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Point Slope Form

In the previous section, we learned about slope-intercept form. The next section will look at standard form. This section focuses on the point-slope equation.

The equation of the line in point slope form is: $$y-y_1=m(x-x_1)$$ Here, $m$ is the slope, and $(x_1,y_1)$ is a point the line passes through.

Graphing

To sketch a graph of the equation, all you need to do is pull out the point the line passes through and the slope.

Remember what we talked about with parallel lines earlier: parallel lines have the same slope but different y-intercepts. That same concept applies here, too! We won't look at parallel lines in this lesson, though.

Let's look at an example. Suppose we want to graph the line: $$y+2=\frac32(x-5)$$ Pulling out the info we need, we have a slope of $m=\frac32$, and the point it passes through is $(x_1,y_1)=(5,-2)$.

In case you forgot, here's the slope formula: $$\text{slope}=m=\frac{\text{rise}}{\text{run}}=\frac{\Delta y}{\Delta x}$$

Next, pick a point and plot it.

Then, use the fact that slope is rise over run, to find another point on the line. Basically, find the slope. If our slope is $\frac32$, then the rate of change is: $\frac{3\text{ rise}}{2\text{ run}}$.

Finally, we draw the line through these points.

To take this a step further, you may be given a word problem of a linear relationship to graph. Follow these steps to solve those problems, too!

(Video) Graphing a Line Using a Point and Slope

by PatrickJMT

This video by Patrick JMT explains how to graph a line using just a single point and the slope of the line.

$\large{\text{Summary}}$

To graph an equation in point slope form, the first step is to extract the known point and slope from the equation.

The equation is: $$y-y_1=m(x-x_1)$$ The slope will be $m$ and the known point will be the coordinate $(x_1,y_1)$.

Once you have the point, plot it, and use the slope to find which way the line extends.

Example

Graph the line $y-2=3(x-1)$.

From the equation we know the slope is $m=3$ and the line passes through the point $(1,2)$.

Next, we plot our point.

Then use the slope to find which way the line extends. Remember that slope is $\frac{\text{rise}}{\text{run}}$. So our slope is $\frac{3\text{ rise}}{1\text{ run}}$.

This gives us another point on our line. All we have to do is connect these points to graph the line.

Solution

In point slope form, the slope, $m$, is the value multiplied by the parentheses. $$y-y_1={\color{#8925ae}{m}}(x-x_1)$$ So in our equation, the slope is $2$. $$y-3={\color{#8925ae}{2}}(x+5)$$

Solution

In point slope form, $x_1$ and $y_1$ are the coordinates of the point the line passes through. $$y-{\color{#8925ae}{y_1}}=m(x-{\color{#8925ae}{x_1}})$$ So in our equation, $y_1=3$ and $x_1=-5$. $$y-{\color{#8925ae}{3}}=2(x\,\color{#8925ae}{{+\,5}})$$ The line passes through the point $(-5,3)$.

Let's graph the line: $y-3=2(x+5)$.

Sketch a Graph Using Point Slope Form

Now that we've learned what point-slope form is, we can learn how to graph it!

Now you know how to graph a function that is given in point-slope form!