What is the difference between -f(x-5) and $f(-x-5)$?

Conditions

• The base function is $y=f(x)$.
• Both expressions include a horizontal shift 5 units right.

Reasoning, step by step

1. Analyze $-f(x-5)$: the negative sign is in front of the entire function, indicating x-axis reflection.
2. Analyze $f(-x-5)$: the negative sign is only on $x$ inside the argument, indicating y-axis reflection.
3. Note that both contain $x-5$ or $-x-5$, which corresponds to a 5-unit rightward shift.
4. Conclude that the placement of the negative sign determines the axis of reflection.

Example

The instructor notes that the last two problems are almost exactly the same except for the reflection type. Part (B) ends as $-f(x-5)$ (x-axis reflection), while part (C) ends as $f(-x-5)$ (y-axis reflection).

Common misconceptions

• Assuming both expressions represent the same type of reflection.
• Confusing the effect of a negative sign outside the function with a negative sign inside the function argument.

Watch the explanation

Explore next

• Where does the negative sign go when reflecting across the x-axis?
• How is reflection across the y-axis written algebraically?