Reflection Across the Y-Axis: y = f(-x)

Reflecting a function across the \(y\)-axis means flipping its graph horizontally, as if the \(y\)-axis were a mirror. If the original function is \(y = f(x)\), the reflected function is written

\(y = f(-x)\)

To build this new function, every \(x\) in the original rule is replaced with \(-x\). The result is a brand new function whose graph is the mirror image of the original graph, using the \(y\)-axis as the mirror line.

The easiest way to picture this reflection is through coordinates. If a point \((a, b)\) lies on the graph of \(y = f(x)\), then the point \((-a, b)\) lies on the graph of \(y = f(-x)\). Only the \(x\)-coordinate changes sign; the \(y\)-coordinate stays exactly the same.

To find the equation of \(y = f(-x)\) from \(y = f(x)\), follow two steps:

1. Take the original function rule \(f(x)\).
2. Replace every \(x\) with \((-x)\), then simplify.

Be careful with signs: any term with an odd power of \(x\), or any standalone \(x\) term, will flip sign, while terms with even powers of \(x\) (like \(x^2\) or \(x^4\)) stay the same because a negative squared is positive.

Let \(f(x) = x^2 - 4x + 3\). Find \(f(-x)\) and compare the two graphs.

Substitute \(-x\) for \(x\):

\(f(-x) = (-x)^2 - 4(-x) + 3 = x^2 + 4x + 3\)

Notice the \(x^2\) term is unchanged (since squaring removes the negative sign), but the middle term flips from \(-4x\) to \(+4x\). The graph below shows the original parabola, with a vertex at \(x = 2\) and \(x\)-intercepts at \(x = 1\) and \(x = 3\).

After reflecting across the \(y\)-axis, the vertex moves to \(x = -2\) and the intercepts move to \(x = -1\) and \(x = -3\). Every feature lands on the exact opposite side of the \(y\)-axis, at the same height.

Let \(f(x) = \sqrt{x - 1}\). This function only exists for \(x \ge 1\), so its domain is \([1, \infty)\).

Substituting \(-x\) gives \(f(-x) = \sqrt{-x - 1}\). For this square root to be defined, \(-x - 1 \ge 0\), which means \(x \le -1\). The domain has been reflected too, becoming \((-\infty, -1]\).

This example is a good reminder that reflecting across the \(y\)-axis can also change the domain and range of a function, not just its shape, whenever the original domain is not symmetric about \(x = 0\).

A function is called even when \(f(-x) = f(x)\) for every value of \(x\) in its domain. Because reflecting across the \(y\)-axis produces exactly \(f(-x)\), an even function's graph looks completely unchanged after this reflection: it is already symmetric about the \(y\)-axis. Functions like \(f(x) = x^2\) or \(f(x) = \cos(x)\) are classic examples.

It is easy to mix up \(y = f(-x)\) with \(y = -f(x)\). Reflecting across the \(y\)-axis flips the input, changing where a point sits left-to-right. Reflection across the x axis instead flips the output, changing where a point sits up-and-down. Keeping track of whether the negative sign is inside or outside the function is the key to telling these two transformations apart.

Reflection across the \(y\)-axis is one of several ways a graph can be transformed from a parent function. It pairs naturally with ideas like horizontal translation, since both transformations act on the input \(x\) rather than the output. Once reflections and shifts feel comfortable, you can combine them to describe almost any transformed graph you encounter.

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