Gradient of linear graphs

1. Overview

The gradient of a linear graph is a number that tells you how steep the line is. It's the change in the $y$-value divided by the change in the $x$-value. A positive gradient means the line slopes upwards, a negative gradient means it slopes downwards, a zero gradient means it's a horizontal line, and an undefined gradient means it's a vertical line. This concept is crucial for understanding rates of change and relationships between variables.

Key Definitions

• Gradient: The numerical value representing the steepness and direction of a line (often denoted by the letter $m$).
• Rise: The vertical change (change in $y$) between two points on a line.
• Run: The horizontal change (change in $x$) between two points on a line.
• Positive Gradient: A line that slopes upwards from left to right.
• Negative Gradient: A line that slopes downwards from left to right.

Core Content

Finding the Gradient of a Straight Line from a Graph

To find the gradient ($m$) of a line drawn on a grid, we use the "Rise over Run" method.

Steps:

1. Identify two points on the line that cross the grid corners exactly.
2. Draw a right-angled triangle connecting these two points.
3. Count the number of units for the vertical "rise" and the horizontal "run."
4. Apply the formula: $m = \frac{\text{Rise}}{\text{Run}}$

Worked example 1 — Gradient from graph

Question: Find the gradient of the line passing through $(0, 1)$ and $(2, 5)$.

• Step 1: Vertical change (Rise) = $5 - 1 = 4$
• Step 2: Horizontal change (Run) = $2 - 0 = 2$
• Step 3: $m = \frac{4}{2}$
• Step 4: $m = 2$

Answer: The gradient of the line is $\boxed{2}$.

Worked example 2 — Gradient from graph (negative)

Question: Determine the gradient of the line shown on the graph, which passes through the points $(-2, 3)$ and $(1, -3)$.

• Step 1: Identify the coordinates of the two points: $(-2, 3)$ and $(1, -3)$.
• Step 2: Calculate the rise (change in $y$): $-3 - 3 = -6$
• Step 3: Calculate the run (change in $x$): $1 - (-2) = 1 + 2 = 3$
• Step 4: Apply the gradient formula: $m = \frac{\text{Rise}}{\text{Run}} = \frac{-6}{3}$
• Step 5: Simplify the fraction: $m = -2$

Answer: The gradient of the line is $\boxed{-2}$.

Special Cases:

• Horizontal Lines: The rise is 0, so the gradient is always 0.
• Vertical Lines: The run is 0. Since we cannot divide by zero, the gradient is undefined.

Extended Content (Extended Only)

Calculating Gradient from Two Coordinates

In the Extended curriculum, you must be able to calculate the gradient without a graph by using the coordinates of two points: $(x_1, y_1)$ and $(x_2, y_2)$. This is essential for problems where a graph isn't provided or isn't practical to draw.

Method: Use the formula:

$\qquad \displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$

This formula directly calculates the rise over run using the coordinates of the two points. The order of subtraction matters; ensure you subtract the $y$ and $x$ coordinates in the same order.

Worked example 3 — Gradient from coordinates

Question: Calculate the gradient of the line passing through the points $A(-3, 7)$ and $B(5, -1)$.

• Step 1: Label your coordinates to avoid confusion. $(x_1, y_1) = (-3, 7)$ $(x_2, y_2) = (5, -1)$
• Step 2: Substitute values into the formula. $m = \frac{-1 - 7}{5 - (-3)}$
• Step 3: Simplify the numerator and denominator. $m = \frac{-8}{5 + 3}$ $m = \frac{-8}{8}$
• Step 4: Solve. $m = -1$

Answer: The gradient of the line is $\boxed{-1}$.

Worked example 4 — Gradient from coordinates (fractions)

Question: Find the gradient of the line segment joining the points $(\frac{1}{2}, 2)$ and $(\frac{5}{2}, 5)$.

• Step 1: Label the coordinates: $x_1 = \frac{1}{2}, y_1 = 2$ $x_2 = \frac{5}{2}, y_2 = 5$
• Step 2: Substitute into the gradient formula: $m = \frac{5 - 2}{\frac{5}{2} - \frac{1}{2}}$
• Step 3: Simplify the numerator and denominator: $m = \frac{3}{\frac{4}{2}}$
• Step 4: Simplify the fraction: $m = \frac{3}{2}$

Answer: The gradient of the line is $\boxed{\frac{3}{2}}$.

Note on Marks: Always show the substitution step. If you make a small arithmetic error but have shown the correct formula and substitution, you may still earn "Method" (M) marks.

Key Equations

$\qquad \displaystyle m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$

• Symbols:
  • $m$: Gradient (no units)
  • $(x_1, y_1)$: Coordinates of the first point
  • $(x_2, y_2)$: Coordinates of the second point
• Formula Sheet: Note that this formula is NOT provided on the IGCSE formula sheet; it must be memorized.

Common Mistakes to Avoid

• ❌ Wrong: Calculating $\frac{\text{Run}}{\text{Rise}}$ (i.e., $\frac{\text{change in } x}{\text{change in } y}$). ✓ Right: Always put the vertical change ($y$) on top: $\frac{\text{Rise}}{\text{Run}} = \frac{\text{change in } y}{\text{change in } x}$.
• ❌ Wrong: Misreading the graph scales (e.g., assuming one square equals 1 unit on both axes when the $y$-axis might have a scale of 2 units per square). ✓ Right: Check the numbers on the $x$ and $y$ axes carefully before counting squares. Pay close attention to the scale on each axis.
• ❌ Wrong: Forgetting the negative sign for lines sloping downwards. ✓ Right: Do a "sanity check"—if the line goes down from left to right, your answer must be negative.
• ❌ Wrong: Sign errors when subtracting negative coordinates (e.g., $5 - (-3)$ becoming $5 - 3 = 2$ instead of $5 + 3 = 8$). ✓ Right: Use brackets when substituting negative numbers into the formula: $x_2 - (x_1)$. Double-check your signs!
• ❌ Wrong: Not simplifying fractional gradients. For example, leaving an answer as $\frac{6}{4}$ instead of simplifying to $\frac{3}{2}$. ✓ Right: Always simplify your gradient to its simplest form.

Exam Tips

• Command Words: If the question says "Find" or "Calculate," you must show your working. If it says "State," the answer is usually obvious (like 0 for a horizontal line) and requires no calculation.
• Calculator Tip: For the Extended paper, you can enter the entire fraction $\frac{y_2 - y_1}{x_2 - x_1}$ into your calculator at once to avoid intermediate rounding or sign errors. Use the fraction button on your calculator to ensure correct order of operations.
• Check the Scale: markers often use different scales for the $x$ and $y$ axes (e.g., 1 cm = 5 units on the $y$-axis but 1 cm = 1 unit on the $x$-axis). Never just count squares; use the values from the axes.
• Contextual Questions: In physics-based math questions (like distance-time graphs), remember that the gradient represents the speed. In a speed-time graph, the gradient represents acceleration.
• Units: Gradients are just numbers, they have no units. However, in contextual questions, be mindful of the units of the axes. For example, if a distance-time graph has distance in meters and time in seconds, the gradient (speed) will be in meters per second (m/s).