Which form represents a linear function?

Which form represents a linear function?

• A. y = mx + b.
• B. y = x^2 + 1.
• C. y = sin x.
• D. y = e^x.

Answer: (A)

Linear functions have graphs that are straight lines and a constant rate of change. The form y = mx + b shows exactly that: m is the slope, the constant amount by which y changes for each unit increase in x, and b is the y-intercept, where the line crosses the y-axis. The x term is only to the first power, so the graph is a straight line for all x. The other expressions don’t produce a straight-line graph: x^2 + 1 curves into a parabola because x is squared; sin x creates a wavy, repeating pattern because it oscillates; e^x grows rapidly with x and has a changing slope, not a constant one. That’s why y = mx + b is the linear form.

Linear functions have graphs that are straight lines and a constant rate of change. The form y = mx + b shows exactly that: m is the slope, the constant amount by which y changes for each unit increase in x, and b is the y-intercept, where the line crosses the y-axis. The x term is only to the first power, so the graph is a straight line for all x.

The other expressions don’t produce a straight-line graph: x^2 + 1 curves into a parabola because x is squared; sin x creates a wavy, repeating pattern because it oscillates; e^x grows rapidly with x and has a changing slope, not a constant one. That’s why y = mx + b is the linear form.