How is reflection across the y-axis written algebraically?

Conditions

• The reflection is across the y-axis.
• The negative sign is applied only to the input variable x.

Reasoning, step by step

1. Identify the reflection axis as the y-axis.
2. Apply the rule that y-axis reflection negates the input variable.
3. Place the negative sign inside the function argument, directly on $x$.
4. Combine with horizontal shifts, such as turning $f(x-5)$ into $f(-x-5)$.

Example

In part (C), the instructor says that when reflecting across the y-axis, you only make the x negative. The expression changes from $f(x-5)$ to $f(-x-5)$, and the final boxed answer is $y=f(-x-5)$.

Common misconceptions

• Placing the negative sign in front of the entire function, which would reflect across the x-axis instead.
• Applying the negative sign to the shifted constant as well, incorrectly writing $f(-(x-5))$ instead of $f(-x-5)$.

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Explore next

• Where does the negative sign go when reflecting across the x-axis?
• What is the difference between -f(x-5) and $f(-x-5)$?