Skill: Identify reflections across the y-axis

A reflection across the $y$-axis changes the sign of every $x$-coordinate but keeps the $y$-coordinate the same:

$$(x,y)\longrightarrow(-x,y).$$

For a function, this means replace $x$ with $(-x)$:

$$y=f(x)\quad\longrightarrow\quad y=f(-x).$$

Example

Reflect the graph of

$$f(x)=\sqrt{x-2}+1$$

across the $y$-axis.

Step 1: Replace $x$ with $(-x)$.

$$f(-x)=\sqrt{(-x)-2}+1$$

So the reflected function is

$$\boxed{y=\sqrt{-x-2}+1}.$$

Step 2: Check the point movement.

The original function has an endpoint at $(2,1)$. Reflecting it across the $y$-axis changes only the $x$-coordinate:

$$(2,1)\longrightarrow(-2,1).$$

The reflected graph therefore begins at $(-2,1)$ and extends to the left.

Step 3: Identify the domain and range.

For the reflected function, the expression inside the square root must be nonnegative:

$$-x-2\ge 0,$$

so

$$x\le -2.$$

Thus, the domain is $(-\infty,-2]$. The $y$-values have not changed, so the range remains $[1,\infty)$.

Remember:

• $f(-x)$ reflects a graph across the $y$-axis.
• $-f(x)$ reflects a graph across the $x$-axis.

For this example, the answer is

$$\boxed{y=\sqrt{-x-2}+1}.$$

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