Reflections: Symmetry Through the x-axis, y-axis, and the Origin

Reflections as Function Transformations

A reflection flips a graph across a line (like a mirror). In algebra, reflections are created by changing the function notation in specific ways. The key idea is to track what happens to each point (x, y) on the original graph, where y = f(x).

We will connect each reflection to a coordinate mapping and to a function-notation rule:

• Over the x-axis: (x, y) → (x, -y) and y = -f(x)
• Over the y-axis: (x, y) → (-x, y) and y = f(-x)
• Through the origin: (x, y) → (-x, -y) and y = -f(-x)

1) Reflection Over the x-axis: -f(x)

What changes in the notation?

Writing -f(x) means: “take the original output and multiply it by -1.” The input x stays the same; only the output sign flips.

Coordinate mapping

If a point (x, y) is on y = f(x), then after reflecting over the x-axis it becomes (x, -y). So every point keeps its x-coordinate and flips its y-coordinate.

Step-by-step using a small table of points

Suppose f(x) = x + 2. Pick a few x-values and compute outputs.

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Notice the line keeps the same “left-to-right” placement, but it flips vertically.

Examples across families

• Quadratic: If f(x) = x^2, then -f(x) = -x^2. The parabola opens downward instead of upward (same width/shape, flipped vertically).
• Absolute value: If f(x) = |x|, then -f(x) = -|x|. The V-shape flips upside down.
• Reciprocal: If f(x) = 1/x, then -f(x) = -1/x. The branches move to the opposite vertical positions (Quadrants I & III become Quadrants II & IV), but the asymptotes stay the same lines.
• Radical: If f(x) = √x, then -f(x) = -√x. The curve reflects below the x-axis; the domain stays x ≥ 0, but outputs become nonpositive.

Common mistake to avoid

-f(x) is not the same as f(-x). In -f(x), the negative sign is outside the function, so it changes outputs.

2) Reflection Over the y-axis: f(-x)

What changes in the notation?

Writing f(-x) means: “replace the input x with -x everywhere inside the function.” This flips the graph left-to-right.

Coordinate mapping

If (x, y) is on y = f(x), then the reflected point over the y-axis is (-x, y). So y stays the same and x changes sign.

Step-by-step: substitute carefully (parentheses matter)

Example: f(x) = 2x - 5.

Compute f(-x) by replacing x with -x:

f(-x) = 2(-x) - 5 = -2x - 5

Notice only the term containing x changes sign here, because the constant -5 is not multiplied by x.

Examples across families (and what to watch for)

• Linear: If f(x) = 3x + 1, then f(-x) = -3x + 1. The y-intercept stays the same; the slope changes sign (mirror across the y-axis).
• Quadratic: If f(x) = x^2 + 4, then f(-x) = (-x)^2 + 4 = x^2 + 4. The graph does not change. This happens because squaring removes the sign; the function is symmetric about the y-axis.
• Absolute value: If f(x) = |x - 2|, then f(-x) = |-x - 2| = |-(x + 2)| = |x + 2|. The vertex moves from x = 2 to x = -2, showing the left-right flip.
• Reciprocal: If f(x) = 1/(x - 1), then f(-x) = 1/(-x - 1) = -1/(x + 1). The reflection over the y-axis can also introduce an overall negative factor after simplification; the key is that the input is negated first.
• Radical: If f(x) = √(x + 3), then f(-x) = √(-x + 3) = √(3 - x). The domain changes direction: 3 - x ≥ 0 so x ≤ 3, which matches a left-right reflection.

Common sign/parentheses mistakes

• Forgetting parentheses: If f(x) = x^2 - 6x, then f(-x) = (-x)^2 - 6(-x) = x^2 + 6x, not x^2 - 6x.
• Negating only one term: If f(x) = (x - 4)^2, then f(-x) = (-x - 4)^2, not (-x - 4^2) and not (x + 4)^2 unless you simplify correctly: (-x - 4)^2 = (x + 4)^2.

3) Reflection Through the Origin: -f(-x)

What changes in the notation?

-f(-x) combines both operations:

• f(-x) reflects over the y-axis (input sign flips).
• Putting a negative sign outside, - ( ... ), reflects over the x-axis (output sign flips).

Together, every point (x, y) becomes (-x, -y), which is a 180° rotation around the origin (often described as “reflection through the origin”).

Coordinate mapping

Start with a point (x, y) on y = f(x).

• After y-axis reflection: (x, y) → (-x, y)
• Then x-axis reflection: (-x, y) → (-x, -y)

So the combined mapping is (x, y) → (-x, -y).

Step-by-step example (showing the order clearly)

Let f(x) = x^3 - 2.

First compute f(-x):

f(-x) = (-x)^3 - 2 = -x^3 - 2

Now apply the outside negative:

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

This transformation changes both the horizontal and vertical orientation.

Examples across families (shape vs orientation)

• Linear: If f(x) = 2x + 3, then -f(-x) = -(2(-x) + 3) = -(-2x + 3) = 2x - 3. The slope ends up the same as the original, but the intercept changes sign.
• Quadratic: If f(x) = x^2 + 1, then -f(-x) = -( (-x)^2 + 1 ) = -(x^2 + 1) = -x^2 - 1. Because f(-x) = f(x) for this example, the origin reflection behaves like just reflecting over the x-axis.
• Reciprocal: If f(x) = 1/x, then -f(-x) = -(1/(-x)) = -(-1/x) = 1/x. The graph is unchanged. This happens because 1/x already has origin symmetry.
• Absolute value: If f(x) = |x|, then -f(-x) = -| -x | = -|x|. Since |x| is y-axis symmetric, the y-axis reflection does nothing, and the origin reflection reduces to an x-axis reflection.

How to Tell -f(x) and f(-x) Apart (Fast Checks)

Check 1: Which coordinate changes sign?

• -f(x): changes y-values, so (x, y) → (x, -y)
• f(-x): changes x-values, so (x, y) → (-x, y)

Check 2: Use a single point

If you know one point on the graph, you can reflect it without recomputing the whole function.

Example: Suppose (3, -5) is on y = f(x).

• On y = -f(x), the reflected point is (3, 5).
• On y = f(-x), the reflected point is (-3, -5).
• On y = -f(-x), the reflected point is (-3, 5).

Check 3: Parentheses rule for substitution

To compute f(-x), replace every x with (-x) first, then simplify.

f(x) = x^2 - 4x + 7  →  f(-x) = (-x)^2 - 4(-x) + 7 = x^2 + 4x + 7

Practice: Identify the Reflection and Write the New Function

A. Match each notation change to the reflection

• 1) y = -f(x)
• 2) y = f(-x)
• 3) y = -f(-x)

Choose from: (a) over x-axis, (b) over y-axis, (c) through origin.

B. Compute each transformed function (show substitution steps)

Let f(x) = 3x - 4.

• 1) -f(x)
• 2) f(-x)
• 3) -f(-x)

Let g(x) = (x + 2)^2.

• 4) -g(x)
• 5) g(-x)
• 6) -g(-x)

Let h(x) = 1/(x - 5).

• 7) -h(x)
• 8) h(-x)
• 9) -h(-x)

C. Spot the common mistake (fix the algebra)

Each line contains an error. Rewrite correctly.

• 1) If f(x) = x^2 + 6x, then f(-x) = -x^2 + 6x
• 2) If g(x) = (x - 3)^2, then g(-x) = (-x - 3^2)
• 3) If h(x) = √(x - 1), then h(-x) = √(-x) - 1

D. Decide whether the graph changes or stays the same

For each function, determine whether reflecting over the y-axis changes the graph (i.e., whether f(-x) equals f(x) after simplification).

• 1) f(x) = x^2 - 9
• 2) f(x) = x^3
• 3) f(x) = |x + 1|
• 4) f(x) = 1/x