Lecture on Linear Functions and Slope

Objectives

• Understand how to find the slope.
• Write and graph the equation of a line.
• Understand slopes of parallel and perpendicular lines.
• Calculate average rate of change.

Finding Slope

• Formula for Slope (m): [ m = \frac{y_2 - y_1}{x_2 - x_1} ]
  • Alternative: Change in y over change in x.
  • Order of points does not matter.
• Examples:
  • Example 1: Points ((x_1, y_1), (x_2, y_2))
    • ( y_2 = -9, y_1 = 6, x_2 = 2, x_1 = 3 )
    • [ m = \frac{-15}{5} = -3 ]
  • Practice examples demonstrate positive, negative, zero, and undefined slopes.

Types of Slopes

• Negative Slope: Line goes down from left to right.
• Positive Slope: Line goes up from left to right.
• Zero Slope: Horizontal line.
• Undefined Slope: Vertical line.

Writing and Graphing Line Equations

• Point-Slope Formula: [ y - y_1 = m(x - x_1) ]
  • Used to write the equation when given a slope and a point.
• Slope-Intercept Form: [ y = mx + b ]
  • Where ( b ) is the y-intercept.
• Example:
  • Given slope ( m = \frac{2}{5} ) and point ( (4, -1) )
  • Substitute into point-slope formula and simplify.
• Graphing:
  • Start at the y-intercept.
  • Use slope to find another point (rise over run).

Graphing with Two Points

• Process:
  • Plot the points.
  • Use the slope formula to find the slope.
  • Use point-slope formula to write the equation.
• Vertical Line Example:
  • Points ((4, -1)) and ((4, 7)).
  • Slope is undefined; equation is ( x = 4 ).

Parallel and Perpendicular Lines

• Parallel Lines:
  • Have the same slope.
• Perpendicular Lines:
  • Slopes are opposite reciprocals.
• Examples:
  • Parallel: Use same slope; solve for ( y ) to find slope.
  • Perpendicular: Calculate opposite reciprocal of given slope.

Summary Notes

• Vertical lines: Form ( x = \text{constant} ).
• Horizontal lines: Form ( y = \text{constant} ).
• Practice plotting and finding equations reinforces understanding.