Understanding Lines and Their Equations 1.4

Aug 18, 2024

Lecture Notes: Section 1.4 - Lines

Key Topics

  • Graphing lines
  • Determining the slope of a line
  • Applying the slope-intercept form of a line
  • Computing the average rate of change of a function

Slope of a Line

  • Formula: [ m = \frac{y_2 - y_1}{x_2 - x_1} ]
    • Where ((x_1, y_1)) and ((x_2, y_2)) are points on the line.

Example: Sketching and Finding the Slope

  • Points: ((4, 2)) and ((-3, 5))
  • Steps:
    1. Plot points P (4, 2) and Q (-3, 5) on the graph.
    2. Connect the points to sketch the line.
    3. Use slope formula:
      • Assign P as ((x_1, y_1) = (4, 2))
      • Assign Q as ((x_2, y_2) = (-3, 5))
      • Calculate slope: [ m = \frac{5 - 2}{-3 - 4} = \frac{3}{-7} = -\frac{3}{7} ]

Equation of a Line

  • Point-Slope Form: Requires one point and the slope.
    • Equation: [ y - y_1 = m(x - x_1) ]

Deriving Point-Slope Equation

  • Use either point from example:
    • Use Point (4, 2) with slope (-\frac{3}{7}):
      • Equation: [ y - 2 = -\frac{3}{7}(x - 4) ]
    • Use Point (-3, 5) with slope (-\frac{3}{7}):
      • Equation: [ y - 5 = -\frac{3}{7}(x + 3) ]
      • Simplified: [ y - 5 = -\frac{3}{7}x + \frac{9}{7} ]
  • Both equations represent the same line.

Slope-Intercept Form

  • Formula: [ y = mx + b ]
    • Where (m) is the slope, and (b) is the y-intercept.

Example: Finding Slope-Intercept Equation

  • Given point (3, -2) and slope -4:
    • Use Point-Slope Form:
      • Equation: [ y + 2 = -4(x - 3) ]
      • Simplify to find slope-intercept form: [ y = -4x + 10 ]
    • Verify using slope-intercept form directly:
      • Equation: ( y = mx + b ) with known values:
      • Solve (-2 = -4(3) + b )
      • Find (b = 10)
      • Equation: [ y = -4x + 10 ]

Conclusion

  • Two methods to derive the equation of a line:
    • Point-Slope Form
    • Directly using Slope-Intercept Form
  • Both methods yield the same final equation for the line. Special Types of Lines and Their Slopes

Horizontal Lines

  • Definition: A line where all points have the same y-coordinate.
  • Equation Example: y = 2
  • Slope: Horizontal lines have a slope of 0.
    • Calculation Example:
      • Points: (-3, 2) and (2, 2)
      • Slope Calculation: (2 - 2) / (2 - (-3)) = 0/5 = 0
    • Reason: The y-values are constant, leading to a zero difference in y-values.

Vertical Lines

  • Definition: A line where all points have the same x-coordinate.
  • Equation Example: x = 2
  • Slope: Vertical lines have an undefined slope.
    • Calculation Example:
      • Points: (2, 3) and (2, -4)
      • Slope Calculation: (-4 - 3) / (2 - 2) = Undefined (anything/0)
    • Reason: The x-values are constant, leading to a zero difference in x-values.

Parallel and Perpendicular Lines

Parallel Lines

  • Definition: Lines with the same slope running in the same direction.
  • Example Problem:
    • Find the point-slope equation of a line through point (-1, 2), parallel to 2x + 3y - 5 = 0.
    • Steps:
      1. Convert the line equation to slope-intercept form (y = mx + b) to find the slope.
      2. Rearrange: 3y = -2x + 5 → y = (-2/3)x + 5/3
      3. Slope: -2/3
      4. Use point-slope form: y - y₁ = m(x - x₁)
      5. For point (-1, 2): y - 2 = (-2/3)(x + 1)
      6. Solve for y: Distribute and add 2 to both sides.
      7. Equation: y = (-2/3)x + 4/3

Graphing the Line

  • Find y-intercept: 4/3
  • Determine Direction:
    • Slope = -2/3
    • Movement: Down 2 and right 3 from y-intercept.
  • Draw the Line: Connect points using the slope and y-intercept.

Additional Notes

  • Point-Slope Form: Useful for determining equations quickly when slope and a point are known.
  • Slope-Intercept Form: Useful for graphing equations easily by identifying the slope and y-intercept directly.

Lecture Notes: Perpendicular Lines and Average Rate of Change

Perpendicular Lines

  • Definition: Perpendicular lines cross each other to form a right angle.
    • Slope Relationship: The slopes of perpendicular lines are negative reciprocals.
      • If the slope of one line is ( M_1 ), the slope of the perpendicular line is (-\frac{1}{M_1}).

Problem Solving Example

  • Problem Statement: Determine an equation for a line through point (-3, 1) that is perpendicular to the line (2x + 4y + 7 = 0).

    1. Find the Slope of the Given Line:
      • Rewrite in slope-intercept form: (4y = -2x - 7)
      • Divide by 4: (y = -\frac{1}{2}x - \frac{7}{4})
      • Slope ( = -\frac{1}{2})
    2. Determine Slope of Perpendicular Line:
      • Negative reciprocal: (2/1 = 2)
    3. Use Point-Slope Form to Find Equation:
      • Formula: (y - y_1 = m(x - x_1))
      • Substituting values: (y - 1 = 2(x + 3))
      • Simplifying: (y = 2x + 7)
  • Graphing:

    • Intercept: Crosses y-axis at 7
    • Slope: Rise 2, run 1

Average Rate of Change

  • Concept: Similar to slope but applies to any function.
    • Formula: (\frac{f(x_2) - f(x_1)}{x_2 - x_1})

Example Problem

  • Function: (f(x) = x^2 - 1)

  • Calculate Average Rate of Change:

    1. Given Points: (x_1 = -2, x_2 = 0)
    2. Calculate Function Values:
      • (f(0) = 0^2 - 1 = -1)
      • (f(-2) = (-2)^2 - 1 = 3)
    3. Compute Average Rate of Change:
      • (\frac{-1 - 3}{0 + 2} = \frac{-4}{2} = -2)
  • Note: This measures change over an interval, not just for linear functions.

Important Points

  • Ensure comprehension of learning outcomes.
  • Prepare to ask questions if unsure during class discussions.