Analytic Geometry
Symmetry Tests and Curve Sketching
Four algebraic tests that reveal a curve's symmetry before any points are plotted, and how each one halves the work of sketching.
What this page covers
- Test a relation for symmetry about the y-axis, x-axis, origin and y = x
- Use a symmetry to halve a table of values
- Find intercepts and combine them with symmetry to sketch a curve
- Connect symmetry to even and odd functions
The four tests
Each test is a substitution. Make the substitution; if the equation is unchanged, the symmetry holds. The source lists all four in order.
| Symmetry | Substitute | Holds if | Geometric meaning |
|---|---|---|---|
| About the y-axis | x → -x | Equation unchanged | Mirror image left to right |
| About the x-axis | y → -y | Equation unchanged | Mirror image top to bottom |
| About the origin | x → -x and y → -y | Equation unchanged | Half-turn about the origin |
| About the line y = x | Swap x and y | Equation unchanged | Reflection in the diagonal |
The origin test is the one most often got wrong. Both substitutions must be made together; changing only x tests for y-axis symmetry instead. The source states it carefully: put -x in place of x, then y has to be replaced by -y.
The tests applied
y = 4 - x2 — symmetric about the y-axis
So the parabola is its own mirror image across the y-axis. The source draws the consequence immediately: we need half the table of values.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y = 4 - x2 | 4 | 3 | 0 | −5 |
The left half is read off by reflection: y = 3 at x = -1, y = 0 at x = -2, y = -5 at x = -3. The table in the source carries exactly these mirrored values.
x = 4 - y2 — symmetric about the x-axis
The same parabola turned on its side. It is not a function: at x = 0 there are two y values, 2 and -2.
y = x3 — symmetric about the origin
A half-turn about the origin maps the curve onto itself. The source's table shows the antisymmetry directly: (1, 1) and (-1, -1), (2, 8) and (-2, -8), (3, 27) and (-3, -27).
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | −27 | −8 | −1 | 0 | 1 | 8 | 27 |
The source's fourth case is symmetry about the line y = x, illustrated with a curve that reflects onto itself in the diagonal. The test is to swap the two variables.
Symmetry and functions
Two of the four tests correspond exactly to the even and odd classification of functions.
| Symmetry | Function condition | Name |
|---|---|---|
| About the y-axis | f(-x) = f(x) | Even |
| About the origin | f(-x) = -f(x) | Odd |
| About the x-axis | Not possible for a function | Fails the vertical line test |
| About y = x | The function is its own inverse | Involution |
The exclusion in the third row is worth understanding rather than memorising. x-axis symmetry means (x, y) on the curve implies (x, -y) is too, so one input has two outputs — which is precisely what a function forbids. The only escape is y = 0 throughout.
Intercepts
Symmetry tells you the shape; intercepts pin it to the plane. Together they are usually enough for a sketch.
y-intercept: set x = 0 and solve for y. x-intercepts: set y = 0 and solve for x.
Worked example — the source's line
For y = 3 - 2x:
Two points determine a line, so the sketch is complete with no table of values at all.
Worked example — the source's parabola
For y = 4 - x2: the y-intercept is (0, 4), and 4 - x2 = 0 gives x = ±2, so the x-intercepts are (-2, 0) and (2, 0).
Note that the two x-intercepts are symmetric about the y-axis, exactly as the symmetry test predicted. That agreement is a free check.
A sketching procedure
- Test all four symmetries. Each one that holds halves the work.
- Find the intercepts. They are the easiest points to compute and the most informative.
- Compute a small table, using symmetry to cover only half the range.
- Look for excluded regions. If y = √x - 5, nothing exists left of x = 5.
- Check the far-field behaviour. For a polynomial the leading term governs what happens for large |x|.
- Plot and join, reflecting across every symmetry found.
Step 4 is easy to skip and often the most informative. The source's x = 4 - y2 exists only for x ≤ 4, because y2 ≥ 0. Knowing that before plotting prevents a sketch that wanders where the curve cannot go.
Common mistakes
| Mistake | Correct | Why |
|---|---|---|
| Testing origin symmetry with x → -x only | Both substitutions | One alone tests the y-axis |
| (-x)2 = -x2 | = x2 | An even power kills the sign |
| (-x)3 = x3 | = -x3 | An odd power preserves it |
| Expecting x-axis symmetry in a function | Impossible except for y = 0 | Vertical line test |
| Plotting outside the domain | Check for excluded regions | x = 4 - y2 has no points right of x = 4 |
| Assuming every symmetric-looking graph is symmetric | Run the algebraic test | Sketches mislead |
Frequently asked questions
Why test for symmetry before plotting?
Because it halves the arithmetic and improves the sketch. Knowing y = 4 - x2 is symmetric about the y-axis means only the values for x ≥ 0 need computing; the rest are mirrored.
Can a graph have more than one symmetry?
Yes. A circle centred at the origin has all four. Any two of y-axis, x-axis and origin symmetry force the third.
Is x-axis symmetry possible for a function?
Only for the constant function y = 0. A non-trivial x-axis symmetry means some x maps to both y and -y, which fails the vertical line test.
How does symmetry about y = x relate to inverses?
A relation symmetric about y = x is its own inverse. In general, the graph of an inverse function is the reflection of the original in the line y = x.
Source. Handwritten teaching notes, Week 4, pages 8-10.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.