Slope is the backbone of linear equations, showing how steep and which way a line goes. It's calculated using the rise-over-run formula, comparing changes in y and x coordinates between two points on the line.

Horizontal lines have zero slope, while vertical lines have undefined slope. Knowing how to graph using a point and slope is key. Parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals.

Understanding Slope

Calculation of slope

• Slope measures the steepness and direction of a line
  • Represented as $m$ in the slope-intercept form equation $y = mx + b$ (a linear equation)
• Rise-over-run formula calculates slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • "Rise" is the vertical change found by subtracting the $y$-coordinates of two points on the line
  • "Run" is the horizontal change found by subtracting the $x$-coordinates of the same two points
• Counting grid units on a graph to find slope
  • Count the number of units the line rises vertically between two points (rise)
  • Count the number of units the line runs horizontally between the same two points (run)
  • Slope is the ratio of rise to run expressed as a fraction: $\frac{\text{rise}}{\text{run}}$
• Positive slope indicates the line slants upward from left to right (increasing function)
• Negative slope indicates the line slants downward from left to right (decreasing function)

Slope of horizontal and vertical lines

• Horizontal lines have a slope of zero
  • $y$-coordinate remains constant for all points on the line (no vertical change)
  • Equation in the form $y = b$, where $b$ is the $y$-intercept (y-value where line crosses y-axis)
• Vertical lines have an undefined slope
  • $x$-coordinate remains constant for all points on the line (no horizontal change)
  • Equation in the form $x = a$, where $a$ is the $x$-intercept (x-value where line crosses x-axis)

Graphing with point and slope

• Start at the given point on the coordinate plane
• Use the slope to find additional points on the line
  • If slope is a fraction ($\frac{a}{b}$), rise by the numerator ($a$) and run by the denominator ($b$)
  • If slope is an integer ($n$), rise by the integer ($n$) and run by 1
• Plot the additional points and connect them with a straight line using a ruler
• Extend the line in both directions to cover the entire graph (line continues infinitely)

Relationships between lines

• Parallel lines have the same slope
• Perpendicular lines have slopes that are negative reciprocals of each other
• Direct variation is a special case where the line passes through the origin, and y is directly proportional to x