Slope

Definition

Slope is a measure of the steepness or incline of a line, defined as the ratio of the vertical change to the horizontal change between two points on the line. It is often represented by the letter $m$ and is calculated using the formula $m = \frac{\Delta y}{\Delta x}$.

5 Must Know Facts For Your Next Test

1. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are two distinct points on a line.
2. A slope of zero indicates a horizontal line, while an undefined slope indicates a vertical line.
3. Positive slopes indicate lines that rise from left to right, whereas negative slopes indicate lines that fall from left to right.
4. In linear equations in the form $y = mx + b$, $m$ represents the slope and $b$ represents the y-intercept.
5. The concept of slope is fundamental in calculus for understanding derivatives, which represent instantaneous rates of change.

Review Questions

• How would you find the slope between two points $(3, 4)$ and $(7, 10)$?
• What does a slope of -2 signify about the direction of a line?
• Explain how you can determine if a line is horizontal or vertical based on its slope.

Related terms

Derivative: A derivative represents an instantaneous rate of change and is defined as the limit of the average rate of change as the interval approaches zero.

$y$-intercept: $y$-intercept is the point where a graph intersects with the $y$-axis, found by setting $x = 0$ in an equation.

$x$-intercept: $x$-intercept is where a graph crosses the $x$-axis, determined by setting $y = 0$ in an equation.