Example

Understanding and Calculating Gradient

The gradient (or slope) measures the steepness and direction of a line. It represents the rate of change between two points on a graph.

The general approach to calculating the gradient includes:

• Identifying two points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \).
• Applying the gradient formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
• Simplifying the result to find the gradient.

Finding the Gradient of a Straight Line

The gradient formula is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Example:

If the points are (2,3) and (5,7), the solution is:

• Step 1: Substitute values: \( m = \frac{7 - 3}{5 - 2} \).
• Step 2: Simplify: \( m = \frac{4}{3} \).

Finding the Gradient from an Equation

If a line is given in equation form, rewrite it as:

\[ y = mx + c \]

Example:

For the equation \( 3y = 6x + 9 \):

• Step 1: Divide by 3: \( y = 2x + 3 \).
• Step 2: Identify \( m \) as the coefficient of \( x \), so \( m = 2 \).

Finding the Gradient of a Perpendicular Line

The gradient of a perpendicular line is the negative reciprocal of the given gradient.

Example:

If a line has a gradient of \( 4 \):

• Step 1: Take the negative reciprocal: \( m = -\frac{1}{4} \).

Real-life Applications of Gradient

Understanding gradients has many real-world uses, such as:

• Determining the steepness of a road or ramp.
• Analyzing data trends on graphs.
• Calculating rates of change in business and economics.