Question

Consider the Rate of Change (ROC) in linear and quadratic functions. Given a linear function f(x)=mx +b and a quadratic function g(x)=ax2+bx+c, which statement accurately describes the Rate of Change (ROC) for each type of function?

a) For linear functions, ROC is constant, while for quadratic functions, ROC varies with x.
b) For quadratic functions, ROC is constant, while for linear functions, ROC varies with x.
c) Both linear and quadratic functions exhibit constant ROC.
d) Both linear and quadratic functions exhibit varying ROC.

Answer 1

The Rate of Change in linear functions is constant, while in quadratic functions it varies with x. This is because the slope of a linear function is constant and for a quadratic function, the slope changes as x changes. The correct option is: (a) For linear functions, ROC is constant, while for quadratic functions, ROC varies with x.

Step-by-step explanation:

When considering the Rate of Change (ROC) of linear and quadratic functions, it's important to understand how ROC behaves for each function type.

Linear Functions

A linear function is defined as f(x) = mx + b, where m is the slope and b is the y-intercept. The ROC of a linear function is the slope (m), which is constant regardless of the value of x. This implies that for any two points on the graph of a linear function, the rate at which y changes with respect to x is the same.

Quadratic Functions

On the other hand, a quadratic function is expressed as g(x) = ax2 + bx + c. The ROC for quadratic functions is given by the derivative g'(x) = 2ax + b, which varies depending on the value of x. This shows that the ROC varies with x and is not constant. As x changes, the slope of the tangent to the curve at any point changes as well, meaning the rate at which y changes in relation to x is not constant.

Therefore, the correct statement that describes the Rate of Change for linear and quadratic functions is: a) For linear functions, ROC is constant, while for quadratic functions, ROC varies with x.