Which expression represents translating the graph of f(x) downward by k units?

Which expression represents translating the graph of f(x) downward by k units?

• A. g(x) = f(x) + k
• B. g(x) = f(x) - k
• C. g(x) = f(x) * k
• D. g(x) = f(x + k)

Answer: (B)

Vertical translation by a constant is done by adjusting the output values of the function. Moving the graph downward by k units means every y-value decreases by k. So for any x, instead of the original point (x, f(x)), you get the point (x, f(x) − k). That makes the entire graph shift down by k. Keep in mind why other modifications don’t fit: adding k to f(x) would raise the graph, not lower it; multiplying f(x) by k would stretch or compress the graph vertically (and can flip if k is negative); and replacing x with x + k shifts the graph horizontally to the left by k, not vertically. Therefore, the expression that represents translating the graph downward by k units is g(x) = f(x) − k.

Vertical translation by a constant is done by adjusting the output values of the function. Moving the graph downward by k units means every y-value decreases by k. So for any x, instead of the original point (x, f(x)), you get the point (x, f(x) − k). That makes the entire graph shift down by k.

Keep in mind why other modifications don’t fit: adding k to f(x) would raise the graph, not lower it; multiplying f(x) by k would stretch or compress the graph vertically (and can flip if k is negative); and replacing x with x + k shifts the graph horizontally to the left by k, not vertically.

Therefore, the expression that represents translating the graph downward by k units is g(x) = f(x) − k.