1.3 rates of change in linear and quadratic functions

Mac Grediser

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27-11-2024

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Understanding Rates of Change: Linear vs. Quadratic Functions

Rates of change are fundamental to understanding how functions behave. They describe how much the output of a function changes in response to a change in its input. While this concept applies to many types of functions, we'll focus here on the distinct rates of change exhibited by linear and quadratic functions.

1. Linear Functions: Constant Rates of Change

A linear function is characterized by a constant rate of change. This means that for every unit increase in the input (x), the output (y) changes by a fixed amount. This constant rate of change is represented by the slope of the line.

The general equation for a linear function is: y = mx + b, where:

• m is the slope (the constant rate of change)
• b is the y-intercept (the value of y when x = 0)

For example, consider the function y = 2x + 1. The slope is 2, indicating that for every 1-unit increase in x, y increases by 2 units. This consistent increase is visually represented by a straight line with a constant incline.

Calculating the rate of change between two points (x₁, y₁) and (x₂, y₂) on a linear function is straightforward:

Rate of Change = (y₂ - y₁) / (x₂ - x₁) = m (the slope)

2. Quadratic Functions: Variable Rates of Change

Unlike linear functions, quadratic functions exhibit a variable rate of change. This means the rate at which the output changes is not constant; it depends on the value of the input. Quadratic functions are characterized by a curved graph (a parabola).

The general equation for a quadratic function is: y = ax² + bx + c, where:

• a, b, and c are constants. The value of 'a' determines the direction and width of the parabola.

The rate of change at any specific point on a quadratic function is given by its derivative. The derivative of a quadratic function is a linear function, representing the instantaneous rate of change at a given point.

Let's consider the function y = x². The derivative of this function is dy/dx = 2x. This means the rate of change at any point x is 2x. Notice that this rate of change is itself a function of x, confirming the variable nature of the rate of change in quadratic functions.

For instance:

• At x = 1, the rate of change is 2(1) = 2.
• At x = 2, the rate of change is 2(2) = 4.
• At x = 3, the rate of change is 2(3) = 6.

The rate of change is increasing as x increases. This is reflected in the increasing steepness of the parabola as x moves further from the vertex.

3. Comparing Linear and Quadratic Rates of Change

The key difference lies in the consistency of the rate of change:

Understanding these differences is crucial for analyzing real-world phenomena. Linear functions model situations with constant growth or decay (e.g., simple interest), while quadratic functions describe situations with accelerating or decelerating change (e.g., projectile motion). By recognizing the type of function involved, we can accurately interpret and predict the rate of change.