is applied only on the appropriate interval. The complete graph appears to the right, with the points f(–4) = 6 and f(2) = 4 noted in particular. Also note the use of a closed circle at (–1,0) to emphasize that this point is part of the graph, and the use of an open circle at (–1,1) to indicate that this point is not part of the graph.
Example: Commonly Occurring Functions
Sketch the graph of the function
2
2 2 if 1
if 1.
x xf x
x x
, 1 1, .
Example: Variation Problems
For the following phrases, write the general formula that applies.
a.“y varies inversely as the n P Q power of x”
b.“y is directly proportional to the n P Q power of x”
c.“y is inversely proportional to the n P Q power of x”
d. “y varies directly as the n P Q power of x”
ny kx
ny kx
n
ky
x
n
ky
x
Shifting, Stretching and Reflecting Graphs
Horizontal ShiftingLet f(x) be a function whose graph is known, and let h be a fixed real number. If we replace x in the definition of f by x – h, we obtain a new function . The graph of g is the same shape as the graph of f, but shifted to the right by h units if h > 0 and shifted to the left by h units if h < 0.
g x f x h
Shifting, Stretching and Reflecting Graphs
Vertical ShiftingLet f(x) be a function whose graph is known and let k be a fixed real number. The graph of the function
is the same shape as the graph of f, but shifted upward if k > 0 and downward if k < 0. g x f x k
Shifting, Stretching and Reflecting Graphs
Reflecting With Respect to the AxesLet f(x) be a function whose graph is known.1. The graph of the function g(x)= –f(x) is the reflection
of the graph f with respect to the x-axis.2. The graph of the function g(x) = f(–x) is the
reflection of the graph of f with respect to the y-axis.
Shifting, Stretching and Reflecting Graphs
Stretching and CompressingLet f(x) be a function whose graph is known, and let a be a positive real number.1. The graph of the function g(x) = a f(x) is stretched
vertically compared to the graph of f if a > 1.2. The graph of the function g(x) = a f(x) is compressed
vertically compared to the graph of f if 0 < a < 1.
Combining Functions Arithmetically
Addition, Subtraction, Multiplication and Division of Functions1.
2.
3.
4.
The domain of each of these new functions consists of the common elements (or the intersection of elements) of the domains of f and g individually.
f g x f x g x
f g x f x g x
f g x f x g x
prov, ided that 0 f xf
x g xg g x
Composing Functions
Composing FunctionsLet f and g be two functions. The composition of f and g, denoted , is the function defined by
.
The domain of consists of all x in the domain of g for which g(x) is in turn in the domain of f. The function is read “f composed with g,” or “f of g.”
f g f g x f g x
f g
f g
Example: Composing Functions
Given f(x) = x2 + 2 and g(x) = x + 5 , find:
= (x + 5)2 + 2
= x2 +10x + 25 + 2
= x2 +10x + 27
f g x f g x
5f x
Again, we know by definition that . f g x f g x
Finding Inverse Function Formulas
To Find a Formula forLet f be a one-to-one function, and assume that f is defined by a formula. To find a formula for , perform the following steps:1. Replace in the definition of f with the variable y. The result is an equation in x and y that is solved for y.2. Interchange x and y in the equation.3. Solve the new equation for y.4. Replace the y in the resulting equation with .
1f
1f
1f x
f x
Long and Synthetic Division
Compare the division of by below, using long division on the left and synthetic division on the right.
3 22 8 9 7x x x 2x
3 22 8 9 72x x x x 2
22 4 1x x
3 242x x 24 9 7x x
2 84x x
7x 2x
5
2 8 9 7 4 8 2
2 4 1 5
Note: the numbers in blue are the coefficients of the dividend and the numbers in pink are the coefficients of the quotient and remainder.
Synthetic Division
Step 1: Write and the coefficients of the dividend. Copy the leading coefficient of the dividend in the first slot below the horizontal line.
Step 2: Multiply this number by and write the result directly below the second coefficient of the dividend.
Step 3: Add the two numbers in that column and write the result in the second slot below the horizontal line.
Step 4: Repeat the process until the last column is completed and the last number written down is the remainder.
k
k
Example: Solving Systems by Elimination
Solve the system by the method of elimination.
To eliminate a variable, you’ll need to multiply each equation by a unique constant. Let’s eliminate y. To do so, notice that we’ll have to multiply the first equation by 3 and the second equation by 2.
5 2 6
2 3 10
x y
x y
15 6 18x y 2 4 6 0x y
19 38x
5 2 6x y 2 3 10x y 32
2x 5 2 2 6y
2y Thus, the solution is . 2, 2
