The gradient of a straight line is a measure of its steepness, indicating how much the line rises or falls for each unit of horizontal movement. This concept is pivotal in coordinate geometry and is extensively covered in the CIE IGCSE maths syllabus.

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Understanding Gradient

The gradient is calculated as the ratio of the vertical change (change in y) to the horizontal change (change in x) between any two points on the line.

Formula:

This formula is crucial for finding the gradient using coordinates of two points on the line.

Worked Examples

Example 1: Calculating Gradient

Given: Points $A(2, 3)$ and $B(5, 11)$.

1. Identify Points: $A(2, 3)$, $B(5, 11)$

2. Calculate Change in Y: $11 - 3 = 8$

3. Calculate Change in X: $5 - 2 = 3$

4. Apply Gradient Formula:

Example 2: Another Gradient Calculation

Given: Points $C(-1, -4)$ and $D(3, -2)$.

1. Points:$C(-1, -4)$, $D(3, -2)$

2. Change in Y: $-2 - (-4) = 2$

3. Change in X: $3 - (-1) = 4$

4. Gradient: $\frac{2}{4} = 0.5$

Practice Questions

Question 1: Gradient Calculation

Find the gradient of the line passing through $E(4, -5)$ and $F(-2, 7)$.

1. Points: $E(4, -5)$, $F(-2, 7)$

2. Change in Y: $7 - (-5) = 12$

3. Change in X: $-2 - 4 = -6$

4. Gradient: $\frac{12}{-6} = -2$

The gradient of the line is -2, indicating the line falls 2 units vertically for every 1 unit it moves horizontally to the right.

Question 2: Finding the Y-intercept

A line with gradient 3 passes through $G(1, 2)$. Determine the y-intercept.

1. Gradient $(m)$: 3

2. Point G Coordinates: $(1, 2)$

3. Equation: $y = mx + c$

4. Substitute and Solve for c (y-intercept):

The y-intercept (c) is -1, indicating that the line crosses the y-axis at $y = -1$.