Negative Slope
AlgebraA negative slope means a line falls from left to right on a graph, indicating that as x increases, y decreases.
Formula
m < 0
Definition
A line has a negative slope when it goes downhill from left to right. As you move right on the graph, the line goes down.
Example
$y = -3x + 6$ has a slope of $-3$, which is negative. Starting at $(0, 6)$, move right $1$ and down $3$ to reach $(1, 3)$. The line drops as you go right.
Key Insight
A negative slope looks like a backslash \. Think of going down a hill as you walk to the right.
Definition
A line has a negative slope when $m < 0$, meaning $y$ decreases as $x$ increases. The more negative $m$ is, the steeper the downward angle. A slope of $-1$ creates a $45$-degree downward angle.
Example
$y = -0.5x + 4$: for every $2$ units right, $y$ drops $1$ unit (gentle decline). $y = -5x + 10$: for every $1$ unit right, $y$ drops $5$ units (steep decline).
Key Insight
Negative slope indicates a negative (inverse) relationship between variables. Examples: as temperature drops, heating costs rise; as speed increases on a fuel-limited trip, range decreases.
Definition
A negative slope $m < 0$ means $f(x) = mx + b$ is strictly decreasing on $\mathbb{R}$, with $f'(x) = m < 0$ everywhere. The line makes an obtuse angle with the positive x-axis: $90^\circ < \theta < 180^\circ$, and $\tan\theta = m < 0$. Perpendicular lines have slopes $m$ and $-1/m$, so a line with negative slope has a perpendicular with positive slope.
Example
Slope $m = -2$: $\theta = \arctan(-2) \approx 116.57^\circ$. The perpendicular has slope $1/2$ and $\theta \approx 26.57^\circ$. Their product: $(-2)(1/2) = -1$.
Key Insight
A strictly decreasing linear function is also invertible. Its inverse has slope $1/m$, which is also negative. The product $m \cdot (1/m) = 1$, not $-1$, so a line and its inverse (reflected over $y = x$) are generally not perpendicular.