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 $m = \frac{\Delta y}{\Delta x}$.
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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.
A slope of zero indicates a horizontal line, while an undefined slope indicates a vertical line.
Positive slopes indicate lines that rise from left to right, whereas negative slopes indicate lines that fall from left to right.
In linear equations in the form $y = mx + b$, $m$ represents the slope and $b$ represents the y-intercept.
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.