Gradient of linear graphs Revision Notes

The slope of a straight line measures how steep the line is and in which direction it goes. It tells us how much the line rises or falls vertically for each unit it moves horizontally. A positive slope means the line rises as you move from left to right, a negative slope means it falls, zero slope means the line is horizontal, and an undefined slope means the line is vertical.

• Slope indicates steepness and direction of a line.
• Positive slope: line rises left to right.
• Negative slope: line falls left to right.
• Zero slope: horizontal line.
• Undefined slope: vertical line.

To calculate the slope, first identify two points on the line. These points should be easy to read from the grid, ideally where the line crosses grid intersections. Label these points as \((x_1, y_1)\) and \((x_2, y_2)\). The order does not affect the slope value but be consistent when subtracting coordinates.

• Choose points exactly on the line at grid intersections.
• Label points clearly as \((x_1, y_1)\) and \((x_2, y_2)\).
• Use points with integer coordinates if possible for simplicity.

Calculate the vertical change (\(\Delta y\)) by subtracting the y-coordinate of the first point from the y-coordinate of the second point: \(\Delta y = y_2 - y_1\). Similarly, calculate the horizontal change (\(\Delta x\)) by subtracting the x-coordinate of the first point from the x-coordinate of the second point: \(\Delta x = x_2 - x_1\). These differences represent how far the line moves vertically and horizontally between the two points.

• \(\Delta y = y_2 - y_1\) (vertical change)
• \(\Delta x = x_2 - x_1\) (horizontal change)
• Use subtraction in the same order for both coordinates.

The slope \(m\) is the ratio of the vertical change to the horizontal change: \[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula gives the steepness of the line. Calculate the numerator and denominator separately, then divide. If \(\Delta x = 0\), the slope is undefined because division by zero is not possible, indicating a vertical line.

• Use formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
• Calculate numerator and denominator carefully.
• If \(\Delta x = 0\), slope is undefined (vertical line).

Consider a line passing through points \((2, 3)\) and \((5, 11)\) on a grid. Calculate the slope:

1. Find \(\Delta y = 11 - 3 = 8\). 2. Find \(\Delta x = 5 - 2 = 3\). 3. Calculate slope \(m = \frac{8}{3} \approx 2.67\).

This means the line rises 8 units vertically for every 3 units it moves horizontally to the right.

• Points: \((2, 3)\) and \((5, 11)\)
• \(\Delta y = 8\), \(\Delta x = 3\)
• Slope \(m = \frac{8}{3} \approx 2.67\)

A slope of \(2.67\) means the line is quite steep and rises as you move from left to right. If the slope were negative, the line would fall. A slope of zero means the line is flat, and an undefined slope means the line is vertical. Understanding these interpretations helps in graph sketching and problem solving.

• Positive slope: line rises.
• Negative slope: line falls.
• Zero slope: horizontal line.
• Undefined slope: vertical line.