Lecture on Linear Functions

Introduction to Linear Functions

• Overview of basic questions related to linear functions.

Slope of a Line

• Problem: Find the slope of the line passing through points (2, -3) and (4, 5).
• Formula: ( m = \frac{y_2 - y_1}{x_2 - x_1} )
  • Assign: ( (x_1, y_1) = (2, -3) ) and ( (x_2, y_2) = (4, 5) )
  • Calculation: ( 5 - (-3) = 8 ) and ( 4 - 2 = 2 )
  • Slope: ( \frac{8}{2} = 4 )

Slope and Y-Intercept

• Problem: Determine slope and y-intercept of ( y = 2x - 3 )
• Slope-Intercept Form: ( y = mx + b )
  • ( m = 2 ) (slope)
  • ( b = -3 ) (y-intercept)
  • Y-intercept can be expressed as ( (0, -3) ).

Graphing Equations

X-Equals and Y-Equals

• Equations: ( x = 2 ) and ( y = 3 )
  • ( x = 2 ): Vertical line at ( x = 2 )
  • ( y = 3 ): Horizontal line at ( y = 3 )

Slope-Intercept Method

• Equation: ( y = 3x - 2 )
  • Slope: 3
  • Y-Intercept: -2
  • Steps:
    • Begin at point ( (0, -2) )
    • From initial point, rise 3 units and run 1 unit for new point ( (1, 1) )
    • Repeat for additional point ( (2, 4) )
    • Connect points with a line.

Using X and Y Intercepts

• Equation: ( 2x - 3y = 6 )
  • X-Intercept: Set ( y = 0 ), solve ( x = 3 )
  • Y-Intercept: Set ( x = 0 ), solve ( y = -2 )
  • Graph from points ( (3, 0) ) and ( (0, -2) ).

Forms of Linear Equations

• Slope-Intercept Form: ( y = mx + b )
• Standard Form: ( Ax + By = C )
• Point-Slope Form: ( y - y_1 = m(x - x_1) )

Writing Equations

Point and Slope

• Problem: Line through point ( (2, 5) ) with slope 3
  • Use Point-Slope: ( y - 5 = 3(x - 2) )
  • Convert to Slope-Intercept: ( y = 3x - 1 )

Two Points

• Points: ( (-3, 1) ) and ( (2, -4) )
  • Calculate slope ( m = -1 )
  • Use Point-Slope: ( y - 1 = -1(x + 3) )
  • Convert to Slope-Intercept: ( y = -x - 2 )

Parallel Line

• Point: ( (3, -2) )
• Line: ( 2x + 5y = 3 ) (Slope: ( -\frac{2}{5} ))
  • Parallel slope is same: ( m = -\frac{2}{5} )
  • Use Point-Slope: ( y + 2 = -\frac{2}{5}(x - 3) )
  • Convert to Slope-Intercept: ( y = -\frac{2}{5}x - \frac{4}{5} )

Perpendicular Line

• Point: ( (-4, -3) )
• Line: ( 3x - 4y = 5 ) (Slope: ( \frac{3}{4} ))
  • Perpendicular slope: ( -\frac{4}{3} )
  • Use Point-Slope: ( y + 3 = -\frac{4}{3}(x + 4) )
  • Convert to Slope-Intercept: ( y = -\frac{4}{3}x - \frac{25}{3} )