Graphs
Graphs are used to visualize what functions look like. There are many different types of graphs but the most common is the Cartesian Coordinate System in a plane. Since it is a plane, it represents two dimensions. It consists of two perpendicular axes: usually the x axis which is horizontal and the y axis which is vertical. Points are plotted on the plane and are represented by dots. The point of intersection of the x and y axes is called the origin and is the center of the graph. The axes also divide the coordinate plane up into four parts called quadrants.
Points in the Plane
Points are in the form of ordered pairs. An ordered pair is two numbers separated by a comma and contained in parenthesis such as: (3,2). It is ordered because the first value always stands for the value of the x-axis while the second number stands for the number on the y-axis.
The way a point is plotted is by drawing a perpendicular line from the value on the x axis which is the first value of the ordered pair. Next a perpendicular line is drawn from the point on the y axis which is the value of the second number of the ordered pair. The intersection of these two perpendicular lines is the location of the point.
The x value for the point is represented by a vertical line intersecting the x axis at the x value.
The y value for the point is represented by a horizontal line intersecting the y axis at the y value.
The place where the perpendicular lines for the x and y value intersect is the location of the point.
A math equation is graphed by calculating each y value from each x value. Each x value and the y value calculated from it form and ordered pair. The point is plotted from the ordered pair. This is done for all x values in the domain. Typically, a table is used to calculate the y value for each x and the points can be plotted from the table.
Distance Formula
To find the distance between two points on a graph, the distance formula is used.
d = √(x2 - x1)2 + (y2 - y1)2
The x and y values for each point are entered into the formula.
 | The distance formula is based on the Pythagorean Theorem. The distance between the x values of the points forms one leg of a right triangle. The distance between the y-values forms the other leg. The hypotenuse of the right triangle is the actual distance. d = √(3 - (-2))2 + (6 - 1)2d = √52 + 52 d = √25 + 25 d = √2 * 25 d = 5√2 The distance between the points is 5√2 |
Midpoint Formula
To find the midpoint between two points, the midpoint formula is used for each x then y to get the midpoint.
| xm | = | (x2 + x1) 2 |
| ym | = | (y2 + y1) 2 |
The midpoint is then (xm, ym).
 | The midpoint is exactly half the distance between the two points. | |
 | The midpoint is calculated by taking the average of the x values of the points for the midpoint x value and then taking the average of the y values of the points for the midpoint y value. The midpoint is (-1,-1). |