Advanced Reflection Function Calculator

Calculator Inputs

Use the responsive grid below. Large screens show three columns, smaller screens show two, and mobile shows one.

Coefficient a

Coefficient of x^4

Coefficient b

Coefficient of x^3

Coefficient c

Coefficient of x^2

Coefficient d

Coefficient of x

Coefficient e

Constant term

Reflection type

Choose the symmetry rule for the reflected function.

Line value

Used only for x = h or y = k reflections.

x minimum

Left graph boundary

x maximum

Right graph boundary

Graph points

Odd values produce a centered sample set.

Example data table

This sample reflects f(x) = x^2 - 4 across the x-axis.

x	Original f(x)	Reflected g(x)
-2	0	0
-1	-3	3
0	-4	4
1	-3	3
2	0	0

Formula used

This calculator models the original curve as a quartic polynomial:

f(x) = ax^4 + bx^3 + cx^2 + dx + e

Reflection rules:

Across the x-axis: g(x) = -f(x)
Across the y-axis: g(x) = f(-x)
Across the origin: g(x) = -f(-x)
Across x = h: g(x) = f(2h - x)
Across y = k: g(x) = 2k - f(x)

Expanded coefficient form used by the calculator:

For reflection across x = h: g(x) = ax^4 + (-8ah - b)x^3 + (24ah^2 + 6bh + c)x^2 + (-32ah^3 - 12bh^2 - 4ch - d)x + (16ah^4 + 8bh^3 + 4ch^2 + 2dh + e) For reflection across y = k: g(x) = -ax^4 - bx^3 - cx^2 - dx + (2k - e)

How to use this calculator

Enter the quartic coefficients a, b, c, d, and e.
Choose a reflection rule from the dropdown menu.
Enter h or k only when the selected rule needs it.
Set the graph domain and number of plotted points.
Press Calculate Reflection to view equations, graph, and sample values.
Download the generated results as CSV or PDF when needed.

Frequently asked questions

1) What does reflecting a function mean?

Reflection flips every point of a graph across a chosen line. The new curve keeps the same overall shape, but its coordinates change according to the selected symmetry rule.

2) What happens when I reflect across the x-axis?

Every y-value changes sign. Positive outputs become negative, negative outputs become positive, and zeros stay fixed. The rule is g(x) = -f(x).

3) What happens when I reflect across the y-axis?

The graph flips left to right. Each point at x moves to the opposite side at -x. Algebraically, the new function becomes g(x) = f(-x).

4) How does reflection across x = h work?

This mirrors the curve around a vertical line instead of an axis. The calculator uses g(x) = f(2h - x), which sends each point the same horizontal distance to the opposite side.

5) How does reflection across y = k work?

This mirrors the graph around a horizontal line. Every output is measured relative to y = k, giving the transformed rule g(x) = 2k - f(x).

6) Why do the reflected coefficients change?

Reflections alter how x and y appear in the equation. When the substitution is expanded, several coefficients can change sign or magnitude, especially for reflections across x = h.

7) Why should I adjust the x-range and point count?

A wider domain helps you see more of the curve. More points create a smoother graph and a denser CSV export, though very large counts can be slower.

8) Can this calculator reflect non-polynomial functions?

This version is built for quartic polynomial input. It is ideal for classroom checks, practice problems, and quick visual comparisons of reflected coefficient models.