Understanding Slope

Overview

The slope of a line is a measure of its steepness and direction. It describes how much the y-coordinate changes for each unit of change in the x-coordinate. In a mathematical sense, slope is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Formula for Slope

The slope \( m \) between two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), can be calculated using the following formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here:

  • \( y_2 - y_1 \): The difference in the y-coordinates (rise)
  • \( x_2 - x_1 \): The difference in the x-coordinates (run)

Types of Slopes

  • Positive Slope: Line rises from left to right.
  • Negative Slope: Line falls from left to right.
  • Zero Slope: Line is horizontal.
  • Undefined Slope: Line is vertical.

Practice Problems

  1. Find the slope of the line passing through the points (2, 3) and (6, 7).
    Solution

    Using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

    \[ m = \frac{7 - 3}{6 - 2} = \frac{4}{4} = 1 \]

  2. Determine the type of slope for each line described:
    • A line that passes through the points (3, 5) and (7, 5):
      Solution

      Zero Slope (horizontal line)

    • A line that passes through the points (-2, -1) and (2, 3):
      Solution

      Positive Slope

    • A line that passes through the points (4, 6) and (4, -2):
      Solution

      Undefined Slope (vertical line)

  3. Calculate the slope of the line passing through points (-5, 2) and (4, -3).
    Solution

    Using the formula:

    \[ m = \frac{-3 - 2}{4 - (-5)} = \frac{-5}{9} \]

    The slope is \( -\frac{5}{9} \), which is a negative slope.

  4. Plot points (1, 1) and (5, 3) on a graph. Determine if the line through these points has a positive, negative, zero, or undefined slope.
    Solution

    Calculate the slope:

    \[ m = \frac{3 - 1}{5 - 1} = \frac{2}{4} = \frac{1}{2} \]

    The slope is positive, so the line rises from left to right.

  5. Challenge: If a line has a slope of 3 and passes through the point (2, -1), find another point on this line.
    Solution

    Using the slope-intercept form, pick a value for x (e.g., \( x = 4 \)):

    \[ y = 3(x - 2) - 1 \rightarrow y = 3 \times 2 - 1 = 6 - 1 = 5 \]

    Another point on the line is (4, 5).

Additional Practice

Try these additional questions for more practice:

  1. What is the slope of the line between the points (0, 0) and (4, 8)?
    Solution

    The slope is \( m = \frac{8 - 0}{4 - 0} = 2 \)

  2. Does the line passing through (-2, 5) and (-2, -3) have a slope? If so, what type?
    Solution

    The slope is undefined (vertical line)

  3. If the slope of a line is -2 and one point on the line is (3, 4), find another point on the line by increasing the x-coordinate by 2 units.
    Solution

    New point: (5, 0) since \( y = 4 - 2 \times 2 = 0 \)

→Slope-Intercept Form of a Line