## Graphing a Line Given a Point and the Slope

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Up to now, in this tutorial, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

One other method we can use to graph lines is called the **point–slope method**. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.

### Example: How To Graph a Line Given a Point and The Slope

Graph the line passing through the point \(\left(1,-1\right)\) whose slope is \(m=\frac{3}{4}\).

### Solution

### Graph a line given a point and the slope.

- Plot the given point.
- Use the slope formula \(m=\frac{\text{rise}}{\text{run}}\) to identify the rise and the run.
- Starting at the given point, count out the rise and run to mark the second point.
- Connect the points with a line.

## Example

Graph the line with *y*-intercept 2 whose slope is \(m=-\frac{2}{3}\).

### Solution

Plot the given point, the *y*-intercept, \(\left(0,2\right)\).

\(\begin{array}{ccccc}\text{Identify the rise and the run.}\hfill & & \hfill m& =\hfill & -\frac{2}{3}\hfill \\ & & \hfill \frac{\text{rise}}{\text{run}}& =\hfill & \frac{-2}{3}\hfill \\ & & \hfill \text{rise}& =\hfill & -2\hfill \\ & & \hfill \text{run}& =\hfill & 3\hfill \end{array}\)

Count the rise and the run. Mark the second point.

Connect the two points with a line.

You can check your work by finding a third point. Since the slope is \(m=-\frac{2}{3}\), it can be written as \(m=\frac{2}{-3}\). Go back to \(\left(0,2\right)\) and count out the rise, 2, and the run, \(-3\).

## Example

Graph the line passing through the point \(\left(-1,-3\right)\) whose slope is \(m=4.\)

### Solution

Plot the given point.

\(\begin{array}{ccccc}\text{Identify the rise and the run.}\hfill & & \hfill m& =\hfill & 4\hfill \\ \text{Write 4 as a fraction.}\hfill & & \hfill \frac{\text{rise}}{\text{run}}& =\hfill & \frac{4}{1}\hfill \\ & & \hfill \text{rise}& =\hfill & 4\phantom{\rule{0.5em}{0ex}}\text{run}=1\hfill \end{array}\)

Count the rise and run and mark the second point.

Connect the two points with a line.

You can check your work by finding a third point. Since the slope is \(m=4\), it can be written as \(m=\frac{-4}{-1}\). Go back to \(\left(-1,-3\right)\) and count out the rise, \(-4\), and the run, \(-1\).