In short

y = -f(x) reflects the graph of f in the x-axis (flip top-to-bottom). y = f(-x) reflects it in the y-axis (flip left-to-right). y = |f(x)| takes every negative output and makes it positive — any part of the curve below the x-axis gets flipped above it. y = f(|x|) keeps the right half of the graph and mirrors it onto the left, creating a symmetric curve. y = [f(x)] maps every output to its greatest integer (floor), producing a step-like version of the curve.

You are standing at the edge of a still lake, looking at a temple on the far bank. The temple and its reflection in the water are identical in shape, but the reflection is upside down — every point above the waterline has a twin the same distance below it. That waterline is the x-axis. The transformation that produces the reflection is y = -f(x).

Now imagine a different kind of mirror — a tall vertical mirror placed along the y-axis. The right side of the temple produces a mirror image on the left. That is y = f(-x).

These two reflections, plus the absolute-value and greatest-integer transformations, complete your toolkit for reshaping graphs.

Reflection in the x-axis: y = -f(x)

Take f(x) = x^2 - 4, a parabola with vertex at (0, -4) and roots at x = \pm 2. What does y = -f(x) = -(x^2 - 4) = -x^2 + 4 look like?

Every output is negated. A point that was at y = -4 (the vertex) moves to y = 4. A point that was at y = 0 (the roots) stays at y = 0. A point that was at y = 5 moves to y = -5. The curve flips upside down, with the x-axis as the mirror line.

Reflection of a parabola in the x-axisTwo parabolas. The original y equals x squared minus 4 opens upward with vertex at (0, negative 4). The reflected curve y equals negative x squared plus 4 opens downward with vertex at (0, 4). They intersect at x equals plus or minus 2 on the x-axis. x y 1 2 3 −1 −2 −3 1 2 −1 −2 4 −4 x² − 4 −x² + 4
The dashed parabola $y = x^2 - 4$ opens upward with vertex at $(0, -4)$. Its reflection $y = -(x^2 - 4) = -x^2 + 4$ (red) opens downward with vertex at $(0, 4)$. They meet at the roots $(\pm 2, 0)$, which sit on the mirror line (the $x$-axis).

Reflection in the $x$-axis

The graph of y = -f(x) is the graph of y = f(x) reflected in the x-axis. Every point (a, b) maps to (a, -b). Points on the x-axis (where f(x) = 0) remain fixed.

This follows the outside-rule pattern from translations and scaling: the negation is applied to the output (y-side), so it acts directly on the y-coordinates. Positive outputs become negative, and negative outputs become positive.

Reflection in the y-axis: y = f(-x)

Take f(x) = \sqrt{x}, defined for x \ge 0, with the curve rising to the right. What does y = f(-x) = \sqrt{-x} look like?

This function is defined when -x \ge 0, that is, x \le 0. At x = -1: \sqrt{-(-1)} = \sqrt{1} = 1. At x = -4: \sqrt{-(-4)} = \sqrt{4} = 2. The curve is a mirror image of \sqrt{x}, extending to the left instead of the right.

Reflection of square root in the y-axisTwo curves. The dashed y equals root x extends to the right from the origin. The red y equals root of negative x extends to the left from the origin, a mirror image across the y-axis. x y 1 2 3 4 −1 −2 −3 −4 1 2 3 √x √(−x)
The dashed curve $y = \sqrt{x}$ extends to the right. Its $y$-axis reflection $y = \sqrt{-x}$ (red) extends to the left. The origin is the only point in common — it sits on the mirror line.

Reflection in the $y$-axis

The graph of y = f(-x) is the graph of y = f(x) reflected in the y-axis. Every point (a, b) maps to (-a, b). Points on the y-axis (where x = 0) remain fixed.

This follows the inside-rule: the negation is applied to the input (x-side), so it acts in the opposite direction on the x-coordinates. Left becomes right, and right becomes left.

Connection to even and odd functions

A function is even if f(-x) = f(x) for all x — its graph is symmetric about the y-axis. Reflecting it changes nothing. Examples: x^2, \cos x, |x|.

A function is odd if f(-x) = -f(x) for all x — reflecting in the y-axis gives the same result as reflecting in the x-axis. The graph has rotational symmetry about the origin. Examples: x^3, \sin x, 1/x.

The absolute value of a function: y = |f(x)|

This transformation keeps all the positive parts of the graph untouched and flips the negative parts above the x-axis. Any portion of the curve that dips below the x-axis is reflected upward — like folding a sheet of paper along the x-axis.

Take f(x) = x^2 - 4 again. Below the x-axis (between x = -2 and x = 2), f is negative. The absolute value flips that portion up:

|f(x)| = |x^2 - 4| = \begin{cases} x^2 - 4 & \text{if } x \le -2 \text{ or } x \ge 2 \\ -(x^2 - 4) = 4 - x^2 & \text{if } -2 < x < 2 \end{cases}
Graph of absolute value of x squared minus 4Two graphs. The dashed y equals x squared minus 4 dips below the x-axis between x equals negative 2 and x equals 2, reaching a minimum of negative 4. The red graph y equals absolute value of (x squared minus 4) is identical outside that interval but has the dip flipped upward to become a hump reaching y equals 4. x y 1 2 3 −1 −2 −3 1 2 4 −1 |x² − 4| x² − 4
The dashed curve is $y = x^2 - 4$. The portion below the $x$-axis (between $x = -2$ and $x = 2$) gets reflected upward in $y = |x^2 - 4|$ (red). The $x$-intercepts at $\pm 2$ become sharp corners — the slope changes abruptly where $f(x) = 0$.

Key observations about y = |f(x)|:

Absolute value of the input: y = f(|x|)

This one is different. Instead of taking the absolute value of the output, you take the absolute value of the input before feeding it to f.

Since |x| = x when x \ge 0 and |x| = -x when x < 0, the function f(|x|) uses the right-half graph of f for x \ge 0 and reflects it onto the left half. The left side of the original is discarded.

Take f(x) = x^2 - 2x, a parabola with vertex at (1, -1) and roots at x = 0 and x = 2. The graph of y = f(|x|) = |x|^2 - 2|x| = x^2 - 2|x|:

For x \ge 0: this is just x^2 - 2x — unchanged. For x < 0: f(|x|) = f(-x) = (-x)^2 - 2(-x) = x^2 + 2x, which is the mirror image of the right half.

Graph of f of absolute x for f(x) equals x squared minus 2xTwo graphs. The dashed original parabola y equals x squared minus 2x has its vertex at (1, negative 1) and is not symmetric about the y-axis. The red graph y equals x squared minus 2 absolute x is symmetric about the y-axis, with the right half identical to the original and the left half being a mirror image. x y 1 2 3 −1 −2 −3 1 2 −1 (1, −1) (−1, −1) f(|x|) f(x)
The dashed curve is $y = x^2 - 2x$ with vertex at $(1, -1)$. The red curve $y = f(|x|)$ keeps the right half and mirrors it to the left, creating a symmetric W-shape with vertices at $(1, -1)$ and $(-1, -1)$.

The result of y = f(|x|) is always symmetric about the y-axis. This is guaranteed because f(|-x|) = f(|x|), so the function values at x and -x are always equal.

The greatest integer function: y = [f(x)]

The greatest integer function [t] (also written \lfloor t \rfloor) returns the largest integer less than or equal to t. For example: [3.7] = 3, [5] = 5, [-1.2] = -2, [-3] = -3.

Applying this to a function f: y = [f(x)] takes each output value and rounds it down to the nearest integer. The smooth curve becomes a series of horizontal steps.

Take f(x) = x, the identity line. The graph of y = [x] is a staircase: on the interval [0, 1), the value is 0. On [1, 2), the value is 1. On [-1, 0), the value is -1. Each step is a half-open interval — closed on the left, open on the right.

Graph of the greatest integer functionA staircase graph of y equals floor of x. Each step is a horizontal segment one unit long. Closed dots mark the left endpoint of each step (included), open circles mark the right endpoint (excluded). The dashed line y equals x is shown for comparison. x y 1 2 3 −1 −2 −3 1 2 −1 −2 [x] y = x
The greatest integer function $y = [x]$ (red staircase). Each step is one unit wide. Filled dots indicate included endpoints; open circles indicate excluded endpoints. The dashed line $y = x$ is shown for comparison — the staircase always sits at or below it.

For a general function, y = [f(x)] produces a stepped version of the curve. Wherever f(x) is between two consecutive integers n and n+1, the output of [f(x)] is n. The steps jump at every point where f(x) is an integer.

Two worked examples

Example 1: Combining reflection with absolute value

Sketch the graph of y = |2 - x^2| and find its range.

Step 1. Rewrite the function. Note that 2 - x^2 = -(x^2 - 2). So |2 - x^2| = |-(x^2 - 2)| = |x^2 - 2|. The absolute value absorbs the leading negative sign.

Why: |{-t}| = |t| for all t. This simplifies the analysis — you only need to study g(x) = x^2 - 2 and then take its absolute value.

Step 2. Analyse g(x) = x^2 - 2. This is a parabola with vertex at (0, -2) and roots at x = \pm\sqrt{2} \approx \pm 1.41. It is negative between -\sqrt{2} and \sqrt{2}, and positive outside that interval.

Why: finding where g is negative tells you exactly which part of the curve gets flipped by the absolute value.

Step 3. Apply the absolute value. For |x| \ge \sqrt{2}: g(x) \ge 0, so |g(x)| = g(x) = x^2 - 2. Unchanged. For |x| < \sqrt{2}: g(x) < 0, so |g(x)| = -(x^2 - 2) = 2 - x^2. This flips the valley into a hill.

Why: the negative portion (the valley between the roots) is reflected upward. The vertex at (0, -2) becomes the peak at (0, 2).

Step 4. Find the range. The function is always \ge 0 (absolute value). The minimum value is 0, occurring at x = \pm\sqrt{2} (the roots of g). The function grows without bound as |x| \to \infty. The range is [0, \infty).

Result: y = |2 - x^2| has a W-shaped graph. It touches the x-axis at x = \pm\sqrt{2}, has a local maximum of 2 at x = 0, and its range is [0, \infty).

Graph of absolute value of 2 minus x squaredA W-shaped graph. The dashed parabola y equals x squared minus 2 dips to negative 2 at the origin. The red curve y equals absolute value of x squared minus 2 flips the negative portion upward, creating a W shape that touches the x-axis at plus and minus root 2 and peaks at (0, 2). x y 1 2 −1 −2 1 2 −2 −√2 √2 (0, 2) x² − 2 |x² − 2|
The dashed parabola $y = x^2 - 2$ has its minimum at $(0, -2)$. Taking the absolute value flips the valley upward, producing the W-shaped red curve $y = |x^2 - 2|$. The roots at $\pm\sqrt{2}$ become the sharp corners where the curve touches the $x$-axis.

The sharp corners at x = \pm\sqrt{2} are characteristic of absolute-value graphs. The slope changes sign abruptly because the curve transitions from the original to its reflection.

Example 2: Reflecting and translating together

The graph of y = f(x) is known. Sketch y = -f(x - 1) + 2 and describe each transformation applied to the point (3, 5) on the original graph.

Step 1. Identify the transformations. Reading inside-out: replace x with x - 1 (shift right 1), then negate the output (reflect in x-axis), then add 2 (shift up 2).

Why: reading from the innermost modification outward ensures each transformation is applied in the correct order.

Step 2. Track the point (3, 5) through each step. Start: (3, 5).

After shifting right by 1: the point that produces f(3) is now at x = 4. So the point becomes (4, 5).

Why: f((4) - 1) = f(3) = 5. The point has moved right by 1.

Step 3. Apply reflection in the x-axis. The y-coordinate is negated: (4, 5) \to (4, -5).

Why: the negative sign in front of f flips 5 to -5.

Step 4. Apply vertical translation up by 2. The y-coordinate increases by 2: (4, -5) \to (4, -3).

Why: adding 2 to -5 gives -3. The final point is (4, -3).

Result: The point (3, 5) on y = f(x) maps to (4, -3) on y = -f(x-1) + 2. The combined transformation shifts right 1, reflects in the x-axis, and shifts up 2.

Transformation of a curve by reflection and translationTwo curves shown. The dashed original curve passes through (3, 5). The red transformed curve y equals negative f(x minus 1) plus 2 passes through (4, negative 3). Arrows show the mapping from (3, 5) to (4, negative 3). x y 1 2 3 4 5 1 2 −1 −2 (3, 5) (4, −3) f(x) −f(x−1)+2
The dashed curve is $y = f(x)$; the red curve is $y = -f(x-1) + 2$. The point $(3, 5)$ maps to $(4, -3)$: shifted right 1, reflected, then shifted up 2. The overall shape is an upside-down, shifted version of the original.

The combination of reflection and translation is common in exam questions. The key is to apply the transformations in order, tracking one or two specific points to verify your work.

Common confusions

Going deeper

If you can sketch y = -f(x), y = f(-x), y = |f(x)|, and y = f(|x|) from a given graph of f, and combine them with translations and scaling, you are well-equipped for transformation questions at any level. The rest of this section covers the formal details.

Composition of reflections

Reflecting in the x-axis and then in the y-axis (or vice versa) produces a rotation by 180° about the origin:

y = -f(-x)

This maps every point (a, b) to (-a, -b). The graph looks like the original rotated half a turn. For odd functions, this rotation brings the graph back to itself — which is why odd functions have rotational symmetry about the origin.

Interactive: reflection explorer

Drag the toggle below to switch between y = f(x), y = -f(x), y = f(-x), and y = |f(x)|.

Interactive reflection explorerAn interactive graph where a point can be dragged along a cubic curve. The readout shows the coordinates and the reflected coordinates in real time. x y 1 2 −1 −2 drag the point
Drag the red point along the curve. The readout shows the coordinates — mentally negate the $y$-value for the $x$-axis reflection, or negate the $x$-value for the $y$-axis reflection.

The fractional part function

The fractional part of x, written \{x\} = x - [x], measures how far x is above its floor. For x = 3.7: \{3.7\} = 3.7 - 3 = 0.7. For x = -1.2: \{-1.2\} = -1.2 - (-2) = 0.8.

The graph of y = \{x\} is a sawtooth wave: it rises linearly from 0 to 1 on each interval [n, n+1), then drops back to 0. The range is always [0, 1).

This connects to the greatest integer function: x = [x] + \{x\}, so the floor and the fractional part together reconstruct the original number.

Where this leads next

You now have the complete set of graph transformations: translations, scaling, and reflections. Combined, they let you sketch any function of the form y = A\,f(B(x - h)) + k from the parent graph.