Algebra Final Review Outline

Unit 1: PatternsInvestigation 1: Representing Patterns Recursive Rules: Explain an algebraic pattern using words. It starts with a given amount and then how much the amount changes for each occurrence.

Explicit Rules: Explains an algebraic pattern using an equation. Sometimes you need to go back to the zero term (when the x value is zero) if it is not given.

Example: Write a recursive rule and explicit rule from the data in the table:

Time Number of Birds

1 5

2 8

3 11

4 14

For recursive and explicit rule, determine how much the time is changing by and how much the number of birds is changing. The time is changing by 1 minute The number of birds is changing by 3.Also - if you are not starting at time of zero, use how the birds are changing to find the zero term (when time is zero). In this example, the zero term is number of birds equals 2 ( 5 - 3).

The recursive rule is: Start with 2 birds then add 3 more birds each minute. The explicit rule is: 2 + 3x where x represents the time Word Problem Example:

Nate has $323.47 in his piggy bank. He is saving for a used car that costs $1,500. How many months will it take him to have enough to buy the car if he saves an additional $124.82 each month

Nate starts with $323.47 (this is the zero term since this is what he starts with when the time variable is 0 months) then he is saving $124.82 per month (this is how much his piggy bank total is changing each month)

So the recursive rule is: Start with $323.47 and add $124.82 each month explicit rule is: 323.47 + 124.82x where x is the number of months he has been saving. The question asks how long will it take to save $1,500 to buy the car so set the explicit rule to $1,500 323.47 + 124.82x = 1500 then solve

Investigation 2: Patterns with Integers Reviewed topics such as operations with integers (adding, subtracting, multiplying and dividing), combining like terms, order of operations, creating algebraic expressions from word problems Integer Rules:

Operation Both Positive Numbers

Both Negative Numbers

One Negative and One Positive Number

Addition add numbers together and answer is positive.Ex. 3 + 5 = 8

add numbers together and answer is negative.Ex. - 5 + - 6 = -11

Subtract the smaller number from the larger number then keep the sign of the larger number.Ex. - 7 + 3 = -4( 7 - 3 = 4 & the sign of the larger number is - so the answer is -4)

Subtraction Same- Change-Change -the first number stays the same, change the subtraction sign to a plus sign then change the sign of the last number to its opposite then use the rules for adding IntegersEx. 4 - 3 → 4 + -3 = 1 -4 - 3 → -4 + -3 = -7 -4 - (-3) → -4 + (+3) = -1 4 - (-3) → 4 + (+3) = 7

Multiplication and Division

Multiply or divide the numbers as usual - if the numbers have the same sign (both positive or both negative) then the answer is positive Ex. 3 x 4 = 12 -3 x -4 = 12If the numbers have different signs (one positive, one negative) then the answer is negative Ex. - 3 x 4 = -12 5 x - 4 = -20

Combining Like Terms: If the terms have the same variable and are on the same side of the equal sign, you can add or subtract the terms. Ex. -3x + 5 + 4y - 3 + 5x - You can combine the x’s ⇒ -3x + 5x = 2x - You can also combine the numbers ⇒ 5 + -3 = 2 - You can’t do anything with 4y since there is no other term with a y so your final answer is 2x + 2 + 4y

Order of Operations: When working with a problem that has several operations you need to do them in a specific order: Parentheses - simplify whatever is in the parentheses first - use order of operations within the parentheses Exponents - raise any numbers or variables to the exponent Multiplication and Division - done at the same time from left to right Add and Subtract - done at the same time from left to right

Ex. -5(7 x 5 - 4)2 + 5 x 6 -5 (35 - 4)2 + 5 x 6 First - Parentheses - do multiplication first then subtract -5 (31)2 + 5 x 6 next - Exponents (31 x 31) -5(961) + 5 x 6 next Multiplication and/or Division -4805 + 30 finally addition and/or subtraction - 4775 final answer

Investigation 3: Arithmetic Sequences Arithmetic Sequences - add or subtract the same amount each time ( multiply )

Ex. Complete the arithmetic sequence:

7, 12, 17, 22, 27, 32, _____, _____,_____

First find out what the difference is between each number. In this case, it is 5 (12 - 7 = 5, 17 - 12 = 5, 22 - 17 = 5 and so on). This means we need to add 5 to 32 to get the next number so the next number is 37. Then add 5 to 37 to get 42. Finally add 5 to 42 to 47. The final sequence is

7, 12, 17, 22, 27, 32, __37___, ___42__,___47__

Word problem example:

A fire truck’s water tank holds 4,500 gallons of water. If the water flows out of the tank at 550 gallons per minute, how much will be left in the tank after the trunk pumps water for five minutes?

The fire truck starts with 4,500 gallons of water (this is our zero term since this is how much water is in the pool when the time variable is 0 minutes). Water flows out at 550 gallons per minute (since the water is flowing out it is going to be a negative number). So our arithmetic sequence can be represented by the explicit rule of 4500 - 550x (where x is the number of minutes the water has been flowing out) To answer the question - put the 5 in for x 4500 - 550(5) = 4500 - 2750 = 1750 There are 1,750 gallons of water left in the tank after 5 minutes.

Investigation 4: Geometric Sequences & Investigation 5: Patterns with Fractals Geometric Sequences - multiply or dividing by the same amount each time (uses exponents) Ex. Finish the Geometric Sequence 5, 30, 180, ______, ______, ______ - looking at this sequence, you can tell that it does not go up by the same amount each time so we need to think about what we would multiply or divide by to get the next number. 5 x 6 = 30 then 30 x 6 = 180 so to get the next number, you will need to multiply 180 x 6 then multiply that answer by 6 so the sequence will be: 5, 30, 180, _1080_, _6480_, _38880__ the explict rule for this is 5 x 6n - the 5 is the value we start with (zero term) - we are multiplying by 6 to get the next term and the n represents the stage in the sequence. Since 5 is the zero term, 30 is the 1st term so the equation is 5 x 61, 180 is the second term so the equation is 5 x 62 and so on.

Word Problem example: Congratulations! You have just won first prize in a poetry writing contest. If you take the $100 you won and invest it in a mutual fund earning 5% interest per year, about how long will it take for your money to triple? The zero term is 100 since that is what you started with when the time variable is 0 years. You are earning 5% per year so that is your change per year (or what you are multiplying by) except you need to change it to 1.05 (.05 for the 5% and 1 to keep the 100 from getting smaller) The number of years is the exponent part of this. The explicit rule would be: 100 x 1.05n

Using your calculator – 1) Put in 100 then hit Enter2) Then put in * 1.05 and hit Enter 3) then continue to hit Enter and count how many times you hit Enter until the

amount is greater than or equal to 300 for this problem For this problem – I hit enter 23 times so it means that it would take 23 years to triple the original $100.

Unit 2: Equalities and Inequalities Investigation 1: Understanding Algebraic Expressions Story of x - being able to read an algebraic expression to create words or from words create an algebraic expression

Investigation 2: One-Step and Two-Step Linear Equations Solving One Step Linear Equations:

Goal is to get the variable by itself so what ever is on the same side as the variable needs to be moved to the other side using the opposite operation

Example: x + 3 = 18 => goal: get x by itself - need to move 3 to other side - 3 -3 => opposite of addition is subtraction & do the same to x = 15 both sides

x - 6 = -4 => goal: get x by itself - need to move 6 to other side +6 +6 => opposite of subtraction is addition & do the same to x = 2 both sides

2x = 6 => goal: get x by itself - need to move 2 to other side ÷2 ÷2 => 2 is mulitplying x so divide both side by 2 x = 3

x ÷ 4 = 12 => goal: get x by itself - need to move 4 to other side x4 x4 => since x is being divided by 4 - need to multiply both x = 48 sides by 4

Solving Two Step Linear Equations - Start with adding or subtracting whatever can be from the side with the variable

to the other side then multiply or divide to get x by itself Example: 2x + 5 = 13 => goal: get x by itself, start by subtracting 5 from both sides - 5 -5 2x = 8 => then divide by 2 ÷2 ÷2 x = 4

Investigation 3: Combining Like Terms to Solve Equations This builds on solving 2 step equations. Before beginning to solve the equation, combine any like terms, either with variable or just constants Example: 3x + 7 - 5x + 4 = 12 ⇒ add together the terms that have x’s in them and the 3x + -5x = -2x terms that only have numbers - make sure when you 7 + 4 = 11 are adding the terms together to take the sign with -2x + 11 = 12 the term - don’t combine across the equal sign

Now solve the two step equationInvestigation 4: Solving Equations Using the Distributive Property This builds on solving 2 step equations also. Before beginning to solve the equation, use the distributive property then combine like terms. Example: 3(2x + 4) - x = 20 ⇒ Use the distributive property to multiply everything in 3(2x) + 3(4) - x = 20 parentheses by 3 6x + 12 - x = 20 next combine like terms, in this example the x’s 5x + 12 = 20 6x - 1x = 5x - now solve - subtract 12 from both side - 12 -12 5x = 8 then divide each side by 5 ÷ 5 ÷ 5 x = 8/5

Investigation 5: Formulas and Literal Equations Solving Formulas means putting the information you are given into the formula and solving it. Example: If converting a Celsius temperature to Fahrenheit – use the formula below and put in the value you have for Celsius where the C is

F=95C+32

If the temperature is 45˚C, put 45 in for C and solve. F = 9 x 45 + 32 multiply first 5 1 F = 81 + 32 = 113˚F

Solving Literal Equations means to solve an equation for a variable in the equation. You will not get a number for an answer. Example: 2a + 4b = 10; Solve for a. Use the same steps you use for solving an equation so the first step is to add or subtract whatever is not associated with the variable a for this example. 2a + 4b = 10 ⇒ subtract 4b -4b -4b since you can not combine 10 & 4b they will stay separate 2a = 10 - 4b now divide both sides by 2 so a is all alone ÷ 2 ÷ 2 you need to divide everything on both side by 2 to get a = 10 - 4b which can be simplified by dividing the 10 & -4b by 2 2 a = 5 - 2b

Investigation 6: Linear Inequalities Solve Inequalities the same way you solve an equation. Inequalities mean that that there is more than one answer to the equation. Any of the values that satisfy the inequality is a possible answer. At the end of solving the equation, if you multiplied or divided by a negative number then you flip the inequality sign. Example: 2x + 4 < 12 - solve as usual - 4 -4 - subtract 4 from both sides 2x < 8 ÷ 2 ÷2 - divide by two x < 4 - any number less than 4 is a possible answer - the final step is to graph the answer ( if the symbol is less than or greater than, use an open unfilled in circle) ←--|----|----|---|----|----|----|----|---> 0 1 2 3 4 5 6 7 Example 2: -5x - 6 ≤ 34 - solve as usual + 6 + 6 - add 6 -5x ≤ 40 ÷-5 ÷ -5 - divide by -5 x ≤ - 8 - since you divided by a negative number you need to flip the ≤ x ≥ -8 - any number greater than or equal to -8 is a possible answer - the final step is to graph the answer ( when graphing a greater than or equal or a less than or equal - the circle is filled in)

←--|----|----|----|----|----|----|----|----|----|----|----> -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Unit 3: FunctionsInvestigation 1: Relations and Functions

RelationsA relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. For example, the relation can be represented as:

Mapping Diagram of Relation Graph of Relationy is not a function of x (x = 0 has multiple outputs)

FunctionsA function is a relation in which each input x (domain) has only one output y(range).To check if a relation is a function, given a mapping diagram of the relation, use the following criterion:1. If each input has only one line connected to it, then the outputs are a function of the inputs.2. The Vertical Line Tests for GraphsTo determine whether y is a function of x, given a graph of a relation, use the following criterion: if every vertical line you can draw goes through only 1 point, y is a function of x. If you can draw a vertical line that goes through 2 points, y is not a function of x. This is called the vertical line test.

Investigation 2: What Is a Function?

The independent variable corresponds to the input values or x values. The dependent variable corresponds to the output values or y values. We say that the dependent variable is a function of the independent variable. This means that the dependent variable’s value is dependent on the independent variable’s value.The set of all possible inputs is called the domain of the function. The set of all possible outputs is called the range of the function. When we present a relation in a table the left column always contains the inputs, also called the independent variable. The dependent variable is always in the right column.

Example: Month is the input, also known as the independent variable or X. The domain for this example is (1,2,3,4,5,6,7,8,9,10,11,12) Inches is the output, also known as the dependent variable or Y. The range for this example is ( 2.65, 3.44, 3.61, 3.66, 3.79, 3.82, 3.83, 3.91, 3.93, 3.99)

Table 1Average Precipitation for Hartford, Connecticut

Month Inches

1 3.66

2 2.65

3 3.61

4 3.82

5 3.99

6 3.83

7 3.93

8 3.83

9 3.83

10 3.91

11 3.79

12 3.44

Investigation 3: Function Notation and Evaluating Functions

Function notation There is a special notation, that is used to represent this situation: if the function name is f , and the input name is x ,then the unique corresponding output is called f(x) (which is read as " f of x ".)We can also use letters: g(x), h(x) or simply y

Question: What does the function notation g(7) represent? Answer: the output from the function g when the input is 7 Question: Suppose f(x) = x + 2 . What is f(3) ? Answer: f(3) = 3 + 2 = 5 (simply substitute number 3 for the variable x) Question: Suppose f(x) = x + 2 . What is f(x+5) ? Answer: f(x+5) = (x + 5) + 2 = x + 7

Example: Ben is trying out for the school’s basketball team. He has been shooting free throws after school for an entire week, trying to get better. He shot 100 free throws on the first day, and then each day after that he shot 10 more free throws than the day before. Create a function that models the number of free throws f taken each day according to the day number d.

(a) Independent variable: Days throwing free throws

(b) Dependent variable: Number of free throws

(c) Write the equation for this function:

f = 90 + 10d

(d) Use function notation to express the function: f(d) = 90 + 10d

(e) We can say the number of free throws is a function of the number of days he has be practicing.

(f) Find the number of free throws Ben shot on day 4. Use function notation. f(4) = 90 + 10(4) = 130

(i) Complete the table below. (Started at day zero by subtracting 10 from his first day total for my equation

InputDay

OutputFree throws

0 90

1 100

2 110

3 120

4 130

(g) Find the day that Ben shot 160 free throws. 160 = 90 + 10d - solve for d -90 -90 70 = 10d ÷10 ÷10 d = 7

(h) What are the domain and range of this function? Domain is the number of days ( 0, 1, 2, 3, 4, …..) or positive whole numbers Range is the number of free throws thrown ( 90 and above )

Unit 4: Equations -

Linear equations have a constant rate of change. Rate of change is also known as slope. Non-linear equations do not have a constant rate of change.

Unit 5: Scatter Plots and Trend Lines Investigation 1: One Variable Data Measures of Center:

Three Measures of Center

The mean is the average that you're used to, where you add up all the data values and then divide by the number of values.

The median is the "middle" value in a list of numbers. To find the median, first list the numbers in numerical order. Then, if the number of values is odd, the median is the number in the middle. If the number of values is even, the median is the mean of the two numbers in the middle

A mode is a value that occurs most often. If no number is repeated, then there is no mode for the list. Some lists of numbers may have more than one mode.

Histograms - A histogram is like a dot plot, except that the values are grouped into intervals called bins. To create a histogram, we must have a frequency table. A frequency table contains a set of bins and the number of data values contained in each bin. The number of data values in a bin is called the frequency of the bin. We can draw a graph and represent each bin with a bar. The height of a bar shows the frequency of the bin.

The Five-Number SummaryWe often are interested in how much spread there is in a data set. The spread, or variability, of data describes how far apart the data values are. A set of statistics that help us see the amount of spread in a data set is the five-number summary. The five-number summary consists of the minimum, Q1, median, Q3, and maximum. Q1 and Q3 are the first and third quartiles. The median equals Q2. Quartiles divide a data set into four quarters.

To create the five-number summary of a data set, start by ordering the data set into increasing or decreasing order. Then, find the median (middle) of your data set. The median divides the data set into two halves. To find the quartiles, find the median of the lower half and find the median of the upper half.

Example: Below are the arm-spans (in cm) of 15 Algebra I students:

Minimum Q1 Median Q3 Maximum

148 152 152 152 154 154 154 162 163 164 164 170 172 180 181

Solution: The data values are already ordered. There are an odd number of values, so the median is the middle number in the list. The lower half and upper half are shown in boxes. Each half has 7 data values. The median of the lower half is 152, and the median of the upper half is 170.

148 152 152 152 154 154 154 162 163 164 164 170 172 180 181

Median of the lower half (Q1) Median of the upper half (Q3)

Middle of the data set (Median = Q2)

Rules for Finding the Median & Quartiles

When you have an even number of data values, the median equals the average of the middle two numbers. If the lower half and upper half of the data set also have an even number of values, Q1 and Q3 will be the average of the middle two numbers in the lower half and upper half, respectively.

IQR - Interquartile Range - Q3 - Q1 = IQR

Outliers:Calculating Upper and Lower Fences

Q1 - 1.5x IQR = lower fence - anything less than this number is considered an outlier

Q3 + 1.5 x IQR = upper fence - anything greater than this number is considered an outlier

A box-and-whisker plot is a convenient way to display the five-number summary. To draw a box-and-whisker plot:

a. Mark the minimum, maximum, median, Q1, and Q3 above the numbers on your number line.

b. Draw a box that represents the middle 50% of the data by drawing a box from Q1 to Q3. The length of the box represents the interquartile range (IQR).

c. Draw a vertical line segment inside the box to show the median.d. Draw “whiskers” to represent the lowest 25% (by connecting Q1 to the minimum value) and

highest 25% (by connecting Q3 to the maximum value) of the data.

Investigation 2: Introduction to Scatterplots and Trend Lines

Scatterplots are graphs that have the points graphed but they do not form a nice straight line so we need to find a line of best fit (also called a trend line). To draw a line of best fit, try to draw a line that best represents how the points are on the graph with approximately the same amount of points below the line and above the line. Once you have drawn the line, take two points that are on the line you have drawn and use them to find the slope then to find an equation. Example: I have drawn a line and choosen the points ( 26, 150) and (46, 300) (These are made up for use in this example) First I will find the slope using y2 - y1 = 300 - 150 = 150 = 7.5 ← slope x2 - x1 46 - 26 20

Second I will use the point slope form of the equation to get an equation point slope form is y - y1 = m(x - x1) and substitute in one of the points from above into x1 and y1 and the slope for m. It does not matter which point you use. y - 150 = 7.5(x - 26) - next use the distributive property y - 150 = 7.5x - 195 - then add 150 to both sides + 150 +150 y = 7.5x - 45 → now it is in slope intercept form (7.5 is the slope and -45 is the y intercept)

Make sure that you can explain what the slope and the y-intercept mean in the context of the problem

2. Solve using substitution              a.)  Look at the two equations. Is there one equation that has a variable that is alone on one side of the equation?  If so, use that and substitute that value in the other equation.                    Example:      2x + 3y = 12                                           x = 3y + 1 => you will substitute the 3y + 1 in for x in the first equation                                                                     to get  2(3y + 1) + 3y = 12              b.)  If there is not one equation that equals a value then you can solve for one of the variables (get one of the variables on a side by itself)  and then use that equation as you did in the step before.                      Example:      3x + 5y = 25                                            4x + 2y = 10  =>   solve for y                                                 4x + 2y = 10                                            -4x             -4x                                              2y = -4x + 10                                                 2             2                                             y = -2x + 5 <=   use this to substitute for y in the first equation                                            3x + 5(-2x + 5) = 25               c.) Once you have made the substitution, use the distributive property if needed and solve the equation for the variable.              d.)  Use your answer from part c and put it back into one of the equations and solve for the other variable.                 e.)  Put your answer in the form of an ordered pair  (x, y).

3. Solve using elimination               a.)  First, make sure the variables are lined up in the same order.               b.)   Add the equations together.   If you eliminate one of the variables, you can solve for the variable that is left.                                   x + 2y = 10                                   3x - 2y =   4                                   4x + 0 = 14   - solve for x    

               c.)   If you add the two equations together and one variable is not eliminated, you will need to multiply one or both of the equations by a number that will help you to eliminate one of the variables.     Multiply the entire equation by the number then add the equations together and solve for the variable that is left.

                                5x + 7y = 51   =>  multiply by 3  =>   15x + 21y = 153                                -3x + 4y = 28   =>  multiply by 5  =>   -15x + 20y = 140                                                                   Now add                  0   + 41y = 293

               d.)  Once you have solved for the first variable, put it back into either equation and solve for the other variable.

Example:        2x - y = 9              if you add them together neither variable is eliminated so                      3x + 4y = -14                we need to multiply each equation so they will eliminate                                                               one variable

                 -3(2x - y = 9)          =>     -6x +  3y =  -27                   2(3x + 4y = -14)    =>      6x   +8y   =   -28                              Now add them                  0  + 11y = -55     then solve for y                    divide each side by 11               y = -5                                   Use y = -5 and put into either equation.   I am going to use the first equation.                             2x - (-5) = 9                              2x  +  5  = 9             -  subtract 5 from both sides                                    2x  =   4             -  now divide by 2                                      x = 2                   the final answer is  (2, -5)   

Exponents:

Basic Rule Example 1 Example 2

PRODUCT RULE

QUOTIENT RULE

NEGATIVE EXPONENT

ZERO EXPONENT

RAISING EXPONENT TO POWER

When working with exponents: 1) Raise any numbers or variables that are inside parenthesis 2) Multiply or divide the coefficients first3) Work with the same variables (multiply, add or subtract exponents) 4) If you have any negative exponents – change them to fractions then multiply everything

together

Example: (2x2y)33x4y-8 - first distribute the 3rd power to everything in parenthesis 23x2*3y1*3 3x4y-8

8 x6y3 3x4y-8 - now multiply coefficients 8 * 3 24x6y3 x4y-8 - now work with the variables 24x10y-5 - since y is to the -5 power – change to fraction 24x10 (1 ) - now multiply them together y5

24x 10 y5

Factoring using GCF GCF is Greatest Common Factor For a given polynomial - find the common factors that each term has then take it out of the term by dividing. Example: 4x4y5 + 12x3y3 + 8xy2 - look at what each term has in common the coefficients have 4 in common, all terms have at least one x and y2 so the GCF for all of these is 4xy2. From each term, you will divide by the GCF. 4xy2 (x3y3 + 3x2y + 2) to check if you are correct – multiply the answer together and you get the original problem Multiplying Binomials (expressions with two terms) In multiplying binomials, you have to multiply each term to each term in the second set of parenthesis. Use of F O I L (acronym which stands for F- First, O –Outer, I – Inner, L – Last) Example: ( 3x + 4) (2x – 5)

First “F”– multiply the first terms in each parenthesis 3x(2x) = 6x2

Second “O” - multiply two outer terms 3x(-5) = -15x Third “I” - multiply the two inner terms 4(2x) = 8x Last “L” - multiply the last terms in each parenthesis 4(-5) = -20

Now put all the terms into 1 expression and combine any like terms 6x2 – 15x + 8x - 20 you can combine the -15x + 8x The final answer is: 6x2 – 7x – 20