Why does translating right use x-5 instead of $x+5$?

Conditions

• The transformation is a horizontal translation of the graph of $y=f(x)$.
• The shift amount is a positive constant.

Reasoning, step by step

1. Recognize that horizontal shifts affect the input variable $x$ inside the function argument.
2. Understand that to move the graph right, the function must receive an input that is $h$ units smaller to produce the same output as the original function.
3. Apply the rule: right by $h$ units corresponds to replacing $x$ with $x-h$.
4. Note the instructor's emphasis that the sign inside the parentheses is opposite the verbal direction.

Example

The instructor explicitly stresses that the sign is opposite the intuitive direction: right corresponds to subtraction inside the parentheses, writing $f(x-5)$ for a 5-unit right shift.

Common misconceptions

• Assuming that a rightward shift on a number line corresponds to a plus sign inside the function argument.
• Confusing horizontal shifts with vertical shifts, where the sign matches the direction.

Watch the explanation

Explore next

• Why does moving left use plus inside the function argument?
• How do you write the equation for translating 5 units right and reflecting across the x-axis?