Cartesian Coordinate System (aka rectangular coordinate system) is a horizontal number line (the x-axis) intersecting with a vertical number line (the y-axis) at right angles at the zero coordinates of each line (the Origin).

Quadrants are the four areas of the Cartesian coordinate system formed by the intersecting number lines. Quadrants are designated by Roman numerals from I to IV beginning in the upper right and proceeding counterclockwise.

x-axis is the horizontal number line. From 0 to the left is negative, from 0 to the right is positive.

y-axis is the vertical number line. From 0 down is negative, from 0 up is positive.

The Origin is the intersection of the two axes at their zeros, thus its coordinates are (0,0).

A Point is any location on the Cartesian coordinate system. Every point has a horizontal and a vertical component that establish its position on the coordinate plane in relation to the Origin.

An Ordered Pair is the pair of coordinates that specify the location of a point on the coordinate plane in relation to the Origin. The ordered pair gives the directions to the point from the Origin. Ordered means that the x-coordinate ALWAYS comes first and the y-coordinate ALWAYS comes second, separated by a comma: (x, y).

The x-coordinate gives the distance and direction of the point from the origin along the horizontal number line, the x-axis. The x-coordinate will ALWAYS be listed first in an ordered pair.

The y-coordinate gives the distance and direction of the point from the origin along the vertical number line, the y-axis. The y-coordinate will ALWAYS be listed second in an ordered pair.

Plot: to locate a point on the coordinate system starting at the origin and using the ordered pair of coordinates, first x then y.

Linear equation: an equation in one or more variables in which no exponent has a power other than one. Called linear because the graph of a linear equation in two variables is a line.

The Standard Form of a Linear equation in two variables is: ax + by = c, where a, b, and c are Real Numbers and x and y are variables in alphabetical order.
Ex: 3x - 2y = 18

The Solution of a linear equation in two variables is the set of all ordered pairs that satisfy (make a true statement of) the equation. When we try to graph all the ordered pairs, we will get a line.

To graph a line: using one of three methods, establish two or more points on the line and draw the line through those points. Lines on the coordinate system are straight and extend to infinity in both directions.

Three Methods to graph a line:

1. Abitrary values for x method (aka the Pick Three method). A value is chosen for the x-coordinate, the independent (input) variable, substituted into the equation, and the value for the y-coordinate is calculated. Repeat twice until three ordered pair are generated. Plot the points and hope the line up.
2. Intercepts method: Find the value of the coordinate of both the x-intercept, (x1, 0), and the y-intercept, (0, y1); plot the intercepts, and draw the line through them.
3. Point-slope method: Using the slope-intercept form of the equation, plot the y-intercept and use the slope to find another point on the line, then draw the line through those two points.

The graph of a line: the representation of the solution set of a linear equation in two variables on the coordinate system.

An ordered pair is on the line when its coordinates are a solution to the equation. To find out, substitute the x-coordinate for the variable x and the y-coordinate for the variable y and simplify. If the statement is true, then the point is on the line. This is the same as checking to see if the numbers are solutions.

Intercepts: the point where the line crosses one of the axes. The name of the intercept specifies which axis is crossed and which coordinate will probably have a value other than 0. The only time both coordinates are 0 is when the line intercepts the Origin.

The x-intercept is where the line crosses the x-axis and has coordinates (x, 0).
The name is the x-intercept so we are looking for a value for the x-coordinate and the y-coordinate is ALWAYS 0 (because the y-coordinate tells us how far above or below the x-axis the point lies, why MUST the y-coordinate always be 0 for an x-intercept?).

The y-intercept is where the line crosses the y-axis and has coordinates (0, y).
The name is the y-intercept so we are looking for a value for the y-coordinate and the x-coordinate is ALWAYS 0 (because the x-coordinate tells us how far right or left the y-axis the point lies, why MUST the x-coordinate always be 0 for a y-intercept?).

Slope: the change in the y-coordinates between two points on the same line divided by the change in the x-coordinates of the same two points. We use the letter m to represent slope because it is French. The slope tells us the RATE of Change between points on the same line. It also gives directions from a point on a line to another point on the same line. The slope is often referred to as the Rise over the Run.

Rise: the difference in the y-coordinates between two points on the same line, usually written as y2 - y1.

Run: the difference in the x-coordinates between two points on the same line, usually written as x2 - x1.
Ex: Find the slope of the line through (3, 7) and (6, 9).
First, calculate the Rise: 9 - 7 = 2
Next, calculate the Run in the same direction: 6 - 3 = 3
Last, divide Rise by Run and reduce to lowest terms: m = 2/3

Slope - Intercept Equation: y = mx + b
m is the slope and b is the y-coordinate of the y-intercept (0, b)
What is the slope and y-intercept of the line y = 3x - 2
m = 3 and the y-intercept is (0, -2)

Two lines graphed on the same set of axes will be parallel, perpendicular, or neither.
Parallel lines have the same slopes and different y-intercepts.
Perpendicular lines intersect at right angles and their slopes are negative reciprocals (product is a negative 1)
If not parallel or perpendicular, then neither. This means the two equations could be graphed with the same line or their intersection does not form right angles.

The graph of a linear equation will be one of four possible lines:

• Rising line: line slants up from left to right on the graph. The slope is ALWAYS positive. IS a function.
• Falling line: line slants down from left to right on the graph. The slope is ALWAYS negative. IS a function.
• Horizontal line: line is straight across the graph from left to right, neither rising nor falling. The slope is ALWAYS zero, or no slope. IS a function.
• Vertical line: line is straight up and down the graph. The slope is ALWAYS undefined (see Division Involving Zero). IS NOT a function!!

Point - Slope Equation Form: y - y1 = m(x - x1)
When we know the slope, m, and the coordinates of a point (x1, y1), we can use the Point - Slope form to write the equation, usually in slope - intercept form (y = mx + b).

Input: the value typed in or used for x in the expression or function being evaluated.

Output: the calculated value, Y1 on the graphing calculator, of the expression or function using the input value.

function: a special case of mathematical statement where an input is matched to only one output.

function notation: ƒ(x) = ax + b
ƒ is the name of the function
x tells us what value to substitute for the variable
ax + b (an expression, just like in Unit 1) tells us how to calculate the value of the function (evaluate the function for the given value)
x is the input, the calculated value ofƒ(x) is the output.
Ex. forƒ(x) = 3x + 2, findƒ(7)
this means that we want to evaluate 3x + 2 when x = 7, so we write:
ƒ(7) = 3(7) + 2 Now evaluate.
ƒ(7) = 21 + 2 = 23.
The input is 7 and the output is 23. We can write this result as an ordered pair, input first, then output, thusly:
(7, 23)
The function named f has matched an input of 7 with an output of 23.
Now try this: forƒ(x) = 3x + 2, findƒ(9)

domain of a function: the set of all values that may be input to the function. All the numbers that are allowed to be used for the input variable, usually x. All the numbers that are allowed to replace x. In the example above, the 7 and any other Real number that we may use.

range of a function: the set of all of the possible values that will result from evaluating the function for an input. All the possible outputs of the function. What we get when we replace x and evaluate to find f(x). In the example above, the 23 and any other output that would result from a different input.

Table: a set of ordered pairs presented in tabular format; paired input and outputs listed as X and Y1 on the graphing calculator (Press [2nd] [GRAPH]).

Input: the value typed in or used for x in the expression or function being evaluated.

Output: the calculated value, Y1 on the graphing calculator, of the expression or function using the input value.

formula: a mathematical statement where the variables represent values of practical interest.

Percent Change formula: To find the percent change between two values, divide the difference of the new (N) value and the previous (P) value by the previous value, then multiply times 100: %change = (N - P)/P * 100

Thickness formula: To find the thickness (T), divide the volume (V, ALWAYS in cubic units) by the area (A, ALWAYS in square units): T = V/A

Geometric formulas:   Polygons     Circles

Slope formula: the slope is the rate of change between two points. Divide the difference of the y-coordinates by the difference of the x-coordinates: 

Distance formula: To find the distance between two points use the Pythagorean Theorem to find the length of the hypotenuse where the length of one leg is the difference in the x-coordinates and the length of the other leg is the difference in the y-coordinates: m = (y₂ - y₁) / (x₂ - x₁)

Polya’s Method: a structured approach to solving applications (word problems). has four steps: (1) Understand the problem; (2) Plan your approach; (3) Carry out the plan; (4) Check your result. See link on my web site. Polya's Method