Straight Line Equation

Introduction

• Linear Equation: Defines a straight line using the formula $y = mx + c$, where:

  • Slope (m): The rate at which y changes with respect to x, indicating the line's steepness and direction.
  • Y-Intercept (c): The point at which the line intersects the y-axis.

• Application: Such equations are useful in real-life applications like budgeting and forecasting trends.

Graphing Linear Functions

• Cartesian Plane: The foundation for graphing functions, intersecting at the origin.
  infoNote

  The Cartesian Plane is a critical tool for plotting and analysing mathematical functions.

• Plotting Steps:
  • Determine the Y-intercept ($c$) and slope ($m$).
  • Highlight: the significance of accurately plotting each element.
• Worked Example:
  • Graph $y = 2x + 3$:
    1. Plot the y-intercept: (0, 3)
    2. Use the slope (2) to find another point: move 1 unit right and 2 units up to (1, 5)
    3. Draw a straight line through these points

Direct Variation and Special Cases

• Direct Variation: Occurs when $c = 0$, expressed as $y = mx$, indicating a direct proportionality between $x$ and $y$.

Slope ($m$) and Y-Intercept ($c$) Characteristics

• Slope Variability:
  • Positive: Line ascends.
  • Negative: Line descends.
  • Zero: Horizontal line.
  • Undefined: Vertical line.
• Visual Aids:

Y-Intercept ($c$)

• Definition: The point where the line intersects the y-axis, influencing the line's vertical orientation.
• Impact: Modifying $c$ moves the line vertically without affecting its slope.

Deriving Equations through Points and Slopes

Point-Slope Form

• Formula: $y - y_1 = m(x - x_1)$
• Used for finding: A line's equation from a known point $(x_1, y_1)$ and the slope ($m$).

Example Derivation Steps:

• Given Point: $(2, 3)$ and $m = 4$
  • Substitute: $y - 3 = 4(x - 2)$
  • Expand: $y - 3 = 4x - 8$
  • Rearrange to: $y = 4x - 8 + 3$
  • Simplify to: $y = 4x - 5$

Equation through Two Points

• Gradient Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Example: For points $(3, 4)$ and $(5, 6)$, derive:
  • Calculate gradient: $m = \frac{6 - 4}{5 - 3} = \frac{2}{2} = 1$
  • Substitute into point-slope form: $y - 4 = 1(x - 3)$
  • Expand: $y - 4 = x - 3$
  • Simplifies to: $y = x + 1$