Identify reflections across the y-axis

A reflection across the $y$ -axis maps each point $(x,y)$ to $(-x,y)$ : horizontal position reverses while vertical position remains unchanged, so the reflected graph of $y=f(x)$ is $y=f(-x)$ , with the domain reflected and the range preserved. This distinguishes $f(-x)$ , a $y$ -axis reflection, from $-f(x)$ , an $x$ -axis reflection, and supports interpreting symmetry and combining function transformations.

Detailed Explanation: 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,

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}.

Learn by doing: Identify reflections across the y-axis

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Cartesian Grid - Reflection of Point (Coordinates to Coordinates) across Axis

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