Algebra 1 EOC Review
Percents
15% can also be written as 0.15, or as 15/100 which then must be reduced to 3/20.
When do you flip an inequality symbol?
Anytime you multiply or divide by a negative number.
The product of a rational number(other than zero) and an irrational number is irrational
(2)(3.14) = 6.28.... (rational)(irrational) = irrational
Point-Slope Form
(y - y*) = m (x - x*) When given a point and the slope, plug in the m, x, and y into the equation. You can then convert it into slope-intercept form to find the graph of the line.
25% of what is 150?
0.25 * x = 150. To solve you divide by 0.25.
How many solutions can a system with 2 equations have?
1) no solutions, if the lines are parallel and have the same slope. 2) one solution, if the lines intersect at one point. 3) infinite many solutions, if the two equations are the same and touch each other everywhere.
Perfect Squares
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, etc. The resulting number when you multiply a number times itself. EX: 4 X 4 = 16
The sum of a rational number and an irrational number is irrational. An irrational number is one that has decimal places that keep going forever such as pi= 3.14......
2 + 3.14.... = 5.14.... and is still an irrational number.
What is a system of equations or system of inequalities?
2 or more equations or inequalities graphed on the same coordinate plane
36% of 500 is what?
36/100 * 500 = x To solve you multiply 500 times 36/100.
3 "is more than" a number
3>n
Distinguish between situations that can be modeled with linear functions and with exponential functions.
A linear function would be best for situation in which something is increasing at a constant rate. An exponential function is best for situations in which the rate increase over time. EX. The growth of bacteria cells in a petri dish would be exponential growth.
Table
A way to represent ordered pairs.
Seven "is less than" a number
7<n
Histogram
A bar graph that has a range of values on the x-axis. In the example, each bar represents an age range of 5 years.
Outliers
A data point that does not fit in with the rest of the data.
Step-Function Graph
A function that is not continuous. It has levels which represent the y-value for a set of x-values.
What is a line of best fit?
A line drawn through the points on a scatterplot to estimate the best line through the data.
Correlation Coefficient
A number between -1 and 1 that tells you if you have a strong or weak correlation. It also lets you know if it is a positive or negative correlation.
Arithmetic Sequence
A pattern in which you add a number each time.
Geometric Sequence
A pattern in which you multiply or divide each term by the same number.
Integer
A whole number, both positive and negative numbers, including zero. ... ,-3, -2, -1, 0, 1, 2, 3, ......
Compound Interest Formula
A=(P+r/n)^nt Common n values are 12=monthly, 4=quarterly, 2=semiannually, 1=annually. t is the time in years.
How do you MULTIPLE exponents of like variables?
ADD them
Sum
Add
How do you combine like terms?
Add their coefficients.
Mean (Average)
Add up all the data points, then divide by however many data points there were. Ex. 3,6,9 3+6+9= 18 /3 = 6
How to tell when a function is increasing or decreasing?
Always read a graph from left to right, just like the words on a page. If it goes up, it is increasing. If it goes down, it is decreasing.
Discounts
Are subtracted from the cost of the item. To find the amount to subtract, multiply the cost by the item times the percent being discounted.
Standard Form of a line
Ax + By = C NO fractions, NO decimals, A has to be positive!
Mapping
Bubbles that represent the x-values and the y-values.
Correlation vs. Causation
CORRELATION is just a recognizable pattern: EX. In the summer ice cream sales increase and so do the number of shark attacks. CAUSATION is the reason something occurs: EX. The number of ice cream sales is NOT the cause for the increase of shark attacks.
How do you multiply FRACTIONS?
Cancel or reduce if possible, then multiply straight across. Some people like to multiply straight across and then reduce. Either way, if you can reduce, you MUST REDUCE.
Two-way Frequency tables and Relative Frequency Tables
Comparing more than one thing in a table. Use the information in a table to answer questions.
Conditional Relative Frequencies
Comparing the individual data to the whole, represented as a percentage.
Natural numbers
Counting numbers: 1, 2, 3, ......
End Behavior
Describes what is happening to the graph overall as the x-values change. As the x-values decrease, does the graph seem to go up or down? As the x-vales increase, does the graph seem to increase or decrease? This represents the end behavior as being positive or negative.
Quotient
Divide
Fractions to Decimals
Divide the numerator by the denominator. 1/4 = 1 divided by 4 = 0.25
The sum of two rational numbers is rational. The product of two rational numbers is rational A rational number is any number that can be written as a fraction. Whole numbers are rational numbers because the number 2 can be written as 2/1.
EX: 1/5 + 2/5 = 3/5 EX. 2/3 * 1/3 = 2/9
Dot plot
Each x or dot represent one person that has that number of pets. EX: One person has 5 pets.
How to solve when written in function notation.
Ex: f(x) = 3x + 4, solve for f(-2). All you do is replace the x with a (-2) and solve. 3(-2) +4 = -2
Use the structure of an expression to identify ways to rewrite it.
Example: See x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as the difference of two squares that can be factored as (x^2-y^2)(x^2+y^2).
Interpret parts of an expression
Examples such as: C= 250 + 35m, Where C is the cost for renting a car. They pay $250 plus $35 for every extra mile they drive. You have to interpret what the numbers and variables mean in the form of a word problem.
Factoring with 2 terms
Factor out the number or variable that all the terms have in common.
How do you Reduce FRACTIONS?
Find one number that you can divide into both the numerator and denominator that is the same number. Then divide them both by that number.
Estimate the Rate of Change from a graph
Find the change in y-values, then divide by the change in x-values.
Flow Charts
Follow the lines by answering questions until you get to your answer!
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal description.
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Calculate and interpret the average rate of change (symbolically or from a table)
How fast is it changing? Sometimes you find out how much each one changes, then take their average.
Improper Fractions to Mixed Numbers
How many times will the denominator go into the numerator without going over. Then how many little pieces are left over.
Solid or Dotted Line? Open or Closed Circle?
If it has the line under it (or equal to) it will be a closed circle or solid line.
How to use the Discriminant when solving the Quadratic Formula?
If the discriminant is positive, you will have 2 solutions or x-intercepts. If the discriminant is zero, you will have 1 solution or x-intercept. If the discriminant is negative, you will have No Solution (or really a complex solution (imaginary) in the form (a + bi)
SUBSTITUTION
If you know what one variable equals, like y=35-10x, then you can plug that value in for y in the second equation, 4x - 7y = -23 becomes 4x -7(35-10x)=-23. Distribute and solve for x. Then you can plug back in to one of the original equations to solve for y.
Quadratic Formula
In standard form, ax^2 + bx + c, where a, b, and c (Just the numbers) can be plugged into the quadratic formula to find the x-intercepts, roots, zeros, or solutions.
Sales Tax
Is added on to the cost of the item. To find the amount to add, multiply the cost of the item by the percent of tax.
Rate of Change
Is another name for slope usually used in word problems.
Completing The Square
Isolate x^2 + bx on one side, move the number to the other side of the equal sign. Take b, divide by 2, then square the result. Add that number to both side. It will then factor into a perfect square.
what is a linear inequality?
It is an equation with a <, >, < or =, > or = in the place of the =.
Joint and Marginal Frequencies
Joint is the individual data in the middle. Marginal is the sum of the data for each row or column.
How do you Divide FRACTIONS?
Keep, Change, Flip! Then multiply fractions!
Decimals to Fractions
Know your decimal places. Tenths, hundredths, thousandths. 0.36 say "36 hundredths" So you write the fraction 36/100. Then reduce divide top and bottom by 4 to get 9/25.
Normal Bell Curve
Let's you know where you should find most of your data.
Box Plots
Make sure the numbers are in order from least to greatest. Find the median, the Q1 and Q3 by finding the median again of the upper and lower sections.
Maximum or Minimum
Maximum is at the highest point of a parabola. Minimum is the lowest point.
Product
Multiply
Distributive Property
Multiply a number or variable to everyone inside parenthesis.
Mixed Number to Improper Fractions
Multiply, then add! Keep the same denominator.
Absolute value of a negative number. IxI = -3
NO SOLUTION
variable
NOT A NUMBER. A letter or symbol that stands for a number.
How to solve absolute value inequalities?
Once the absolute value bars are on one side of the equal sign, by itself, you work two separate problems (1) Looks exactly the same as the original equation without the absolute value symbols. (2) Is the same on one side, but flip the inequality symbol and make the number negative.
Linear functions
Over equal intervals, the y-values increase by adding or subtracting the same number each time. *Note that the slope in the example would be 3/1= 3
Exponential functions y=2^x Where the variable is the exponent.
Over equal intervals, the y-values increase by multiplying or dividing by the same number each time.
PEMDAS "Order of Operations"
Parenthesis Exponents Multiply/Divide Add/Sub
Correlation
Positive Correlation, Negative Correlation, or No Correlation.
Interquartile Range
Q3 - Q1
Simplifying Radicals
Radical is the square root symbol. sqrt(20) = sqrt(4*5) = 2 sqrt (5)
y= and f(x)=
These are the same. They can be interchanged with one another. f(x) simply lets you know the equation is a function. Can also be written with other variables like g(x), p(x), etc.
Rounding Rules
Round to the nearest TENTH means you keep ONE decimal place EX 0.5648 rounds to 0.6 The 6 behind the five tells the 5 to go up. Round to the nearest HUNDRETH means to keep TWO decimal places EX 0.56 The 4 tells the 6 to stay the same.
How do you DIVIDE exponents of like variables?
SUBTRACT them
Direct Variation
Set up a proportion and solve. y1/x1 = y2/x2
What is the "rate of change" on a graph?
Slope
How do you solve Proportions?
When two fractions are set EQUAL to one another. This is the only times you Cross multiply!
How to rearrange variables to solve for a specific variable.
Solve A=P+PRT for T. You perform the opposite operation on each variable until you get the one you are solving for all by itself. Subtract P, then divide by P and R.
Factoring with 4 terms
Solve by grouping. Group two sets and then factor those two groups.
Whole numbers
Start with 0, 1, 2, ......
Difference
Subtract
Axis of Symmetry
The equations x=0 is the axis of symmetry for a parabola in standard form.
The solution to a system of linear inequalities
The intersection of the two half-planes.
Mode
The number that shows up the MOST
How do the slopes of perdendicular lines compare?
They are opposite and fraction is flipped. Example: lines with the slopes -2 and 1/2 are perpendicular.
Associative Property
The way numbers are grouped is different, but the result is the same.
Exponent Rules
There are a bunch!
Geometric Sequences
There is a pattern in which you multiply or divide by a common number.
How do the slopes of parallel lines compare?
They are the same
The Square Root Property to solve Quadratics
To get rid of the exponent 2, you can take the square root of both sides. Remember the plus or minus symbol, which can give you two answers.
Cube Root
What number multiplies times itself 3 times to get the number under the house?
Range
When talking about data sets, make sure the numbers are in order from smallest to largest, to find the RANGE you SUBTRACT the smallest from the largest.
Skewed Bell Curve
When the data is not evenly distributed.
Median
When the data is put in order from smallest to largest, it is the number in the middle. If there are two numbers in the middle, you add them together then divide by 2. (You take their average)
Piecewise-defined function Graph
When the graph is doing two (or more) different functions depending on what the x-vales are. EX. If x<-1, then y=1. If x>(or equal to) -1, then y=x^2 -3
Standard Deviation
When using a bell curve to represent data, the standard deviation tells you how far away a number is from the center of the data.
Communitive Property
With addition and multiplication, it does not matter what order the numbers are written, you still get the same result.
Absolute Value when solving order of operations problems.
Work any math that you need to within the absolute value bars. Then always give the positive of that number as the answer.
Simple Interest Formula
You can solve for any of the variables when given the other 3. The rate is the percentage rate usually written as a decimal. EX. 13%= 0.13
ELIMINATION
You have to get one of the variables to eliminate by getting them to be the same number just one positive and one negative, so they zero out. Sometimes in order to do this you have to multiply one or both of the equations by a number. Then add straight down and continue to solve.
How do you Add and Subtract FRACTIONS?
You have to have a common denominator.
Explain each step in solving an equation.
You may have to list the steps you performed explaining what you did, step by step. Know how to justify your solutions.
coordinate plane
a plane formed by 2 perpendicular lines called axes (the x-axis goes sided to side, the y-axis goes up and down).
vertical line test
a relation is a function and passes the vertical line test if when a vertical line is drawn through the graph, it touches the graph only ONE time.
What is a quadratic equation or function?
an equation of function that has "x-squared" as its highest order exponent.
Where are the solutions of a quadratic function?
at the x-intercepts. They are also known as zeros and roots.
Factoring with 3 terms
ax^2 + bx + c = 0, where a=1 (x )(x )=0 Find the two numbers that multiply to get you c, AND they add/sub to get you b. If asked to SOLVE by factoring, then set each factor equal to zero and solve.
continuous function
data appears as a smooth curve on a graph.
discreet function (scatter plot)
data appears as individual points on a graph. Whatever is being graphed cannot be broken down in to smaller pieces.
other names for "x"
domain, independent variable, input
Which quantity increases fastest?
exponential functions > quadratic functions > linear functions
If y=f(x) and y=g(x), then.....
f(x) = g(x) meaning you can set the two equations equal to each other in order to solve for the variable. Then to find y you plug the variable back into one of the original equations and solve.
words for "<"
is less than, is smaller than, shade below, shade to the left
words for ">"
is more than, is greater than, is larger than, shade above, shade to the right
how is slope important for linear equations and lines?
linear equations and lines have CONSTANT slopes. The slope is the SAME all the way up and down the line.
What are "like terms"?
mathmatical terms that have the SAME variables raised to the SAME exponents.
changing the "b" in the slope-intercept form of the equation of a line changes what?
moves the line up or down the y-axis.
3 "more than" a number
n + 3 Add 3 to the number
Seven "less than" a number
n - 7 You subtract 7 from the number.
other names for "y"
range, dependent variable, output
domain
set of all of the possible x-values. INPUTS
range
set of all the possible y-values. OUTPUTS
relation
set of ordered pairs
in y = mx + b, "m" stands for?
slope
what slope does a HORIZONTAL line have?
slope = 0
function
special relation where each x-value is paired with ONLY ONE y-value. (x-values DO NOT repeat, it passes the vertical line test)
Square Root
sqrt(9) = 3 It means what number multiplies times itself to get the number under the house.
independent variable
the INPUT or CAUSE in a situation or problem. It's the variable YOU CAN control. It's the X variable.
dependent variable
the OUTPUT or RESULT in a situation or problem. It's the variable YOU CANNOT control. It's the Y variable. In a equation, its the variable BY ITSELF on one side of the = sign.
what is the x-intercept?
the point where the graph crosses the x-axis. It has the coordinates (x, 0)
what is the y-intercept?
the point where the graph crosses the y-axis. It has the coordinates (0, y)
define slope
the steepness of a line, vertical change/horizontal change, (y-y)/(x-x), rise over run
changing the "m" in the slope-intercept form of the equation of a line changes what?
the steepness of the line
ordered pair
the x and y coordinates of a point on the coordinate plane (x,y)
in y = mx + b, "b" stands for?
the y-intercept
what slope does a VERTICAL line have?
undefined slope
What is the SOLUTION of a system of equations?
where the lines touch or cross
What percent of 200 is 40?
x/100 * 200 = 40. To solve you would divide by two hundred, then multiply by 100.
What is Inverse Variation?
y = 1/x. As one quantity increases, the other decreases
What is Direct Variation?
y = mx, or y = ax, or y = kx. m, a, or k is just the coefficient in front of the x. They are also the SLOPE of the direct variation line. Direct variation lines have a y-intercept ot (0, 0)
what is the linear parent function?
y = x f(x) = x
Identify the effects of k on the graph y=f(x)
y=f(x) + k moves the graph up k units or down k units. y=f(x+k) moves the graph right if k is negative OR left if k is positive. Remember this is opposite of what's on the number line. y=k f(x) vertical stretch if k is bigger than 1 and a vertical compression if k is less than 1.
Inverse Variation
y=k/x
Slope-intercept form
y=mx +b