4D Gradient Notes

Learning intentions

• To understand that the gradient is the ratio of the vertical change of a graph to its horizontal change between two points
• To understand that the gradient of a straight line is constant
• To know that the gradient can be positive, negative, zero or undefined
• To be able to find the gradient of a line using a graph or two given points

Key ideas

• Gradient (m) is the ratio of rise to run: $m = \frac{\text{rise}}{\text{run}}$
• Rise = vertical change, Run = horizontal distance
• When you work from left to right, the run is considered positive
• Gradient can be positive, negative, zero or undefined
• A vertical line has an undefined gradient (division by zero)
• A horizontal line has a gradient of zero (rise = 0)
• The gradient is constant for a straight line; it does not change along the line
• If you know two points ((x1,y1)) and ((x2,y2)), the gradient is: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

Building understanding

• How to calculate gradient from two points:

  • Given two points ((x1,y1)) and ((x2,y2)), compute
    $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
  • If the line slopes upwards as you move left to right, the gradient is positive
  • If it slopes downwards, the gradient is negative
  • If the line is horizontal, the gradient is zero (rise = 0)
  • If the line is vertical, the run is zero and the gradient is undefined

• Example computations (illustrative values from the transcript):

  • From ((3,6)) to ((2,3)):
    $m = \frac{3 - 6}{2 - 3} = \frac{-3}{-1} = 3.$
  • Gradient is positive.
  • From ((0,2)) to ((1,2)):
    $m = \frac{2 - 2}{1 - 0} = \frac{0}{1} = 0.$
  • Gradient is zero (horizontal line).
  • From ((-3,-1)) to ((-2,3)):
    $m = \frac{3 - (-1)}{-2 - (-3)} = \frac{4}{1} = 4.$
  • Gradient is positive.
  • From ((-1,0)) to ((7,0)):
    $m = \frac{0 - 0}{7 - (-1)} = \frac{0}{8} = 0.$
  • Gradient is zero (horizontal).

• Worked example from the activity prompts (gradients from given rise/run data):

  • a) If m = 4 and rise = 8, then
    $4 = \frac{8}{\text{run}} \quad\Rightarrow\quad \text{run} = 2.$
  • b) If m = 6 and rise = 3, then
    $6 = \frac{3}{\text{run}} \quad\Rightarrow\quad \text{run} = \frac{1}{2}.$
  • c) If m = \text{run} = 4, rise = ?
    $4 = \frac{\text{rise}}{4} \quad\Rightarrow\quad \text{rise} = 16.$
  • d) If run = 15 and m is negative (e.g. m = -3 as implied by the shorthand),
    $\text{rise} = m \times \text{run} = (-3) \times 15 = -45.$

Example problems (Example 7: Finding the gradient of a line)

• For each graph, state whether the gradient is positive, negative, zero or undefined, then find the gradient where possible.

  • a) Positive gradient (rises left to right). If rise = 4 and run = 2, then
    $m = \frac{4}{2} = 2.$
  • b) Negative gradient (decreases left to right). If rise = -3 and run = 2, then
    $m = \frac{-3}{2} = -\frac{3}{2}.$
  • c) Zero gradient (horizontal line). Gradient: $m = 0.$
  • d) Undefined gradient (vertical line). Gradient: undefined.

• You try: The line is horizontal → gradient = 0. The line is vertical → gradient = undefined.

Practical activity insights

• Dragging endpoints on a grid demonstrates that the gradient remains the same under translation along the line: the slope is determined by direction, not by position.
• It is possible to drag endpoints while keeping the same gradient value (the line remains parallel to its original orientation).
• To achieve gradient zero, make the line horizontal (rise = 0). To achieve gradient undefined, make the line vertical (run = 0).

Connections and implications

• Real-world relevance: gradient corresponds to rate of change (e.g., distance vs. time gives speed; price vs. quantity shows rate of change in revenue or cost).
• Foundational principle: the gradient is constant for a straight line; changing the line’s position does not change its slope.
• Ethical/practical note: misinterpreting gradient signs can lead to incorrect conclusions about trends; always verify with coordinates or a graph.

Quick reference formulas

• Gradient from two points: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
• Gradient as rise/run: $m = \frac{\text{rise}}{\text{run}}$
• Special cases:
  • Horizontal line: rise = 0 ⇒ $m = 0$
  • Vertical line: run = 0 ⇒ $m\text{ is undefined}$

Summary

• The gradient measures steepness and direction of a straight line.
• It is the ratio of vertical change (rise) to horizontal change (run).
• The gradient can be positive, negative, zero, or undefined depending on the line’s orientation.
• From two points, the gradient is computed as $m = \dfrac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$, and from rise/run notation as $m = \dfrac{\text{rise}}{\text{run}}.$