Skill: Graph equations in slope-intercept form

An equation in slope-intercept form is

$$y=mx+b.$$

• $m$ is the slope, or the change in $y$ for each 1-unit change in $x$.
• $b$ is the $y$-intercept, the point where the line crosses the $y$-axis: $(0,b)$.

Example

Graph

$$y=-\frac{2}{3}x+4.$$

Step 1: Identify the slope and $y$-intercept.

Compare the equation with $y=mx+b$:

$$m=-\frac{2}{3}, \qquad b=4.$$

The $y$-intercept is $(0,4)$.

Step 2: Plot the $y$-intercept.

Plot the point $(0,4)$ on the coordinate plane.

Step 3: Use the slope to find another point.

The slope is

$$-\frac{2}{3}=\frac{\text{rise}}{\text{run}}.$$

This means:

• Move right 3 units.
• Move down 2 units.

Starting at $(0,4)$, this gives the point

$$(3,2).$$

Repeat the same movement to get another point:

$$(6,0).$$

Step 4: Draw the line.

Draw a straight line through the points $(0,4)$, $(3,2)$, and $(6,0)$. Extend the line in both directions and add arrows.

The graph represents every ordered pair $(x,y)$ that makes the equation true. For example, when $x=3$,

$$y=-\frac{2}{3}(3)+4=-2+4=2,$$

so $(3,2)$ is on the line.

A negative slope means the line decreases as you move from left to right.