Lecture Notes on Finding Slope from a Graph

Understanding Slope

• Slope: Measures how steep a line is.
• Formula: Slope (m) = ( \frac{\Delta y}{\Delta x} )
  • ( \Delta ) (Delta) is a symbol for "change in".
  • ( \Delta y ): Change in the y-coordinate (vertical change).
  • ( \Delta x ): Change in the x-coordinate (horizontal change).
• Constant Slope: For a straight line, the slope is constant.

Calculating Slope

1. Select Two Points

   • Choose two points on the line, preferably with integer coordinates for simplicity.
   • Example points: (-3, -3) and (0, -1).

2. Determine Change in x

   • Move from the first point to the second point horizontally.
   • Count the steps:
     • Start at x = -3, move to x = 0.
     • ( \Delta x = 0 - (-3) = 3 ).

3. Determine Change in y

   • Move from the first point to the second point vertically.
   • Count the steps:
     • Start at y = -3, move to y = -1.
     • ( \Delta y = -1 - (-3) = 2 ).

4. Calculate Slope

   • ( m = \frac{\Delta y}{\Delta x} = \frac{2}{3} )

Consistency of Slope

• You can choose any two points on the line and get the same slope.
• Example with different points:
  • Starting at (3, 1) and moving to (-3, -3):
    • ( \Delta y = -3 - 1 = -4 )
    • ( \Delta x = -3 - 3 = -6 )
    • ( m = \frac{-4}{-6} = \frac{2}{3} )

Key Points

• Consistency: Regardless of points chosen, the slope remains ( \frac{2}{3} ).
• Direction: The slope calculation is consistent as long as the direction (positive or negative changes) is accounted for properly.
• Simplification: Slope fractions should always be simplified.