Here’s a standard problem that could be found in any Algebra I textbook.
Find the equation of the line between and .
The first step is clear: the slope of the line is
At this point, there are two reasonable approaches for finding the equation of the line.
Method #1. This is the method that was hammered into my head when I took Algebra I. We use the point-slope form of the line:
For what it’s worth, the point-slope form of the line relies on the fact that the slope between and is also equal to .
Method #2. I can honestly say that I never saw this second method until I became a college professor and I saw it on my students’ homework. In fact, I was so taken aback that I almost marked the solution incorrect until I took a minute to think through the logic of my students’ solution. Let’s set up the slope-intercept form of a line:
Then we plug in one of the points for and to solve for .
Therefore, the line is .
My experience is that most college students prefer Method #2, and I can’t say that I blame them. The slope-intercept form of a line is far easier to use than the point-slope form, and it’s one less formula to memorize.
Still, I’d like to point out that there are instances in courses above Algebra I that the point-slope form is really helpful, and so the point-slope form should continue to be taught in Algebra I so that students are prepared for these applications later in life.
Topic #1. In calculus, if is differentiable, then the tangent line to the curve at the point has slope . Therefore, the equation of the tangent line (or the linearization) has the form
This linearization is immediately obtained from the point-slope form of a line. It also can be obtained using Method #2 above, so it takes a little bit of extra work.
Topic #2. In statistics, a common topic is finding the least-squares fit to a set of points . The solution is called the regression line, which has the form
In this equation,
- and are the means of the and values, respectively.
- and are the sample standard deviations of the and values, respectively.
- is the correlation coefficient between the and values.
The formula of the regression line is decidedly easier to write in point-slope form than in slope-intercept form. Also, the point-slope form makes the interpretation of the regression line clear: it must pass through the point of averages .