This tutorial covers graphing a line using standard form, through the exploration of:
1. Graphing an Equation: A Review
As you may recall, it’s often useful to graph equations to visually represent the relationship between variables. To graph a line, you only need to find at least two points on the line. There are several ways to graph a line:
- One strategy is to pick two values for x and find the corresponding y values to plot on a graph.
- Another strategy is to write an equation in a certain form to easily identify important information about the line.
2. Standard Form of an Equation
Equations in standard form are expressed as A times x plus B times y equals C:
-
- Standard Form of a Line
The benefit of having an equation in standard form is that it makes finding the x- and y-intercepts relatively easy. To find the y-intercept, you substitute 0 for x, making the whole A times x term 0, and leaving B times y equals C, which can be easily solved for y.
Similarly, to find the x-intercept, you substitute 0 for y, making the B times y term 0 and leaving A times x equals C, which is easily solved for x. The standard form of a line is important when studying systems of linear equations, which will be covered later in this course.
3. Graphing Equations in Standard Form
There are several methods for graphing an equation in standard form. In the first method, you can use the variables in the standard form equation to easily identify the x- and y-intercepts, which will give you the minimum two points needed to graph a line.
-
EXAMPLE
Suppose you want to graph the equation:
Start by finding the x-intercept, which is at a point (x, 0), where y is 0.
You can substitute 0 for y into your equation, which gives you the expression shown below. 4 times 0 is 0, so you now have 3x equals 12. Divide both sides by 3, which simplifies to x equals 4. Therefore, the x-intercept is at the point (4, 0).
Now, find the y-intercept. The y-intercept is at a point (0, y), where x is 0. You can substitute 0 for x into your equation. 3 times 0 is 0, so you now have -4y equals 12. Divide both sides by -4, which simplifies to y equals -3. Therefore, the y-intercept is at the point (0, -3).
You can now plot both of your intercepts on the graph. The x-intercept is (4, 0), and the y-intercept is (0, -3), so you have the two points needed to graph a line. Finally, you can connect your points to create a line.
In the second method, instead of graphing the x- and y-intercepts, you can rewrite the standard form equation in slope intercept form and graph the equation using the y-intercept and slope of the line. Slope intercept form may be preferred for graphing since y is on one side of the equation and the variables representing slope (m) and y-intercept (b) are easily identifiable on the other side.
-
EXAMPLE
Suppose you want to graph the equation:
To write your equation in slope intercept form, you need to isolate the y variable. Start by adding 2x on both sides. On the right side, 10 and 2x are not like terms, so you cannot actually combine them.
You’ll note that it is written as 2x+10 instead of 10+2x, because you should place the x term in front of the constant term. This is because you want the format to reflect the slope-intercept form.
Divide both sides by 5. On the right side, you divide
both terms by 5.
Your final equation is:
To graph, start at the y-intercept, which is at 2. You can then use your slope, 2 over 5, to find a second point. Therefore, from the y-intercept, you move up 2 and over 5 to place your second point. Finally, you connect your points to create a line.
Today you learned that it is often useful to graph equations to visually represent the relationship between variables, and that to graph a line, you need to find at least two points on the line. You also learned how to identify equations in standard form. Lastly, you learned two methods of graphing equations in standard form: one, solving for the x- and y-intercepts; and two, rewriting a standard form equation in slope intercept form.
