Graphing Linear Equations Study Guide
Introduction to Graphing Linear Equations
Mathematical genius and artistic genius touch one another.
—Gösta Mittag-Leffler (1846–1927) Swedish Mathematician
In this lesson, you'll learn about the coordinate plane and how to graph the equation of a line.
The equation of a line is a linear equation. A linear equation is an equation that can contain constants and variables, and the exponent of the variables is 1. We can graph linear equations on the coordinate plane. The coordinate plane is a two-dimensional surface with an x-axis and a y-axis. The x-axis is a horizontal line along which y = 0, and the y-axis is a vertical line along which x = 0. The coordinate plane is shown on the following page.
The first few lines to each side of the axes are labeled. The x values increase as we move to the right, and the y values increase as we move up. Points in the upper right corner, or quadrant I, of the plane have positive x values and positive y values. Points in the upper left corner, or quadrant II, have negative x values and positive y values. Points in the lower left corner, quadrant III, have negative x values and negative y values, and points in the lower right corner, quadrant IV, have positive x values and negative y values.
We plot points on the coordinate plane according to their x and y values. These values are called an ordered pair or a coordinate pair. In an ordered pair, the x value is listed first, and then the y value, like this: (4,2). The x value of this point is 4 and the y value of the point is 2.
Each row in the input/output tables from the preceding lesson represents an ordered pair. Following is the input/output table from the equation
y = 3x – 1.
We can plot each of these points on the coordinate plane. The first row shows that x is –2 and y is –7. This is the ordered pair (–2,–7). To plot this point, start at the origin of the coordinate plane. The origin is the place where the x-axis and y-axis cross, where x = 0 and y = 0. Because the x value of this point is –2, move two units to the left. The y value of the point is –7, so move seven units down. Place a dot where x = –2 and y = –7, and label it (–2,–7). Do the same for all five rows of the table. The points in rows three, four, and five of the table will be plotted in quadrant I, because the x and y values are both positive.
Finally, connect the dots with a solid line, and put arrows on both ends of the line to show that the line continues in both directions. Label the line with its equation, and you have finished. That's all there is to graphing the equation of a line!
How do we graph a line if we do not have an input/output table? We create our own input/output table. Choose a few x values, and use the equation of the line to find their y values.
When you are making an input/output table, it is helpful to choose a few positive x values and a few negative x values. This will give you points on both sides of the y-axis. Also, include x = 0 in your table, so that the y-intercept is plotted.
To graph the line y = 2x – 6, start with an input/output table:
Now, you have five points. Plot these points on the graph. The first two points, (–2,–10) and (–1,–8), will be in quadrant III, since the x and y values are both negative. The last two points, (1,–4) and (2,–2) will be in quadrant IV. Connect the dots to form the line y = 2x – 6.
If the slope of the equation is a fraction, choose x values for your input/output table that are multiples of the denominator of the fraction. The y values will be integers, which are easier to graph than fractions.
Find practice problems and solutions for these concepts at Graphing Linear Equations Practice Questions.
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