Where does the negative sign go when reflecting across the x-axis?

Conditions

• The reflection is across the x-axis.
• The negative sign applies to the whole function value.

Reasoning, step by step

1. Identify the reflection axis as the x-axis.
2. Apply the rule that x-axis reflection negates the output values.
3. Place the negative sign outside the function, in front of $f$.
4. Combine with any horizontal shifts, such as turning $f(x-5)$ into $-f(x-5)$.

Example

In part (B), the instructor explains that reflecting across the x-axis means putting a negative sign in front of the entire function, changing $f(x-5)$ to $-f(x-5)$. An orange sketch shows a curve above a horizontal line flipping to below it.

Common misconceptions

• Placing the negative sign inside the function argument, which would reflect across the y-axis instead.
• Forgetting to apply the negative sign to the entire function when combining it with horizontal shifts.

Watch the explanation

Explore next

• How is reflection across the y-axis written algebraically?
• What is the difference between -f(x-5) and $f(-x-5)$?