Final answer:
f(x) = 4x^2 is not a linear function because it is a quadratic function, which means its degree is 2, not 1 as required for linear functions. The correct answer to whether f(x) = 4x^2 is a linear function is a) The degree of the polynomial is 2.
Step-by-step explanation:
The student's question asks how we can determine if the function f(x) = 4x^2 is a linear function. The definition of a linear function is crucial to answering this question. A linear function is one that can be written in the form y = mx + b, where m represents the slope, and b denotes the y-intercept. In a linear function, the graph is a straight line, which reflects a constant rate of change.
Looking at the function f(x) = 4x^2, we see that the highest power of x is 2, which makes it a second-order polynomial, also known as a quadratic function. This differs from a linear function where the highest power of x should be 1. Quadratic functions have graphs that curve and are shaped like parabolas, not straight lines. Therefore, they do not have a constant rate of change and can not be described by a linear equation.
In terms of the options given in the student's question, the correct option that indicates why f(x) = 4x^2 is not a linear function is a) The degree of the polynomial is 2. Options concerning the rate of change and passing through the origin are not relevant in determining the linearity of this function since those aspects are secondary to the fact that the degree of the function dictates its overall shape.
The mention of correct option answer in the final answer is option a) The degree of the polynomial is 2.