College Algebra FINAL EXAM Review, Fall...

College Mathematics FINAL EXAM Review of 43

Section 1.1: Fractions

Section 1.1.1: Meaning of a Fraction A fraction is meant to represent the number of pieces of an object you take or use when you divide it

into equal pieces. So, if you cut a pizza into eight equal-size pieces and then take five of them, we are

talking about five of the eight equal-sized pieces. We write this amount of pizza as the fraction (of

the pizza).

There are three pieces to writing a fraction. Their names are:

Fractions are usually called “rational numbers” in mathematical circles because a ratio is a name for when you compare how many pieces you have to a total number of pieces.

Section 1.1.2: Equivalent Fractions

Fractions that have different numbers but that represent the same amount are called “equivalent fractions”.

To convert one fraction to another equivalent fraction, you can either: Multiply both the numerator and denominator by the same number, or Divide both the numerator and denominator by the same number.


Reducing Fractions: “Reducing a fraction” involves writing the fraction as an equivalent fraction with smaller numbers in the numerator and denominator. Reducing a fraction until the numbers are as small as possible is called reducing the fraction to “lowest terms”.

One way to reduce a fraction is just to do your best to see what you can divide top and bottom by.

Another method for reducing fractions is to do a prime factorization of the numerator and denominator, then divide out (or “cancel”) all the common factors.

Here is a list of the first twenty prime numbers to aid your factoring:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71

Equivalent fractions are also useful for comparing fractions when you don’t have a calculator.To compare fractions:

Write each fraction as an equivalent fraction, but with the same denominator. The bigger numerator belongs to the bigger fraction.

Section 1.1.3: Improper Fractions and Mixed Numbers

To write a fractional amount that is between numbers bigger than one, we use either “improper fractions” or “mixed numbers”.

An improper fraction is a fraction where the numerator is larger than the denominator. To see how to

write pizzas as an improper fraction, you can think of the first pizza as being of a pizza (i.e., five

of the five equal pieces), the second pizza as more, and then you also have the remaining of a

pizza. So, you have 5 + 5 + 3 = 13 pieces of pizza, each of which is an equal fifth of a pizza. Since a fraction is the number of pieces we have over the number of equal pieces in a whole, we can write the

mixed number as the improper fraction .

To convert a mixed number to an improper fraction: Multiply the denominator and the whole number part. Add that answer to the numerator to get the improper numerator (the denominator stays the same).

To convert an improper fraction to a mixed number: Divide the denominator into the numerator. The quotient is the whole number part; the remainder is the numerator.


Section 1.1.4: Adding and Subtracting Proper Fractions

The key to adding or subtracting fractions is to remember that you can only add together pieces that are the same size.

If the denominators are not the same, then you have to change your fractions to equivalent fractions so that they have the same denominators. Thus you will be adding or subtracting pieces that are the same size.

You can multiply the denominators together to get a common denominator, but you will do the least amount of work in the long run if you use the Least Common Denominator (LCD).

The LCD has to have all the factors from every denominator:


Section 1.1.5: Adding and Subtracting Mixed Numbers

To add or subtract mixed numbers, first deal with the whole numbers, then deal with the fractions.

If the fractions add up to an improper fraction, convert the improper fraction to a mixed number, and then combine the whole numbers again.


If subtracting the fractions would result in a negative fraction, borrow from the whole number part to make the positive fraction bigger.

Section 1.1.6: Multiplying and Dividing Proper and Improper FractionsMultiplying fractions: The rule for multiplying proper (and improper) fractions is to take numerator times numerator and denominator times denominator. Example:

One way to be efficient is to cancel common factors before you multiply:

Dividing fractions: One way to help understand what happens when dividing with fractions: If you divide by a number that is bigger than the number one, then you get a smaller number,

because you are dividing something into smaller pieces.

If you divide by a number that is smaller than one, you get a bigger number, because when you divide something into smaller pieces, then you get more pieces overall.

The rule for dividing proper (and improper) fractions is to take the reciprocal of the second fraction, then multiply. Example:


Section 1.1.7: Multiplying and Dividing Mixed NumbersTo multiply or divide mixed numbers, you have to convert them to improper fractions, then use the regular rules for multiplying and dividing fractions just covered.

Section 1.2: Decimals

Section 1.2.1: The Decimal Representation (for Terminating Decimals) Positional notation as increasing factors of 10… Positional notation as decreasing factors of 10… once get past 1 10, you’re in the decimal range… The positional notation can be used to write a fraction equivalent:

Or you can use the trick of writing a fraction with the digits after the decimal in the numerator and the next larger power of 10 in the denominator:


Section 1.2.2: General Conversion between Fractions and Decimals To convert from fraction to a decimal, you can try to write the fraction as an equivalent fraction with a

denominator that is a power of 10. Or you can do long division, remembering to put a decimal at the end if quotient and dividend. Stop when

you get zero for a remainder… For some fractions, you never get a remainder of zero no matter how long you do the long division. These

fractions have a decimal representation that repeats forever, and are called non-terminating or repeating decimals. You know you have a repeating decimal when the long-division pattern keeps on repeating.

A bar over the repeating pattern is frequently used to represent a repeating decimal.

Section 1.2.3: Rounding Decimals A place to round to will be specified for you with phrases like “round to two decimal places” or “round

to the nearest hundredth”. The idea is that you will re-write the number, but only up to that place, and with the last digit rounded according to the following rules:

o If the next digit is less than 5, leave the last digit aloneo If the next digit is 5 or more, add 1 to the last digit.

For an example, here is how to round the number 10,547.395 to different specified decimal places:

10,547.395 rounded to Decimal Place of Rounding Result2 places hundredth’s place 10,547.401 place tenth’s place 10,547.4the nearest unit one’s place 10,547.the nearest ten ten’s place 10,550the nearest hundred hundred’s place 10,500the nearest thousand thousand’s place 11,000

Section 1.2.4: Adding and Subtracting Decimals

To add: Line up the decimal points and add the digits from right to left, carrying as necessary. To subtract: Line up the decimal points and subtract the digits from right to left, borrowing as necessary.

Section 1.2.5: Multiplying Decimals Multiply as if there were no decimals present The answer will have as many decimal places as all the decimal places of both factors added up.

Section 1.2.6: Dividing Decimals

Move the decimal point the same number of places in the divisor and dividend until the divisor is a whole number.

Then do standard long division, putting a decimal point in the quotient right above the decimal point in the dividend.


Section 1.3: Significant Digits

Section 1.3.1: Identifying and Writing Significant Digits There is a shortcut that scientists and engineers use to state the precision of a measurement when writing

down the measurement. For example, if you measure the width of a box of chalk with a school-type ruler, you might get an

answer of 6.1 cm. Since the ruler is only marked in increments of 0.1 cm, you really can’t measure any finer than half that increment, which is 0.05 cm. Thus, the best way to state your measurement is as 6.1 cm 0.05 cm. That way, people know exactly how precise your measurement is, and they know not to expect more precision (like 6.139 cm), and that a number of 6 cm is neglecting the full precision of the measurement. Note: 6.1 cm 0.05 cm = 61 mm 0.5 mm…

If you measure the width with a high-precision instrument like a digital caliper, then you might discover the answer to be 61.22 mm. Again, the uncertainty would be one-half of the smallest measurement possible, which would be 0.005 mm. Thus, we’d write the measurement as 61.22 mm 0.005 mm.

However, people have found a more efficient way to write a measurement like 6.1 cm 0.05 cm. Instead, they just write 6.1 cm, and it is understood that the precision is one-half of the farthest-right digit. In this case, significant digits are being used to indicate the precision of the measurement.

The significant digits (also called significant figures and abbreviated sig figs) of a number are those digits that carry meaning contributing to its precision.

That last practice problem embodies one of the most confusing parts of working with significant digits: measurements that have a lot of zeros in them.

o When you want to show that a zero is significant, you go to the extra effort of writing extra stuff that you normally wouldn’t write, like putting extra zeros after the decimal point to show the finest increment of measurement:

Example: 100.00 mo Continuing the theme of writing something a little extra to show what zeros are significant, an

old-fashioned trick for showing which trailing zeros are significant is to put a bar over every significant zero:

Like or for the last two examples above.o A more widely used trick to show when a measurement has been made to the nearest ones place

is to put a decimal after the ones palce:

The rule for identifying the significant digits in a measurement can be stated in two ways:o All digits are significant except leading zeros after a decimal point and trailing zeros before it.

Both types of zeros merely serve as placeholders.

o “point right, otherwise left”: If there is a decimal point present, find the left-most nonzero digit, and then count digits

toward the right. If there is no decimal point in the number, find the right-most nonzero digit and count toward the left. In both cases, keep counting digits until you reach the other end of the number.


Section 1.3.2: Significant Digits after Adding or Subtracting When you add or subtract measurements, then you have to round your answer to the precision of the

least accurate measurement. That’s because the uncertainty in the least precise measurement overwhelms the precision of the other measurements.

Section 1.3.3: Significant Digits after Multiplying or Dividing Rule: Find the measurement with the fewest significant figures. Round your answer to that many

significant figures. One exception to the rule: numbers without a unit of measurement attached are considered exact

numbers that have an infinite number of significant digits. Ignore them when determining the number of sig. figs. in your answer.

1. Example: 100 22.85 cm = 2285 cm

Section 1.4: Signed Numbers

Section 1.4.1: The Concept of Negative Numbers To represent a change that decreases, we use negative numbers (positive numbers represent an increase) The number line is one convenient way to visualize positive and negative numbers:

o Positive (or “ordinary”) numbers are to the right of zero o Negative numbers are to the left of zero.o The number line could represent measurements like yards from the line of scrimmage, dollars in

your checking account, or miles to the east or west of your current location. The first important concept of signed numbers is the “opposite” or “additive inverse” of a number. When a signed number and its opposite are added together, the answer is zero.

o The opposite of 5 is –5 since 5 + (-5) = 0 For example, if you make $5 then lose $5, you’re back to zero dollars

o The opposite of –5 is 5 For example, if a running back loses 5 yards on first down, then gains 5 yards on second

down, then the team is back to where it started on the line of scrimmage. To find the opposite of a signed number, just change the sign that is in front of the number.


Section 1.4.2: Adding and Subtracting Signed Numbers

To add a signed number to another number, move that amount in the positive direction for a positive number, or move that amount in the negative direction for a negative number.

To subtract a signed number from another number, move that amount in the opposite direction of the sign of the subtracted number.

Section 1.4.3: Multiplying and Dividing Signed Numbers

A positive number times a positive number has a positive answer. (A friend of a friend is a friend.) A positive number times a negative number has a negative answer. (A friend of an enemy is an enemy;

an enemy of a friend is an enemy.) A negative number times a negative number has a positive answer. (An enemy of an enemy is a friend.)

o If I say “I am not dishonest”, the double negative makes the sentence equivalent to saying “I am honest”.

The signs for dividing are the same as for multiplying, because a division problem can always be re-written as a multiplication problem:

Section 1.5: Exponents

Section 1.5.1 Definition of Positive Integer ExponentsExponents were originally developed as a shortcut notation for repeated multiplication:

Section 1.5.5 Simplifying Expressions Involving Two or More of These FormsSome expressions will involve more than one of the exponent laws. You will need to carefully determine which laws to apply. In general, start by simplifying within any parentheses groups. Work your way outward, applying exponents to the factors of each group, and then finish by multiplying the simplified groups together.


Section 1.5.6 The Meaning of Zero and Negative ExponentsIf we look at the pattern of repeatedly dividing a number that is a perfect power by the base, we can see a meaningful value for a base raised to a power of zero, and for a base raised to a negative power:

Any base (except a base of zero) raised to a power of zero is equal to one. Here is another explanation:

Any base raised to a negative power is the same as that base raised to the positive of that power, but on the other side of the fraction bar.


Section 1.5.10 Scientific Notation Exponents can be used to write very large and very small numbers in a more concise way, called

scientific notation. Scientific notation is a way to write either a very large number or a number very close to zero using a

number between 1 and 10 times a power of ten. Scientific notation is considered simpler because there is no need to write a long string of 0 digits at the

end of a large number, or long string of 0’s in a decimal that is very close to zero.

Writing Numbers in Scientific Notation shortcut: count how many times you have to move the decimal to get it behind the first significant digit; that count is the positive power of ten for large numbers or the negative power of ten for numbers close to zero

Change numbers written in scientific notation to standard (decimal) form shortcut: move the decimal left or right by the power of ten. Positive exponents make a big number, negative exponents make a decimal close to zero.

Section 1.5.11 Multiplying and Dividing Numbers Using Scientific NotationUsing scientific notation can simplify calculations involving large numbers. First, convert numbers to scientific notation, and then use exponent rules to reduce the powers of 10.


Section 1.6: Order of OperationsOrder of Operations:1. Perform all operations that appear in parentheses (or other grouping symbols) first. If grouping symbols

are nested, do the innermost first.

2. Raise all bases to powers in the order encountered moving from left to right.

3. Perform all multiplications/divisions in the order encountered moving from left to right.

4. Perform all additions/subtractions in the order encountered moving from left to right.

Here, “other grouping symbols” means brackets [ ], braces { }, numerators or denominators of fractions, and radicals, like square roots. An example of a nested expression is . The innermost grouping symbol is (4+1) so the result is

.

Note that expressions like 2(3 + 7) can be evaluated in two ways: using either the order of operations, or the “Distributive Property:” a(b + c) = ab + ac.

In addition to parentheses, brackets and braces, certain symbols act as implied grouping symbols. The most important of these are the fraction bar and the square root symbol.

The fraction bar acts to separate the numerator from the denominator. If either or both of the numerator or denominator consist of an expression with operations, these must be performed first before the division indicated by the fraction bar. For example,


Section 1.7: Evaluating and Simplifying Expressions

Section 1.7.1 Evaluating a Variable Expression

When you are asked to evaluate a variable expression, replace the variables with given values, then follow the order of operations to simplify the remaining numeric expression to a single value.

Use the rules of significant digits to correctly round calculations involving measurements.

Section 1.7.2 Simplifying Expressions

Many times, we write down a variable expression to capture a formula for how to perform a multi-step calculation, but that expression is not as efficient as it could be. Simplifying the expression before we have to evaluate it allows for maximum efficiency when we finally do evaluate the expression, especially if we have to evaluate it over and over again.

Section 1.7.2 Addition and Subtraction of Variable TermsWe can only combine terms that have the exact same variable.

4x + 5x = 9x3a + 4b + 7a + 3b = 10a + 7b

The number in front of any variable is called the coefficient of the variable. The coefficient is the number attached a variable term by multiplication. To add or subtract variable terms, we combine the coefficients and keep the same variable (these are called like terms).

If two terms have different variable parts, then we cannot simplify the expression. For example, the expression 12b + 4w has no simpler form. How else could you write “12 bananas and 4 watermelons” without losing information?

Also note that a constant term (one with no variable) is different from any variable term. If terms have different variables, leave them alone. If they have the same variable, you may combine the coefficients.

Variable terms must also have the same exponent to qualify as like terms. 3x2 and 5x2 are like terms and may be added: 3x2 + 5x2 = 8x2.

Section 1.7.3 Multiplication using the distributive property

In general, we can write the distributive property as a(b + c) = ab + bc for all real numbers a, b, and c. In other words, when we have a multiplier in front of a parentheses group, we can choose to simplify and remove the parentheses provided that we multiply through to each term inside the parentheses. This is especially useful when variables are involved, because we may have no other way to simplify an expression with parentheses.

Section 2.1: Solving Linear Equations in One Variable


An equation is a mathematical statement where two algebraic expressions are set equal to each other.Equations are used when we know what we want the answer to a calculation to turn out to be, but we don’t know what number to put in the calculation to get that answer.

The addition property of equality states that you can add or subtract the same number from both sides of an equation without changing the fact that both sides remain equal.

Examples: 3 = 3 2(9-6) = 63 + 5 = 3 + 5 2(9-6) + 10 = 6 + 10

To use the addition property of equality to solve an equation, either add or subtract a number from both sides of the equation to make a zero on the side of the equation with the variable. Usually, the equation gets simpler-looking if you apply the property correctly.

You should always check your answers after solving an equation by putting your solution back into the original equation, and then verifying that both sides evaluate to the same number.

The second piece of knowledge you need to solve an equation is that sometimes you’ll need to simplify expressions in the equation before you use the addition property of equality. There are two simple ways to simplify: (1) Use the distributive law (i.e., multiply through any parentheses).(2) Combine like terms on each side of the equation.

The multiplication property of equality states that you can multiply or divide both sides of an equation by the same number without changing the fact that both sides remain equal.

Examples: 3 = 3 2(9-6) = 63 5 = 3 5 2(9-6) 2 = 6 2

To use the multiplication property of equality to solve an equation, either multiply or divide both sides of the equation by the same number to make a one as a factor in front of the variable. Note: you usually only use this property when you have a number times the variable alone on one side of the equation.

Procedure for solving any linear equation

1) First simplify each side of the equation as much as possible


a) Use the distributive property to remove parentheses groups

b) Combine like terms on each side.

2) Use the addition/subtraction properties of equality to move like terms to the same side of the equation. Your equation will have all variables on one side and all numbers on the other side at this point. Combine those like terms.

3) Use the multiplication property of equality to remove any remaining coefficient on the variable term by dividing both sides by the coefficient.

4) Check all solutions in the original equation to verify solutions. Correct for possible mistakes if your answers do not satisfy the equation.

Solve


Section 2.2: Rearranging Formulas and Solving Literal EquationsSome geometric formulas for perimeters:

Rectangle

Circle(perimeter is called

circumference)

Pi is a numeric constant, and is approximately: 3.14

Triangle

Some geometric formulas for areas:

Rectangle

Circle

Triangle

Trapezoid

Sphere


Box

Cylinder

Some geometric formulas for volumes:

Box

Cylinder

Sphere

PyramidB is the base area.

The base can be any shape (not just a

triangle, square, or rectangle)!

Cone


Section 2.3: Linear Inequalities in One Variable

Big Idea: Inequalities are algebraic expressions related >, <, , and . They are used when you want the result of a calculation to be greater than or less than a certain answer. Linear inequalities are solved exactly the same as linear equalities, except that if you multiply by a negative number, you have to reverse the inequality.

To solve an inequality means to find all values of the variable that satisfy the inequality.Any of these values is called a solution to the inequality, and the set of all possible solutions is called the solution set.Properties of Inequalities: An inequality can be transformed into an equivalent inequality by: adding or subtracting any quantity to both sides (the addition property of inequalities), or multiplying or dividing by any positive quantity (the multiplication property of inequalities). If both sides are multiplied or divided by a negative quantity, then the inequality symbol gets reversed.

Steps for Solving a Linear Inequality:1. Simplify each side separately.2. Isolate the variable terms on one side using the addition property of inequalities.3. Isolate the variable using the multiplication property of inequalities.

Section 2.4: Applied ProblemsSix Steps for Doing Word Problems (Applied Problems):

1. Read the problem carefully, and draw a picture if you can. Label the picture with all given information.2. Assign a variable, and write any other unknown quantities in terms of the variable.3. Write an equation, starting with a verbal equation if it helps.4. Solve the equation.5. State the answer, and verify that it makes sense.6. Check the answer in the words of the original problem.

Mixture Problems: Draw a picture with each container, a label written above each container, the amount held under each container, and the percentage held inside each container.

Interest Problems: Interest = Principal Interest Rate I = pr.

Money Problems: number value of one item = total value.

Speed Problems: distance = rate time d = rt. Draw a picture with all distances, rates (speeds), and times labeled.


Section 2.5: Percent ProblemsDefinition of Percent: A percent (%) indicates the number of hundredths of a whole.

1 part of 100 parts

To convert a decimal fraction to a percent: move the decimal two places to the right, then add the % symbol.

To convert common fractions and mixed numbers to a percent: use a calculator to express the number as a decimal fraction, then convert that decimal to a percent.

To convert percents to decimal fractions: drop the % symbol, then move the decimal two places to the left.

To convert percents to common fractions: express the percent as a decimal fraction, then find the common fraction equivalent of the decimal (and reduce).

Basic Calculations of Percentages, Percents, and Rates

Section 2.6: Percent Problems from FinanceBig Idea: Percents have a huge application in finance because interest is always computed as a percentage of the principal.

When you invest money, the amount you have invested is called the principal. Interest is an additional amount of money paid back to you for the inconvenience of not having your money available to you while it was invested. Interest is usually paid as a percentage of the principal. The longer you leave your money invested, the more interest it accrues, and thus your principal grows larger over time.

The principal you have after an investment period can be calculated by adding the percentage rate for that period to 100%, then multiplying the initial principal by that percentage. Exponents can be used to simplify the calculation when interest accrues over many periods.

Principal after compounding every year for N years at an annual interest rate R:

Principal after compounding every quarter for N years at an annual interest rate R:

Principal after compounding every month for N years at an annual interest rate R:


When you borrow money, the amount you borrow is also called the principal. Interest is the additional amount of money you pay back for the temporary use of the lender’s money.

Simple interest loan: everything is paid in one lump sum.

If we choose the following four parameters variables:1. P = the initial principal2. R = the annual percentage rate (APR) of interest3. N = the number of years over which the loan is paid off (the period of amortization) 4. M = the monthly payment

To calculate M, the monthly payment, knowing P, R, and N use the formula:

To calculate N, the number of years, knowing P, R and M use the formula:

To calculate P, the principal, knowing M, R, and N use the formula:

Section 2.7: Direct and Inverse Variation ProblemsIf two quantities vary directly or are directly proportional, (i.e., we say, ‘distance varies directly with time,’ or ‘sales tax is directly proportional to purchase amount’) then we write the equation describing the relationship as:

the first quantity equals a constant times the second quantitydistance = speed timesales tax = percentage purchase amount

or, in general, if ‘y varies directly with x,’ or ‘y is directly proportional to x,’ then:y = kx

where k is the constant of proportionality or rate of change.

To solve a direct variation problem:1) Write down the direct variation equation: Quantity1 = constant Quantity2

2) Put given numbers in for Quantity1 and Quantity2, then solve for the constant.3) Use the constant in the direct variation equation to find the desired quantity.


Joint variation is the case where one quantity is directly proportional to the product of two or more quantities.To solve a joint variation problem:

1) Write down the direct variation equation: Quantity1 = constant Quantity2 Quantity3 …2) Put given numbers in for the quantities, then solve for the constant.3) Use the constant in the direct variation equation to find the desired quantity.

Inverse variation, or inverse proportion is the case where one quantity is proportional to the reciprocal of another quantity. Examples include: your homework grade is inversely proportional to the number of hours spent watching TV, or the loudness of a sound you hear is inversely proportional to the square of the distance from the source.

To solve an inverse variation problem:1) Write down the inverse variation equation: Quantity1 = constant / Quantity2 2) Put given numbers in for the quantities, then solve for the constant.3) Use the constant in the direct variation equation to find the desired quantity.

Chapter 3: Algebra and the Graph of a Line

Section 3.1: Graphing a Linear Equation Using a Table of Values

Big Skill: You should be able to graph linear algebraic equations by creating a table of values and then plotting the points and connecting the dots to form a straight line. You should also be able to read a given graph for approximate values.

When there is a relationship between two quantities, it is helpful to understand that relationship with a picture. The main way we picture mathematical relationships is with a graph.

A graph is a picture formed by dividing the plane into four regions, called quadrants, with a pair of number lines that intersect at right angles. We put a point on the graph to represent the relationship for a single pair of values by moving horizontally along the first number line, and then vertically parallel to the second number line. The collection of all such points for the relationship is called the graph of the relationship. When we graph “A vs. B”, the vertical axis represents the amount of A, and the horizontal axis represents the amount of B. B is called the independent variable, and A is called the dependent variable.

Technique #1 for graphing a line: USING A TABLE OF VALUES 1. Solve the linear equation for the dependent variable (y if y and x are the only two variables in the

equation.)2. Pick some random values of the independent variable (x if y and x are the only two variables in the

equation).3. Calculate the values for the dependent variable using the equation; make a table of the values.4. Graph the points and connect them with a straight line.


Section 3.2: Graphing a Linear Equation Using the Slope Intercept MethodBig Idea: A shortcut for graphing lines is to understand that when you solve for the dependent variable, then the number multiplying the constant is where the line crosses the vertical axis, and the number multiplying the dependent variable is the “slope,” which tells you how far up and over to move to get to the next point.

One thing to notice when making a table of values to graph a line is that if you pick your “random” values of the independent variable to change by the same amount every time, then the dependent variable also changes by the same amount every time. Here is an example from Section 3.1 that illustrates this:

This constant rate of change for both variables is captured in a ratio called the “slope” of the line. The slope of a line is how much the dependent variable changes divide by how much the dependent variable changes. For the example above:

Notice that the slope is just the rate from the original statement of the problem, and also that it is the number multiplying m, the independent variable. Simply put, the slope converts a change in minutes to a change in dollars in this example.


A second thing to notice from the example above is that when m = 0 minutes, the cost was c = $10, and that $10 was the constant in the equation we got when we solved for the dependent variable:

The point on the graph (0 minutes, $10) is called the “vertical intercept,” or more commonly the “y intercept,” because most math books are so boring that they only ask you to graph x and y as the independent and dependent variables, respectively.

Technique #2 for graphing a line: SLOPE INTERCEPT METHOD Solve the linear equation for the dependent variable (i.e., solve for y if x and y are the only two variables in the equation) The constant term is the vertical intercept, or the y-intercept. Plot it. The factor of the x term is the slope. Use it to count up and over (or down and backward) to plot more points on the line.

Section 3.3: Graphing a Linear Equation Using Intercepts

Technique #3 for graphing a line: INTERCEPT METHOD Set the independent variable to zero, then solve the linear equation for the dependent variable (i.e., set x = 0 and solve for y if x and y are the only two variables in the equation). Plot the point (0, y). This is called the y-intercept. Set the dependent variable to zero, then solve the linear equation for the independent variable (i.e., set y = 0 and solve for x if x and y are the only two variables in the equation). Plot the point (x, 0). This is called the x-intercept. Draw the line between the two intercepts.


Section 3.4: Graphing a Linear Inequality

Big Skill: You should be able to graph inequalities by drawing either a solid or dotted line, and then shading in one side of the line or the other.

To graph a linear inequality:1. Graph the boundary line by solving for the dependent variable.

a. If the inequality uses or , then draw a solid line to show that the line itself satisfies the inequality.

b. If the inequality uses just < or >, then draw a dashed line to show that the line does not satisfies the inequality.

2. Shade the appropriate side.a. If the inequality is < or , shade the graph below the line.b. If the inequality is > or , shade the graph above the line.

Section 3.5: Solving a System of Two Linear Equations by GraphingBig Skill: You should be able to solve a system of two linear equations in two unknowns by graphing the lines and reading the intersection point off the graph.

Example of the type of problem the skills in this chapter help us solve:Suppose that one country club has a $200 membership fee, and the golf costs $36 per round. A second country club has no membership fee and the gold costs $40 per round. How many games would you have to play at each club so that the cost was the same?Let n = the number of games you playLet c = the cost of playing that number of gamesFor club #1: c = 36n + 200For club #2: c = 40n

Graph the lines and see where they cross…

From the graph, it looks like if we play 50 games at either club, the cost will be $2,000.00.


A system of linear equations is a grouping of two or more linear equations, each of which contains the same variables.

Examples:

In previous chapters, when we had just a single linear equation (in one variable) our goal was to find the single number that made the equation be true. That number was called the solution of the equation.Example:The equation 2x – 5 = 15 has a solution of x = 5, because 2(5) – 5 = 10 – 5 = 15

Now for this chapter, our goal is to find the solution to a system of linear equations, which consists of values for both of the variables x and y that make both equations in the system be true.Example:

The system of equations has solution (x, y) = (50, 2000) because:

2000 = 36(50) + 200 AND 2000 = 40(50)2000 = 1800 + 200 2000 = 20002000 = 2000

Solving a System of Two Linear Equations by Graphing Graph both the lines. Read the coordinates of the intersection point off the graph. Check to see if those coordinates are the solution.

We see from these examples that there are three different cases for the solution to a system of two equations in two variables. We describe these cases using the words: Consistent, which means that there is at least one solution (no solutions inconsistent) Dependent, which means that the graphs of the lines are the same (different lines independent)

Three Possible Cases for Solutions of a System of Two Linear Equations in Two Variables:

INTERSECT: The lines intersect at one point, and thus the system has exactly one solution. This type of system is called consistent and the equations are called independent.

PARALLEL: The lines never intersect (i.e., they are parallel to one another), and thus the system has no solutions. This type of system is called inconsistent and the equations are called independent.

COINCIDENT: The lines lie on top of each other, and thus the system has infinitely many solutions. This type of system is called consistent and the equations are called dependent.

We can identify which of these three cases a system of equations will fall into (without graphing) by putting both equations into slope-intercept form, and comparing the slopes and y-intercepts.


Section 3.6: Solving a System of Two Linear Equations by Algebraic Methods

Big Skill: You should be able to solve a system of two linear equations in two unknowns by solving one equation for one of the variables, then replacing that variable in the other equation with its expression.

Yesterday, we saw that the lines cross at the point (50, 2000), which means that if you play

50 games at either club, the cost will be $2,000. Here is how to solve this system using substitution.

Since the first equation already tells us that y = 36x + 200, we can replace the y in the second equation with 36x + 200:

Now that we know x = 50, we can replace the x in y = 36x + 200 to get:

So, we just derived that the solution to this system is (50, 2000).

Solving a System of Two Linear Equations Using the Substitution Method Solve one of the equations for one of the variables; pick the easiest variable to solve for. Replace that variable in the other equation with the expression you just derived. Solve your new equation; it should have only one variable in it. Substitute that answer into the first equation and solve it to find the value of the original variable. Check to see if those coordinates are the solution.

Chapter 4: Measurement

Section 4.1: Linear Measurements

Section 4.1.1: Conversions within the English System

Linear Measure : Area Measure : Volume Measure :


1 ft = 12 in1 yd = 3 ft1 mi = 5,280 ft1 mi = 1,760 yd1 rod = 16.5 ft1 furlong = 220 yd

1 acre = 43,560 sq feet1 acre = 160 sq rods1 sq mile = 640 acres

1 pt = 16 oz1 qt = 2 pt = 32 oz1 gal = 4 qt = 128 oz1 gal = 231 in3 1 ft3 = 7.480 519 gal

Weight : Time : Speed :1 lb = 16 oz1 ton = 2,000 lb1 stone = 14 lb1 slug = 32.174 049 lb

1 min = 60 s1 hr = 60 min1 hr = 3600 s1 day = 24 hours1 day = 1,440 min1 day = 86,400 sec1 year = 365.25 day1 year = 31,557,600 sec

1 mi / hr = 1.466 467 ft / sec

Section 4.1.2: The Metric System

Base Unit Symbol What it Measuresmeter m length

seconds s timegram g massliter L volumewatt w powerjoule j energy

newton N forcehertz Hz frequency

ampere A electric currentvolt V electric potentialohm resistancefarad F electric capacitancehenry H electric inductance


Prefix Symbol Power of Ten Number per Base Unit

nano n

micro

milli m

centi c

deci d

deca da

hecto h

kilo k

mega M

giga G

Conversions within the Metric System:Conversions within the metric system simply require the shifting of the decimal point.

Section 4.1.3: Performing Operations with Measurements

Section 4.1.3a: Addition and Subtraction of Measurements:The key is to add the like type of units.

Section 4.1.3b Multiplying and Dividing Measurements:We not only multiply/ divide the number parts of the measurements, we also multiply/ divide the unit parts of each measurement.

Section 4.2: Measuring Area The measurements in the previous section all involved single “fundamental” units for length, weight,

and time (and electrical charge…). All other measurement units are “compound units” made up of factors of the four fundamental units. The units of measurement for area are some of the most basic compound units there are. Area is used to measure the amount of space occupied by a two dimensional shape. The idea of area allows us to describe with one number the “amount contained” by a piece of cloth, a

piece of sheet metal, a computer screen, a wound, or a piece of land. Remember that measurement involves deciding on a unit of measurement, and then counting up how

many of those units are in the object we want to measure. To measure area, we use square units. A square unit is a square whose sides are each one unit in length.


For example, if we choose to measure area by with a square that is one foot on each side as our unit of measurement, then the area of a 2 ft by 6 ft rectangle is 12 square feet, because exactly twelve of the one-foot by one-foot unit squares fit in the 2 ft by 6 ft rectangle.

We called this unit measurement a square foot because it is a square that is one foot on each side. However, there is another, more algebraic, name for this unit. It is: 1 ft2. That is because we must also be able to connect this unit to the geometry formula for the area of a

square: A = (side length)(side length)

= (1 ft)(1 ft)= (1)(1)(ft)(ft)= 1 ft2

In general, when converting area units, you have to square the conversion fraction because the units of measurement are a dimension squared.


Section 4.3: Measuring Volume Remember that measurement involves deciding on a unit of measurement, and then counting up how

many of those units are in the object we want to measure. To measure area, we use cubic units. A cubic unit is a cube whose sides are each one unit in length. For example, if we choose to measure volume with a cube that is one foot on each side as our unit of

measurement, then the volume of a cube that is one yard on a side is 27 cubic feet, because exactly 27 of the one-foot by one-foot by one-foot unit cubes fit in the 3 ft by 3 ft by 3 ft cube.

We called this unit measurement a cubic foot because it is a cube that is one foot on each side. However, there is another, more algebraic, name for this unit. It is: 1 ft3. That is because we must also be able to connect this unit to the geometry formula for the volume of a

cube: V = (side length)(side length)(side length)

= (1 ft)(1 ft) (1 ft)= (1)(1)(1)(ft)(ft)(ft)= 1 ft3

In general, when converting area units, you have to cube the conversion fraction because the units of measurement are a dimension cubed.


Section 4.4: Conversion Between Metric and English Units

Linear Measure : Area Measure : Volume Measure :1 inch = 2.54 centimeters1 meter = 3.280 840 feet1 meter = 1.093 613 yard1 mile = 1.609 344 kilometer

1 in2 = 6.451 6 cm2

1 m2 = 1.195 990 yd2

1 mi2 = 2.589 988 km2

1 acre = 40.468 5642 are1 are = 100 square meters1 hectare = 100 are=2.471 054 acre

1 oz = 29.573 530 mL1 L = 1.056 688 qt1 gal = 3.785 412 L1 ft3 = 7.480 519 48 gal 1 ft3 = 28.316 846 59 L1 L = 61.023 744 in3

1 m3 = 1000 L1mL = 1 cm3

Weight : Speed : Temperature :1 kilogram = 2.20462262 pounds1 oz = 28.349 523 g

1 m / s = 2.236 936 mph1 m / s = 3.280 840 ft / s1 mph = 1.609344 kph


Chapter 5: Geometry

Section 5.1: Plane geometryBasic Facts About Points and Lines:

Two different points define one and only one line that passes through both of them We shall use capital letters such as A to label points and a pair of letters such as AB to label the segment

of the line between A and B. Line segments are measured in units of length such as feet or meters. The shortest distance between the two points is along the line passing through them. Two lines in a plane either meet (or “intersect”) at a single point or are parallel, meaning that the two

lines never touch. Where the lines meet four “openings” or angles are formed as shown below. The symbol is used for

the word angle. ( See figure below )

Angles

The vertex of an angle is the common point shared by the two line segments or lines that form the sides of the angle.

One way to name angles is to write the “angle symbol” , then the name for a point on one side, the name for the vertex, and then the name of a point on the other side.

Thus BAD is the angle with vertex at A and with sides that include segments AB and AD . This same angle could also be labeled as DAB .

Another way is to write the angle symbol, and then a single letter or number, like 1 in the above diagram.


Measuring Angles The most common unit for measuring angles is called the degree. The symbol for the degree is: The Babylonians decided that there are 360 degrees (360) in one full rotation. A right angle is one-fourth of a full rotation and measures 90. A straight angle is one-half of a full rotation and measures 180. An angle between 0 and 90 is called an acute angle. An angle between 90 and 180 is called an obtuse angle.

Complementary and Supplementary Angles Complementary angles have measures that add up to 90. Supplementary angles have measures that add up to 180.

Fractions of a Degree An old-school technique for talking about angle measures less than a degree is to subdivide the degree in

the same way we subdivide time: into minutes and seconds.

One minute of angle is one-sixtieth of a degree:

One second of angle is one-sixtieth of a minute:

Example of an angle measurement stated in degrees, minutes, and seconds (DMS):

Converting from degrees, minutes, and seconds (DMS) to decimal degrees (DD) Example: convert to DD Divide the number of second by sixty to convert it to an equivalent number in minutes.

o

Add that to the number of minutes, then divide by sixty again to convert to an equivalent number in degrees.

o

Add that number to the number of degrees, and round to an appropriate number of places:o

Note that . Thus, stating DMS measurements to the nearest thousandth of a

degree won’t result in much round-off error, but stating your answer beyond a ten-thousandth of a degree implies a precision not conveyed by mere seconds.

Note: on a graphing calculator, you can type in a DMS measurement using the degree and minute symbols found in the ANGLE menu, and the quotation marks for the seconds.


Converting from decimal degrees (DD) to degrees, minutes, and seconds (DMS) Example: convert to DMS Multiply the decimal portion by sixty to convert it to an equivalent number in minutes.

o

Multiply the resulting decimal portion by sixty again to convert to an equivalent number in seconds.

o

Round to an appropriate number of places:o

Note: on a graphing calculator, you can type in a DD measurement and convert it to a DMS measurement using the DMS function found in the ANGLE menu.

When two lines intersect, the angles that are opposite each other are equal. These equal angles formed by two intersecting lines are called vertex ( or vertical ) angles.

Parallel lines lie in the same plane and do not intersect.A transversal is a line that intersects two parallel lines.

Different angles formed by the transversal are given special names to reflect special relationships between their measures:

o Corresponding angles are on the same side of the transversal and on the same corresponding sides of the parallel lines. Corresponding angles have equal measures

o Alternate interior angles have equal measures.o Alternate exterior angles have equal measures.o Interior angles on the same side of the transversal are supplementary (add to 180).


PolygonsPolygons are closed figures in the plane whose sides are line segments.

TrianglesThe simplest polygon is the three-sided triangle. The points at the corners A , B , and C are the vertices of the triangle, and the angles BAC , BCA , and ABC are called the interior angles of the triangle. The triangle is often then labeled as triangle ABC .

Types of TrianglesTriangles are classified according to their angles and sides.

Angle classifications:o Acute triangles have all angles less than 90.o Right triangles have one angle of 90.o Obtuse triangles have one angle greater than 90.

Side classifications:o Equilateral triangles have all sides the same length. Also, all angles are 60.o Isosceles triangles have two sides of the same length, and the angles opposite those two sides are

equal.o Scalene triangles have no sides of the same length.

Congruent triangles have all the same side lengths and all the same angles. Congruent is the proper way to say that two triangles are “equal” or “the same”. We use the word congruent because there are really six measurements that have to be equal for two triangles to be the same: the three angle measurements, and the three side length measurements.The three rules for congruency when using triangles: Side Angle Side ( abbreviated SAS) , Angle Side Angle ( abbreviated ASA ), and Side Side Side ( abbreviated SSS ).


In the diagram shown, the hypotenuse is c and the legs are a and b . The two acute (less than ) angles are 1 and 2 with 1 opposite to the side of length a and 2 opposite to the

The Pythagorean Theorem: .

To calculate a leg, say a, knowing the hypotenuse and the other leg, b, rearrange the formula to

or .

Other Polygons ( Polygons with more than 3 sides )

In general, if a polygon has n sides, then the sum of the interior angles is (n – 2)(180).


Perimeter and Area

Perimeter is the total linear distance around the boundary of a polygon (or any closed shape).Area is the number of unit squares of measurement it takes to fill a closed shape.

The area of a rectangle, is given by .

Area and Peremiter of a Parallelogram

A parallelogram is a four sided polygon (or quadralateral) with opposite sides parallel. All squares and rectangles are parallelograms, but a general parallelogram does not have to have

angles.

Then,

Area and Perimeter of a Triangle

One easy formula for the area of a triangle comes from measuring one side (the base b in the picture below), and the perpendicular height (h in the picture below) from the base to the top vertex.

.

, where .


Area and Perimeter of a Trapezoid

A quadrilateral with two opposite sides parallel is called a trapezoid. Suppose that the two parallel faces have lengths a and b and are separated by a perpendicular distance h .

Circles


Section 5.2: Radian Measure and its Applications

To convert decimal degree measurements to radians we use the conversion factor .

To convert radian measure to decimal degrees we use the conversion factor .


Section 5.3: The Volume and Surface Area of a Solid

Volume and Surface Area of a Rectangular Prism

Probably the most recognizable prism is the rectangular prism or “box”. The volume is simply the product of the lengths of the three sides.


Volume and Surface Area if a Cylinder

A right circular cylinder is also a prism. Here the circumference of the base circle is used in calculating the lateral surface area.

Volume of a Pyramid and Cone

Right pyramids and cones both have a volume equal to one third that of the corresponding prism having the same base and height.

Volume and Total Surface Area, A, of a Sphere

College Algebra FINAL EXAM Review, Fall...

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