What is the graph of f(x) = |ax + b|?

• A. A horizontal line.
• B. A circle.
• C. A V-shaped graph.
• D. A straight line with positive slope.

Answer: (C)

Taking the absolute value of a linear expression makes any part of the line that would lie below the x-axis flip above it, so the graph becomes a V shape that sits entirely at or above the axis.

For f(x) = |a x + b|, the inner line ax + b crosses the x-axis where ax + b = 0, at x = -b/a (assuming a ≠ 0). On one side of that point the graph follows the line ax + b, and on the other side it follows its reflection, -ax - b. This creates two straight pieces with slopes a and -a meeting at the vertex (-b/a, 0). Since the absolute value never produces negative outputs, the graph opens upward like a V.

If a = 0, the inside is a constant b, so f(x) = |b|, which is a horizontal line. But when a is not zero, the graph is the characteristic V shape.