3.4 Graphing Linear Equations

Terrance Berg

Chapter 3: Graphing

3.4 Graphing Linear Equations

There are two common procedures that are used to draw the line represented by a linear equation. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation.

If the equation is given in the form $y = mx + b$ , then $m$ gives the rise over run value and the value $b$ gives the point where the line crosses the $y$ -axis, also known as the $y$ -intercept.

Example 3.4.1

Given the following equations, identify the slope and the $y$ -intercept.

$\begin{array}{lll} y = 2x - 3\hspace{0.14in} & \text{Slope }(m)=2\hspace{0.1in}&y\text{-intercept } (b)=-3 \end{array}$
$\begin{array}{lll} y = \dfrac{1}{2}x - 1\hspace{0.08in} & \text{Slope }(m)=\dfrac{1}{2}\hspace{0.1in}&y\text{-intercept } (b)=-1 \end{array}$
$\begin{array}{lll} y = -3x + 4 & \text{Slope }(m)=-3 &y\text{-intercept } (b)=4 \end{array}$
$\begin{array}{lll} y = \dfrac{2}{3}x\hspace{0.34in} & \text{Slope }(m)=\dfrac{2}{3}\hspace{0.1in} &y\text{-intercept } (b)=0 \end{array}$

When graphing a linear equation using the slope-intercept method, start by using the value given for the $y$ -intercept. After this point is marked, then identify other points using the slope.

This is shown in the following example.

Example 3.4.2

Graph the equation $y = 2x - 3$ .

First, place a dot on the $y$ -intercept, $y = -3$ , which is placed on the coordinate $(0, -3).$

Graph with intercept at (0,-3)

Now, place the next dot using the slope of 2.

A slope of 2 means that the line rises 2 for every 1 across.

Simply, $m = 2$ is the same as $m = \dfrac{2}{1}$ , where $\Delta y = 2$ and $\Delta x = 1$ .

Placing these points on the graph becomes a simple counting exercise, which is done as follows:

For m = 2, go up 2 and forward 1 from each point.

From each point, go up 2 and forward 1 to find the next point.

Once several dots have been drawn, draw a line through them, like so:

Line with slope of 2. Passes through (−3, 0), (1, −1), (2, 1), and (3, 3).

Note that dots can also be drawn in the reverse of what has been drawn here.

Slope is 2 when rise over run is $\dfrac{2}{1}$ or $\dfrac{-2}{-1}$ , which would be drawn as follows:

For m = 2, go down 2 and back 1 from each point.

Example 3.4.3

Graph the equation $y = \dfrac{2}{3}x$ .

First, place a dot on the $y$ -intercept, $(0, 0)$ .

Now, place the dots according to the slope, $\dfrac{2}{3}$ .

When m = 2 over 3, go up 2 and forward 3 to get the next point.

This will generate the following set of dots on the graph. All that remains is to draw a line through the dots.

Line with slope 2 over 3. Passes through (−3, −2), (0, 0), (3, 2), and (6, 4).

The second method of drawing lines represented by linear equations and functions is to identify the two intercepts of the linear equation. Specifically, find $x$ when $y = 0$ and find $y$ when $x = 0$ .