Which expression represents translating f(x) to the left by 3 units?

Which expression represents translating f(x) to the left by 3 units?

• A. g(x) = f(x + 3)
• B. g(x) = f(x) - 3
• C. g(x) = f(-x)
• D. g(x) = f(x - 3)

Answer: (D)

Translating a graph horizontally changes the input inside the function. A shift to the left by h units is achieved by replacing x with x + h. So for a left shift of 3 units, the expression is g(x) = f(x + 3). Why this is the right form: adding inside moves the graph left because you’re effectively evaluating the function at larger x-values to get the same outputs, which translates the graph leftward. For comparison, f(x) − 3 would move the graph down 3 units (a vertical shift), f(−x) would reflect the graph across the y-axis, and f(x − 3) would shift the graph to the right by 3 units.

Translating a graph horizontally changes the input inside the function. A shift to the left by h units is achieved by replacing x with x + h. So for a left shift of 3 units, the expression is g(x) = f(x + 3).

Why this is the right form: adding inside moves the graph left because you’re effectively evaluating the function at larger x-values to get the same outputs, which translates the graph leftward.

For comparison, f(x) − 3 would move the graph down 3 units (a vertical shift), f(−x) would reflect the graph across the y-axis, and f(x − 3) would shift the graph to the right by 3 units.