Algebra 1 State Test Cram Packet

Created by Mrs. Hamrick 7

A variable is a letter that represents a number. Treat it just like you would treat any #. An algebraic expression is a #, variable, or combination of the two.

Addition: sum, plus, add to, more than, increased by, total

Subtraction: difference of, minus, subtracted from, less than, decreased by, less

Multiplication: product, times, multiply, twice, of

Division: quotient, divide into, ratio

Exponents: Power of, Raised to, to the __power

Real # system:

A. Rational #s

a. Fractions (All terminating fractions & All integers)

b. Integers (…,-3,-2,-1,0,1,2,3)

c. Whole #s (0,1,2,3)

d. Natural #s (1,2,3,…)

B. Irrational #s

a. Anything that doesn’t make sense… (Non-repeating, non-terminating decimals, pi, uneven roots, etc.)

Absolute value: measures the distance that x is away from the origin on the # line. ALWAYS MAKES WHATEVER IS INSIDE THE BARS POSITIVE…

PROPERTIES:

-Commutative: Changes the order of addition or multiplication without changing results.

-Associative: Moves the grouping symbols to change the order without changing the results.

-Distributive-Multiplies everything on the outside of the parenthesis times everything on the inside of the parenthesis. This is usually a monomial times a polynomial.

-Additive Identity: Add 0 to a #, you get what you started with.

-Multiplicative Identity: Multiply 1 times a #, you get what you started with.

-Additive Inverse: Add the opposites together and they cancel out to get 0. a + (-a) = 0.

-Multiplicative Inverse: Multiply the inverses together, and they cancel out to get 1.

-Symmetric: Doesn’t matter what side of the equation you write it on.

-Substitution: Allows you to solve one part of the equation and plug it in to solve another part.

-Transitive: If a = b and b = c, then a = c.

<,> In order are equals, less than or equals to, greater than or equals to, not equals, less than, greater than. The equals to sign is used for equations. All other signs are used for inequalities.

Same signs: ADD & KEEP THE SIGNS. Different signs: SUBTRACT & KEEP THE SIGN OF THE BIGGER #.

When you multiply or divide with negative signs, every two negative signs cancel out to make a positive.

A MINUS A NEGATIVE IS A POSITIVE! -(-X) = +X

Order of Operations: Please Excuse My Dear Aunt Sally. We always multiply/divide and add/subtract from left to right.

ZERO RULES: Anything times zero is zero. Zero divided by anything is zero. Anything divided by zero is UNDEFINED!!!

Polynomial: Algebraic expression containing 1 or more than 1 terms.

Find the Degree of a Polynomial: Find the degree of each term and pick the highest. (Find the degree of each term by adding the exponents on the variables. If there is no variable for a term, then that is a zero degree term.)

Descending order polynomial: The powers of x go from greatest to least. Example: x4+2x3-4x+6.

Ascending order polynomial: The powers of x go from least to greatest. Example: -3+x-2x2+4x3

Monomial: one term. Binomial: two terms. Trinomial: three terms. (Remember a term is a #, a variable, or the product of a # and a variable.)

YOU ALWAYS, ALWAYS, ALWAYS COMBINE LIKE TERMS TO SIMPLIFY. Remember: Like terms have the exact same variables with the exact same exponent. Example: x and 2x are like terms, but x and 2x2 are not like terms.

EXPONENT RULES:

To add polynomials: Just combine like terms…

To subtract polynomials: Change all of the signs in the polynomial that is being subtracted and then combine like terms.

To multiply polynomials: Use the distributive property until every term of one polynomial is multiplied times every term of the other polynomial.

To divide polynomials: See if you can factor out anything and cancel terms. (We seldom see this.)

FOIL: First terms, Outer terms, Inside terms, Last terms

Created by Mrs. Hamrick 7

Algebra 1 State Test Cram Packet

Created by Mrs. Hamrick 7

Solving one step equations:

Use opposite operations to move terms from one side of the equation to the other, so that you get your variable by itself, and positive, on one side.. (Opposite operations: addition/subtraction; multiplication/division; powers/roots.)

Example:

x - 2 = 10

+2 +2

X = 12

Solving Multi-step equations:

Simplify each side first. Use opposite operations to move terms from one side of the equation to the other so that you get your variable by itself, positive, on one side of the equals sign.

Example:

2x – 4 = 16

+4 +4

2x = 20

X = 10

Solving equations with variables on both sides of the equals sign:

Simplify each side first. Use opposite operations to move terms from one side of the equation to the other side. You need to get all of your variables together on one side of the equals sign and all of your constants on the other side of the equals sign. Make sure that you keep using opposite operations to move terms until your variable is completely alone on one side of the equals sign and positive.

Example:

2( x + 2) = 6x – 4

2x + 4 = 6x – 4

-2x -2x

4 = 4x – 4

+4 +4

8 = 4x

2 = x

Created by Mrs. Hamrick 7

We work inequalities the exact same way that we work equations. The only rule is: If you multiply or divide the variable by a negative #, then you have to flip the inequality sign.

Created by Mrs. Hamrick 7

Graphing inequalities on a number line:

Open circles mean it cannot equal a number. Filled in circles mean it can equal a number. We shade in what the solution can be.

Example: x + 4 > 10

-4 -4

x > 6

When graphing compound inequalities: AND means find where the two graphs overlap; If they never overlap, then the answer is no solution. If they always overlap, then the answer is all real #s. OR means to find both graphs simultaneously. If the entire graph ends up being shaded, then the answer is all real #s.

Example: Graph the solution set of x ≥ -2 AND x <3.

Example: Graph the solution set of x ≥ -1 OR x <-4.

Absolute value equations and Inequalities will most always have two answers. We work these by setting up two separate problems. One problem is the positive version of the original problem, and the other is the negative version of the original problem.

Example:

Positive Version Negative Version

+(x – 3) = 5 -(x - 3) = 5

+3 +3 -x + 3 = 5

x = 8 -3 -3

-x = 2

X = -2

So, your two answers would be x = -2 , 8.

If the problem is set up to equal a negative #, then the answer would be no solution. You should remember that absolute value is a measure of distance, which can NEVER be a negative #. An example of this would be if the problem looked like: .

Work absolute value inequalities the exact same way…

Created by Mrs. Hamrick 7

Algebra 1 State Test Cram Packet

Created by Mrs. Hamrick 7

SLOPE: is a measure of the rate of change.

(Remember the subscripts just tell you which ordered pair that letter came from: Point 1 or Point 2.)

Ordered Pairs are always listed in the form ( x, y ).

Slope can be found by:

· Looking at the graph and finding the rise/run.

· By plugging in values from two ordered pairs.

· By looking at the equation in slope-intercept form ( y = mx + b)

Zero slope: a Horizontal line

Undefined slope: a Vertical line.

MIDPOINT of a line segment:

Don’t forget about what the subscripts mean!!! Look above if you can’t remember.

DISTANCE formula (also known as how to find the length of a line segment):

Just plug in the values from the two ordered pairs and

COORDINATE PLANE:

Some examples of slope are shown below.

There are several ways to GRAPH AN EQUATION:

Using a Calculator (EASIEST WAY):

1. Solve your equation for y first by hand.

2. Click Y= .

3. Clear out anything on the screen by using the clear button and the up/down arrows.

4. Type in your equation. For example: if your equation is y = 2 x + 3, then you would click 2 , x,T,θ,n , + , 3 .Then click GRAPH .

5. Choose the graph that most closely matches what shows on your calculator screen.

Using the chart below:

1. Solve your equation for y first by hand.

2. Pick at least three x values and write them down in the x column.

3. Plug those x values in for x into your equation to solve for your y values.

4. Put your results for y in the y column.

5. Put your x and y values into the ordered pair column.

6. Plot these ordered pairs on a coordinate plane.

7. Choose the graph that most closely matches the sketched graph.

x values / Plug in x / y values / (x, y)

Using x- and y- intercepts:

1. Solve the equation for y first.

2. Plug in 0 for x. You should get y = some #. This is your y-intercept.

3. Solve the original equation for x.

4. Plug in 0 for y. You should get x = some #. This is your x intercept.

5. On your coordinate plane, graph your y-intercept on your y axis. Graph your x-intercept on your x axis. Draw a line to connect the two points.

6. Choose the graph that most closely matches the sketched graph.

To Solve systems of equations:

Algebraically:

1. Solve the first equation for y. You should get y = some expression.

2. Plug this expression in for y into the second equation. Solve this for x. You should get x = some #.

3. Plug your x value back into your first equation. Solve for y. You should get y = some #.

Using a Calculator (Easy way):

1. Graph both equations simultaneously on your calculator using Y1 and Y2 on the Y= button.

2. Click GRAPH . Click 2nd . Click TRACE . Click 5. Hit enter three times. The answer is shown at the bottom of the screen.

Created by Mrs. Hamrick 7

Algebra 1 State Test Cram Packet

Created by Mrs. Hamrick 7

FUNCTIONS:

To determine if a list of ordered pairs are functions, just make sure that none of the x values repeat. To determine if a graph is a function, use the vertical line test, which says if you can draw a vertical line anywhere on the graph and it touches in more than one spot, then it is NOT a function.

The DOMAIN of a list of ordered pairs is all of the x values. To find the domain of an equation, you need to ask yourself “What can x be?” You can graph it on the calculator to check. Can it be negative? Can it be zero? Can it be positive? If the answer to all three questions is YES, then your answer is ALL REAL NUMBERS. If the answer to all three questions is NO, then your answer is NO SOLUTION. If your answer to some questions is yes and some questions is no, then you must list out the conditions. For example, if all you have are positive x values, then your domain would be x > 0.

The RANGE of a list of ordered pairs is all of the y values. To find the range of an equation, you need to ask yourself the same questions as the domain but about y instead of x.

Function Notation: f(x) is another name for y. It is just the name of a function. If the problem asks you to find f(something), then all you do is plug in whatever is in the parenthesis everywhere there was an x. For f(2), you would plug in a 2 everywhere there was an x. (If it is a constant function, we treat it differently.

Example f(x) = 3, find f(2). There isn’t a x in the problem, so the answer stays 3.

HOW TO WRITE AN EQUATION:

Given two ordered pairs:

1. Find the slope using: .

2. Plug into point slope form of an equation: ***Remember you need to pick one of the two points given to plug in for x1 and y1. The other x and y stay x and y, so that you are left with an equation.

3. Look at the problem to see how you need your answer. If it asks for point-slope form, you are done. If it asks for another form, you need to do some simplification.

Given one ordered pair and slope.

1. Plug into point slope form of an equation: ***Remember you need to plug in the point that was given for x1 and y1. The other x and y stay x and y, so that you are left with an equation.

2. Look at the problem to see how you need your answer. If it asks for point-slope form, you are done. If it asks for another form, you need to do some simplification.

3. If it asks for slope-intercept form, you need to get y by itself on one side of the equals sign. .If it asks for standard form, you need to get x and y on one side of the equals sign with the constant on the other side of the equals sign.

Given a table:

1. Pick two ordered pairs out of the table and then follow the rules as if you were given two ordered pairs. See above.