Graphing Linear Equations on a Coordinate Plane
Graphing Linear Equations on a Coordinate Plane (Basic)
A coordinate plane is a plane containing the x– and y-axes. The x-axis is the horizontal line on a graph where y = 0. The y-axis is the vertical line on a graph where x = 0.
The x-axis and y-axis intersect to create four quadrants. The first quadrant is in the upper right, and other quadrants are labeled counterclockwise using the roman numerals I, II, III, and IV. Points, or locations, on the graph are written as ordered pairs, (x, y), with the point (0, 0) called the origin. Points are plotted by counting over x places from the origin horizontally and y places from the origin vertically.
The most common way to write a linear equation is slope-intercept form:
\(y = mx\ +\ b\)
In this equation, m is the slope, and b is the y-intercept. The y-intercept is the point where the line crosses the y-axis, or where x equals zero. Slope is often described as “rise over run” because it is calculated as the difference in y-values (rise) over the difference in x-values (run).
\(m = \frac{y_2\ −\ y_1}{x_2\ −\ x_1} = \frac{rise}{run}\)
Use the phrase begin, move to remember that b is the y-intercept (where to begin) and m is the slope (how the line moves).
To graph a linear equation, identify the y-intercept and place that point on the y-axis. Then, starting at the y-intercept, use the slope to go “up and over” and place the next point. The numerator of the slope is the number of units to go up (or down if the slope is negative). The denominator of the slope is the number of units to go right. Repeat the process to plot additional points. These points can then be connected to draw the line.
To find the equation of a line, identify the y-intercept, if possible, on the graph and use two easily identifiable points to find the slope.
Two or more parallel lines never intersect, and they have the same or equal slopes. Perpendicular lines intersect to form right angles. The slopes of two perpendicular lines are negative reciprocals of each other.
Another way to express a linear equation is in standard form: Ax + By = C. To graph such an equation, it can be converted to slope-intercept form, or the slope and intercepts can be found from the standard form:
- \(m = −\frac{A}{B}\)
- \(\frac{C}{A}\)
- \(\frac{C}{B}\)
It is easy to find the x– and y-intercepts from this form. To find the x-intercept, simply set y = 0 and solve for x. Similarly, to find the y-intercept, set x = 0 and solve for y. Once these two points are known, a line can be drawn through them.
The point-slope equation can be used to find the equation of a line using the slope and one point (x1, y1):
y – y1 = m(x – x1).