Understanding Slope: Rise Over Run

Meaning of Slope

• Slope indicates a line's steepness and direction relative to the x-axis.
• It represents the rate of change, indicating how quickly and in what direction a line moves.
• Slope shows the rate at which "y" changes with respect to "x" along the line.
• Two points on a line are needed to calculate its slope.
• Slope (m) between points (x1, y1) and (x2, y2) is calculated using the formula: m = (y2 - y1) / (x2 - x1)

Rise Over Run

• The term (y2 - y1) is called the rise, representing the vertical difference between two points on the line.
• The term (x2 - x1) is called the run, representing the horizontal difference between the two points on the line.
• Slope is often described as "rise over run".

Properties of Slope

• Slope is a numerical value describing a line's incline or slant.
• It's the change in y-coordinates divided by the change in x-coordinates for two distinct points on the line.
• Slope can be any real number value.
• A line with a slope of zero is horizontal.
• A positive slope indicates that the graph increases as it moves to the right.
• A negative slope indicates that the graph decreases as it moves to the right.

Examples of Slope Values

• A slope of 0.1 indicates a very slow increase, appearing almost horizontal.
• A slope of -0.1 indicates a very slow decrease, appearing almost horizontal.
• An irrational slope (e.g., √2/2) is possible since irrational numbers are real numbers.
• A slope of -9/2 means moving 9 units down and 2 units to the right to get from one point to another point on the line.
• A slope of 1,000 indicates a very steep increase, appearing almost vertical.

Calculating Slope from a Graph

• Select two points on the graph.
• Count the units moved vertically (rise) and horizontally (run) to get from one point to the other.
• The slope is the rise divided by the run.
• If the coordinates of two points are known, the slope formula can be used directly: m = (y2 - y1) / (x2 - x1)

Slope from the Equation of a Line

• The equation of a straight line can be used to determine the slope of the line.

Point Slope Form

• The point-slope form of a linear equation is: y - y1 = m(x - x1)
• (x1, y1) are the coordinates of a point on the line.
• "m" represents the slope.
• If a line is in point-slope form, the slope is the number being multiplied by (x - x1).
  • Example: y - 3 = 2(x + 1) has a slope of 2.
  • Example: y + 1 = -3(x - 2) has a slope of -3.

Slope Intercept Form

• The slope-intercept form of a line is y = mx + b
• "m" is the slope of the line.
• "b" is the y-intercept of the line.
• In this form, the slope is the coefficient of the x-variable.
  • Example: y = (3/2)x + 1 has a slope of 3/2.
  • Example: y = -2.5x - 4 has a slope of -2.5.

Parallel Lines

• Parallel lines are distinct lines that never intersect.
• Parallel lines have the same slope.
• If two lines have the same slope but different y-intercepts, they are parallel.

Conceptual Questions

• Slope of a Vertical Line: The slope is undefined because the denominator of the slope formula is zero.
• Slope of a Horizontal Line: The slope is zero because the numerator of the slope formula is zero.
• Slopes of Perpendicular Lines: The slopes are negative reciprocals of each other. If one line has a slope of m, a line perpendicular to it has a slope of -1/m.

Numerical Problems and Solutions

• Problem: What is the slope of the line between the points (1, 2) and (4, 3)?
  • Solution: Using the slope formula m = (3 - 2) / (4 - 1) = 1/3.
• Problem: What is the slope of the line y - 4 = 5(x + 2)?
  • Solution: The equation is in point-slope form, so the slope is 5.
• Problem: What is the slope of the line y = 6x - 2?
  • Solution: The equation is in slope-intercept form, so the slope is 6.
• Problem: What is the slope of the line containing the points (3, 4) and (3, 7)?
  • Solution: Using the slope formula m = (7 - 4) / (3 - 3) = 3 / 0, which is undefined, so the slope is undefined.
• Problem: Given the line y = 7x + 3, what is the slope of a line parallel to it? Perpendicular to it?
  • Solution: The slope of the given line is 7.
    • The slope of a parallel line is also 7.
    • The slope of a perpendicular line is -1/7.