relation
set of ordered pairs
ordered pair
the x and y coordinates of a point on the coordinate plane (x,y)
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).
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)
Mapping
Bubbles that represent the x-values and the y-values.
Table
A way to represent ordered pairs.
domain
set of all of the possible x-values. INPUTS
range
set of all the possible y-values. OUTPUTS
variable
NOT A NUMBER. A letter or symbol that stands for a number.
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.
other names for "x
domain, independent variable, input
other names for "y
range, dependent variable, output
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.
discreet function (scatter plot)
data appears as individual points on a graph. Whatever is being graphed cannot be broken down in to smaller pieces.
continuous function
data appears as a smooth curve on a graph.
graph of a linear function
Linear Function Parent Graph
graph of a quadratic function
Quadratic Parent Function
Absolute Value Parent Function
graph of exponential decay function
graph of exponential growth function
Graph a parabola using a table
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
in y = mx + b, "m" stands for?
slope
in y = mx + b, "b" stands for?
the y-intercept
changing the "m" in the slope-intercept form of the equation of a line changes what?
the steepness of the line
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.
what is the linear parent function?
y = x
f(x) = x
define slope
the steepness of a line, vertical change/horizontal change, (y-y)/(x-x), rise over run
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 slope does a HORIZONTAL line have?
slope = 0
what slope does a VERTICAL line have?
undefined slope
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)
what is a linear inequality?
It is an equation with a <, >, < or =, > or = in the place of the =.
How do the slopes of parallel lines compare?
They are the same
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.
What is a line of best fit?
A line drawn through the points on a scatterplot to estimate the best line through the data.
What is a system of equations or system of inequalities?
2 or more equations or inequalities graphed on the same coordinate plane
What is the SOLUTION of a system of equations?
where the lines touch or cross
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.
How do you MULTIPLY exponents of like variables?
ADD them
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.
What are "like terms"?
mathmatical terms that have the SAME variables raised to the SAME exponents.
How do you combine like terms?
Add their coefficients.
How do you DIVIDE exponents of like variables?
SUBTRACT them
What is the "rate of change" on a graph?
Slope
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
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)
Mode
The number that shows up the MOST
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.
Integer
A whole number, both positive and negative numbers, including zero.
... ,-3, -2, -1, 0, 1, 2, 3, ......
Whole numbers
Start with 0, 1, 2, ......
Natural numbers
Counting numbers: 1, 2, 3, ......
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.
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.
Maximum or Minimum
Maximum is at the highest point of a parabola.
Minimum is the lowest point.
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.
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 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.
When do you flip an inequality symbol?
Anytime you multiply or divide by a negative number.
Absolute value of a negative number. IxI = -3
NO SOLUTION
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 f
Factoring with 4 terms
Solve by grouping. Group two sets and then factor those two groups.
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
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 do
The solution to a system of linear inequalities
The intersection of the two half-planes.
Percents
15% can also be written as 0.15, or as 15/100 which then must be reduced to 3/20.
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.
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
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.
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.
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
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.
Axis of Symmetry
The equations x=0 is the axis of symmetry for a parabola in standard form.
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.
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 compres
Estimate the Rate of Change from a graph
Find the change in y-values, then divide by the change in x-values.
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 g
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.
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.
Which quantity increases fastest?
exponential functions > quadratic functions > linear functions
Rate of Change
Is another name for slope usually used in word problems.
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.
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.
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.
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.
Normal Bell Curve
Let's you know where you should find most of your data.
Skewed Bell Curve
When the data is not evenly distributed.
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.
Outliers
A data point that does not fit in with the rest of the data.
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.
36% of 500 is what?
36/100 * 500 = x To solve you multiply 500 times 36/100.
What percent of 200 is 40?
x/100 * 200 = 40. To solve you would divide by two hundred, then multiply by 100.
25% of what is 150?
0.25 * x = 150. To solve you divide by 0.25.
Seven "less than" a number
n - 7 You subtract 7 from the number.
Seven "is less than" a number
7<n
3 "more than" a number
n + 3 Add 3 to the number
3 "is more than" a number
3>n
Fractions to Decimals
Divide the numerator by the denominator. 1/4 = 1 divided by 4 = 0.25
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.
How do you solve Proportions?
When two fractions are set EQUAL to one another. This is the only time you Cross multiply!
Exponent Rules
There are a bunch!
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
Square Root
sqrt(9) = 3
It means what number multiplies times itself to get the number under the house.
Cube Root
What number multiplies times itself 3 times to get the number under the house?
Slope-intercept form
y=mx +b
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.
Standard Form of a line
Ax + By = C NO fractions, NO decimals, A has to be positive!