Drawing the Line
Where do you draw the line?
OBJECTIVE: To graph a line on the coordinate plane using different forms of linear equations
There's a postulate in Geometry that 'two points determine a line.' So if you're given two points, just plot them, draw a straight line passing through both points, and your job is done! In a previous class, you also learned how to graph a line using two points derived from the STANDARD FORM of the line, Ax + By = C. You did it in two ways. Method 1. You assigned a value to x and solved for y to get an ordered pair (x₁, y₁); then you repeated the process to get another ordered pair (x₂, y₂). Once you got your two points, you graphed them on the coordinate plane and drew a line passing through both points. Method 2. You solved for the intercepts: x-intercept = C/A and y-intercept = C/B. This process gave you the coordinates of the x-intercept (x, 0) and the y-intercept (0, y). You then graphed the intercepts on the coordinate plane and drew a line passing through both points. NOW you're going to explore yet another way of graphing a line, this time using the SLOPE-INTERCEPT FORM, y= mx + b, where m is the slope and b is the y-intercept. The idea is to use the y-intercept (b) as a starting point and then use the slope (m) to come up with a second point. Once you have two points, you can draw the line. As a preliminary step, you need to convert the STANDARD FORM into the SLOPE-INTERCEPT FORM using these formulas: m = -A/B and b = C/B. On the upper left-hand corner of the applet below, you'll see the STANDARD FORM of the equation of a line. Your task is to rewrite this equation into the SLOPE-INTERCEPT FORM. When entering ratios (fractions), be sure to enter EXACT VALUES. (Example: Enter "1/3" for , and NOT 0.33). You may use the original (given) values of A, B, and C. The applet is smart enough to reduce your fractions to lowest terms. If you type in a correct equation, you'll see a big "CORRECT !!!" sign appear. Repeat at least 10 times.
Now let's start graphing lines using the SLOPE-INTERCEPT FORM.
On the upper right-hand corner of the applet below, you'll find a linear equation in SLOPE-INTERCEPT FORM. Graph the line as follows: 1. Drag the BLUE DOT to the position of the y-intercept (b). 2. Drag the BLACK DOT to the position of the BLUE DOT and use the slope (m) to locate the final position of the second point. Remember 'rise over run'? Repeat at least 10 times.
Now you know that you can also draw a line using just one point and the slope.
In the previous task, you used the y-intercept as you starting point, but it doesn't always have to be that way. If you're given one point and the slope, you can follow the same process to come up with a second point and draw your line. TRY THIS! GIVEN: P = (2, 1) and m = -1/2. Use the empty Geogebra Sheet below to do this task. Plot the point and use the slope to get a second point. Draw a line through the two points.
What's the linear equation that's now in your Algebra window?
Let's pause to review what we've covered so far.
In the study of linear equations, you're usually asked to do two things: one, graph a line; and two, find the equation of a line. You've learned how to graph a line given the following: 1. two points: (x₁, y₁), (x₂, y₂) 2. the standard form of a line (Ax + By = C): yielding two points: (x₁, y₁), (x₂, y₂) or the intercepts: (x, 0), (0, y) 3. the slope-intercept form of a line (y = mx + b): yielding the y-intercept (0, b); m yields another point (x₂, y₂) 4. one point and the slope: (x₁, y₁); m yields another point (x₂, y₂) Do you notice a common element in all 4 cases? Just stick to the basics: 'two points determine a line.'
In our next trip to the slopes, we'll address the second task that Algebra students are usually asked to do: how to find the equation of a line.
Hasta la vista!