Skill: Interpret positive, negative, zero, and undefined slopes

Slope tells how much $y$ changes when $x$ changes:

$$m=\frac{\text{change in }y}{\text{change in }x}=\frac{\text{rise}}{\text{run}}$$

Use these rules:

• Positive slope: The line rises from left to right. As $x$ increases, $y$ increases.
• Negative slope: The line falls from left to right. As $x$ increases, $y$ decreases.
• Zero slope: The line is horizontal, so $y$ stays constant.
• Undefined slope: The line is vertical, so $x$ does not change. The slope would require division by $0$.

Worked example

Classify the slope of each line:

$$y=2x+1,\qquad y=-x+3,\qquad y=4,\qquad x=2$$

For an equation in the form $y=mx+b$, the number $m$ is the slope.

1. Line $y=2x+1$

   The slope is $m=2$, which is positive. Therefore, the line rises from left to right.

   Slope: positive

2. Line $y=-x+3$

   The slope is $m=-1$, which is negative. Therefore, the line falls from left to right.

   Slope: negative

3. Line $y=4$

   The $y$-value is always $4$, so the line is horizontal. There is no vertical change as $x$ changes.

   Slope: zero

4. Line $x=2$

   The $x$-value is always $2$, so the line is vertical. The horizontal change is $0$, and division by $0$ is undefined.

   Slope: undefined

Remember: a zero slope is a horizontal line, while an undefined slope is a vertical line.