Which statement describes a linear function?

Which statement describes a linear function?

• A. Its graph is a curve with changing slope
• B. Its graph is a line but the rate of change is not constant
• C. It is not a function
• D. A function whose graph is a line; rate of change is constant

Answer: (D)

A linear function has a constant rate of change, which shows up as a straight-line graph. This means as x changes by equal amounts, y changes by the same amount each time, so the slope stays the same everywhere. The statement that describes a function whose graph is a line and whose rate of change is constant captures both ideas: it is a function (each x has exactly one y) and its graph is a line with constant slope. Therefore this is the description of a linear function. A curve with changing slope isn’t linear, and a line with a nonconstant rate of change contradicts the idea of a straight line. A claim that it isn’t a function would conflict with the fact that a straight, non-vertical line gives a unique y value for every x.

A linear function has a constant rate of change, which shows up as a straight-line graph. This means as x changes by equal amounts, y changes by the same amount each time, so the slope stays the same everywhere. The statement that describes a function whose graph is a line and whose rate of change is constant captures both ideas: it is a function (each x has exactly one y) and its graph is a line with constant slope. Therefore this is the description of a linear function.

A curve with changing slope isn’t linear, and a line with a nonconstant rate of change contradicts the idea of a straight line. A claim that it isn’t a function would conflict with the fact that a straight, non-vertical line gives a unique y value for every x.