Learning Objectives
- Understand point-slope form as an application of the idea of the slope of a line.
- Determine the point-slope form of a line from given information.
- Convert equations of lines from point-slope form to slope-intercept form.
Chapter 14 Point-Slope Form
Section 14.1 Start Here, Go There
The slope-intercept form of a line is useful when using them in practical applications. However, when it comes to writing down the equation of a line, it is often inconvenient. The challenge is that for the slope-intercept form, the location of the initial point is restricted to being the and that’s not always true of the available information.
We are going to consider the situation where we know one point on the line and we know what the slope of the line is. The known point will be labeled as \((x_0, y_0)\text{.}\) We are also going to pick another point on the line and call it \((x,y)\text{.}\) The goal is to find a mathematical relationship between these two points.
The only piece of information that has not been used is the slope. The slope is the same value no matter which two points on the line are chosen, so we will use these two points. The rise is the change in the \(y\)-coordinates and the run is the change in the \(x\)-coordinates. We calculate these by taking the differences between the corresponding coordinates.
We know that the slope is the change \(y\) in divided by the change in \(x\) so we can plug this in and then manipulate the formula.
\begin{equation*}
m = \frac{y - y_0}{x - x_0} \implies y - y_0 = m(x - x_0)
\end{equation*}
Definition 14.1. Point-Slope Form.
The point-slope form of the line that passes through the point \((x_0, y_0)\) with slope \(m\) is \(y - y_0 = m(x - x_0)\text{.}\)
Activity 14.1. Using the Point-Slope Form.
Once you have the formula, some problems involving the point-slope form of a line simply require to plug in values. For example, the point-slope form of the line that passes through the point \((1, -2)\) with slope \(\frac{4}{3}\) is \(y + 2 = \frac{4}{3} (x - 1)\text{.}\)
Try it!
Determine the point-slope form of the line that passes through the point \((-2, 5)\) with slope \(-2\text{.}\)
Solution.
\begin{equation*}
y - 5 = -2(x + 2)
\end{equation*}
Activity 14.2. Lines Have Multiple Representations in Point-Slope Form.
It turns out that a line has multiple representations using the point-slope form, depending on which point is considered to be the initial point. This is much easier to identify from a graph. Notice that the lines in the graphs below are identical.
Try it!
Find at least one more point-slope form for the line above.
Solution.
\begin{equation*}
y - 3 = \frac{2}{3} (x - 4) \qquad \text{ or } \qquad y + 3 = \frac{2}{3} (x + 5)
\end{equation*}
Activity 14.3. Rewriting Point-Slope Form in Slope-Intercept Form.
Because there are multiple representations of a single line, we say that the point-slope form of a line is not unique. This means that two different people may end up getting different answers to the same question, and they can both be correct. One way to check that the equations represent the same line is to rewrite the equations in slope-intercept form by solving for \(y\text{.}\)
Try it!
Write the equation \(y - 3 = \frac{2}{3} (x - 4)\) in slope-intercept form.
Solution.
\begin{equation*}
\begin{aligned}
y - 3 \amp = \frac{2}{3} (x - 4) \\
y - 3 \amp = \frac{2}{3} x - \frac{8}{3} \\
y \amp = \frac{2}{3} x + \frac{1}{3}
\end{aligned}
\end{equation*}
Section 14.2 Worksheets
PDF Version of these Worksheets
1
external/worksheets/14-Worksheets.pdfWorksheet Worksheet 1
1.
Determine the point-slope form of the line that passes through the point \((1, 2)\) with slope \(-2\) then convert that equation to slope-intercept form.
2.
Determine the point-slope form of the line that passes through the point \((-4, 1)\) with slope \(\frac{5}{2}\) then convert that equation to slope-intercept form.
3.
Determine the point-slope form of the line that passes through the point \((2, -3)\) with slope \(-\frac{1}{4}\) then convert that equation to slope-intercept form.
4.
Determine the point-slope form of the line that passes through the point \((-3, -2)\) with slope \(\frac{5}{3}\) then convert that equation to slope-intercept form.
Worksheet Worksheet 2
1.
Determine the equation of the line that passes through the point \((-1, 3)\) with slope \(-\frac{2}{3}\text{,}\) then graph it.
2.
Identify the point and slope used to create the equation \(y - 2 = \frac{3}{2}(x - 4)\text{,}\) then graph the line.
Worksheet Worksheet 3
1.
Find two point-slope equations for the line that passes through the points \((-2, 1)\) and \((5, 5)\text{.}\)
2.
Find two point-slope equations for the line that passes through the points \((-3, 4)\) and \((4, -3)\text{.}\)
3.
Find two point-slope equations for the line that passes through the points \((2, -3)\) and \((-1, 4)\text{.}\)
4.
Find two point-slope equations for the line that passes through the points \((-2, -1)\) and \((3, -1)\text{.}\)
Worksheet Worksheet 4
1.
Find three different point-slope forms for the given line.
2.
Find three different point-slope forms for the given line.
Worksheet Worksheet 5
1.
Find a point-slope form of the line that passes through the point \((0, b)\) with slope \(m\text{.}\)
2.
Find a point-slope form of the line that passes through the point \((a, 0)\) with slope \(m\text{.}\)
3.
Two lines are parallel if they have the same slope. Find the point-slope form of the line that passes through the point \((2, -1)\) that is parallel to the line \(y = 2x - 3\text{.}\)
4.
Two lines are parallel if they have the same slope. Find the point-slope form of the line that passes through the point \((-1, -3)\) that is parallel to the line \(y - 2 = \frac{4}{3} (x + 1)\text{.}\)
Section 14.3 Deliberate Practice: Point-Slope Form
Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.
Focus on these skills:
- Present your work legibly.
Worksheet Worksheet
Instructions: Find a point-slope form of the line that meets the specified conditions, then rewrite it in slope-intercept form.
1.
The line that passes through the point \((2,-1)\) with slope \(\frac{1}{3}\text{.}\)
2.
The line that passes through the point \((-1,2)\) with slope \(\frac{3}{2}\text{.}\)
3.
The line that passes through the point \((3,0)\) with slope \(-3\text{.}\)
4.
The line that passes through the points \((2,4)\) and \((-1,-1)\text{.}\)
5.
The line that passes through the points \((1,-2)\) and \((3,2)\text{.}\)
6.
The line that passes through the points \((0,3)\) and \((2,0)\text{.}\)
7.
The line that passes through the point \((-1, -2)\) that is parallel to \(y = - \frac{1}{2} x + 4\)
8.
The line that passes through the point \((-1, -2)\) that is parallel to \(y = \frac{2}{3} x - 1\)
9.
The line that passes through the point \((2, -3)\) that is parallel to \(y + 2 = \frac{5}{2} (x - 2)\)
10.
The line that passes through the point \((2, -3)\) that is parallel to \(y - 1 = -2(x + 3)\)
Section 14.4 Closing Ideas
In the last two sections, we have seen two different forms for the equation of a line. Why would we need two different versions of the same thing?
- The slope-intercept form is very useful in practical applications. It has a form that can be interpreted in useful ways.
- The point-slope form is more flexible, and can be applied in both practical and theoretical applications. However, it does not give a unique formula for each line and is often harder to intuitively visualize.
One of the keys to mathematical thinking is having the flexibility to see the same thing in different ways. This is not the only time this theme will come up. Later on, we’re going to look at several ways of understanding another concept, and see that each one has their own strengths and weaknesses.
As you continue to learn more mathematics, do not assume that simply because you know how to do things in a certain way that it is the "correct" way of looking at it. It may be that a different approach will yield even better results.
