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C. Polynomial and Rational Functions · Polynomial Functions and Equations

Even and Odd Polynomial Functions: Symmetry

C1.9 C1.3About 60 minutes
Learning goals
  • I can test algebraically whether a polynomial function is even, odd or neither by finding f(−x).
  • I can connect even functions with line symmetry in the y-axis and odd functions with point symmetry about the origin.
  • I can recognize even and odd polynomials from their exponents, and know that "even function" is not the same as "even degree".
  • I can use symmetry to find unknown values and complete graphs.

Words to know

even function
A function with f(−x) = f(x) for every x in its domain. Its graph is symmetric about the y-axis.
odd function
A function with f(−x) = −f(x) for every x in its domain. Its graph has point symmetry about the origin: it looks the same after a 180° turn.
line symmetry
A graph has line symmetry if folding along a line makes the two halves match.
point symmetry
A graph has point symmetry about a point if turning it 180° about that point leaves it unchanged.

Two kinds of balance

Some graphs are perfectly balanced. The graph of y = x² looks the same in a mirror placed along the y-axis. The graph of y = x³ looks the same if you turn the page upside down (a 180° rotation about the origin). Mathematicians call these functions even and odd, and the names come from the exponents: x², x⁴, x⁶ are even functions and x, x³, x⁵ are odd functions.

What matters is what happens when you replace x by −x:

  • Even: f(−x) = f(x). The points (x, y) and (−x, y) are both on the graph: line symmetry in the y-axis.
  • Odd: f(−x) = −f(x). The points (x, y) and (−x, −y) are both on the graph: point symmetry about the origin.
  • Most functions are neither.
xy−3−2−1123−3−2−112345(2, 1)(−2, 1)xy−3−113−4−3−2−11234(1, −1.5)(−1, 1.5)even: f(−x) = f(x)odd: f(−x) = −f(x)mirror image in the y-axissame after a half-turn about the origin
Left: f(x) = 0.5x⁴ − 2x² + 1 is even, so (2, 1) and (−2, 1) are both on the graph. Right: g(x) = 0.5x³ − 2x is odd, so (1, −1.5) and (−1, 1.5) are both on it, opposite each other through the origin.

The exponent shortcut for polynomials

Since (−x)ⁿ = xⁿ when n is even and (−x)ⁿ = −xⁿ when n is odd:

  • A polynomial is even exactly when every term has an even exponent. A constant term counts as even (it is c·x⁰).
  • A polynomial is odd exactly when every term has an odd exponent. So an odd polynomial has no constant term, and f(0) = 0: its graph passes through the origin.
  • If both kinds of exponents appear, the polynomial is neither.

Careful: an even function is not the same as a polynomial of even degree. f(x) = x⁴ + x has even degree but is neither even nor odd.

f(x) = 2x⁴ − 3x² + 1powers 4, 2, 0: all evenEVENg(x) = x³ − 4xpowers 3, 1: all oddODDh(x) = x³ + 1powers 3 and 0: mixedNEITHER
The exponent shortcut. Expand any factored form first, then look at every exponent, including the 0 on a constant term.
Worked example

Example 1: Testing with f(−x)

Decide whether each function is even, odd or neither.

(a) f(x) = 2x⁴ − 3x² + 1 (b) g(x) = x³ − 4x (c) h(x) = x³ + 1

  1. (a) f(−x) = 2(−x)⁴ − 3(−x)² + 1 = 2x⁴ − 3x² + 1 = f(x). Even.
  2. (b) g(−x) = (−x)³ − 4(−x) = −x³ + 4x = −(x³ − 4x) = −g(x). Odd.
  3. (c) h(−x) = −x³ + 1. Compare: h(x) = x³ + 1 (not equal) and −h(x) = −x³ − 1 (not equal). Neither. Shifting the odd function x³ up by 1 moves its centre of symmetry from the origin to (0, 1), so it is no longer odd.

(a) even (b) odd (c) neither

Worked example

Example 2: Factored forms

Decide whether each is even, odd or neither: (a) m(x) = x(x − 2)(x + 2) (b) k(x) = (x² − 1)(x + 2) (c) q(x) = (x − 3)²(x + 3)²

  1. (a) m(x) = x(x² − 4) = x³ − 4x. Exponents 3 and 1: odd. (Its zeros −2, 0, 2 are symmetric about 0, and it has the factor x.)
  2. (b) k(x) = x³ + 2x² − x − 2. Mixed exponents: neither.
  3. (c) q(x) = ((x − 3)(x + 3))² = (x² − 9)² = x⁴ − 18x² + 81. All even: even.

(a) odd (b) neither (c) even

Worked example

Example 3: Using symmetry

(a) f is an even polynomial function with f(3) = −7 and f(−1) = 4. Find f(−3) and f(1). (b) g is an odd polynomial function with g(2) = 5. Find g(−2) and g(0).

  1. (a) Even means f(−x) = f(x). So f(−3) = f(3) = −7 and f(1) = f(−1) = 4.
  2. (b) Odd means g(−x) = −g(x). So g(−2) = −g(2) = −5.
  3. For x = 0: g(−0) = −g(0), so g(0) = −g(0), which forces g(0) = 0.

f(−3) = −7, f(1) = 4, g(−2) = −5, g(0) = 0.

Why this is useful

Symmetry halves your work. If you know a function is even, you only need to graph x ≥ 0 and reflect. If it is odd, graph x ≥ 0 and rotate. In the unit Combining Functions and Modelling you will also predict the symmetry of sums and products of functions, and in calculus, symmetry is a shortcut for many calculations.

Just for fun

Why do even functions take such good selfies?

Their left side always looks exactly like their right side.

Hands-on activity

Fold and spin tests

You need: grid paper, a pencil, tracing paper (or thin printer paper held against a window) and a pin or a sharp pencil point (be careful with the pin).

  1. Make a table and plot f(x) = x⁴ − 4x² for x = −2.5 to 2.5 in steps of 0.5, and g(x) = x³ − 3x for the same x-values, on two separate grids.
  2. Fold test: fold each grid along the y-axis and hold it up to a light. Which graph matches itself?
  3. Spin test: trace each graph and its axes onto tracing paper. Put the pin through the origin and turn the tracing paper half a turn (180°). Which graph lands on itself?
  4. Now try h(x) = x³ − 3x + 2. Does it pass either test?

What to look for: f passes the fold test (even). g passes the spin test (odd). h passes neither; its centre of symmetry has moved to (0, 2).

Your turn

1

Is f(x) = x⁴ + 5x² even, odd or neither?

2

Is g(x) = −2x⁵ + x³ − 7x even, odd or neither?

3

Is h(x) = x⁴ + x even, odd or neither?

4

Show algebraically that p(x) = 3 − x² is even.

5

f is odd and f(2) = 5. Find f(−2).

6

f is even and f(−4) = 10. Find f(4).

7

Is q(x) = (x − 1)²(x + 1)² even?

More practice: warm-up

1

What kind of symmetry does the graph of an even function have?

2

What kind of symmetry does the graph of an odd function have?

3

Is y = x odd?

4

Is the constant function y = 5 even?

5

If f is an odd polynomial function, what is f(0)?

More practice: on your own

1

Is f(x) = x(x² − 9)(x² − 1) even, odd or neither?

2

Is f(x) = (x + 2)³ even, odd or neither?

3

An odd function has f(1) = 3, f(2) = −4 and f(3) = 0. Give f(−1), f(−2), f(−3) and f(0).

4

Explain why the graph of every odd polynomial function passes through the origin.

5

Find k so that f(x) = x⁴ + (k − 3)x³ + 2x² is an even function.

Challenge

1

Prove that the product of two odd functions is even.

2

Can a polynomial function be both even and odd? Explain.

3

Write f(x) = x³ + 2x² − 5x + 4 as the sum of an even function and an odd function, and show the even part equals (f(x) + f(−x))/2.

Check your understanding

Quiz

1.

Which function is odd?

2.

Which function is even?

3.

The graph of an odd function contains the point (3, −8). Which point must also be on it?

4.

Which statement is true?

5.

What is f(−x) for f(x) = 2x³ − x² + 5?

6.

f is an even function with f(−2) = 6. What is f(2)?

For parents and teachers

Big idea (C1.9): a function is even if f(−x) = f(x) (mirror symmetry in the y-axis) and odd if f(−x) = −f(x) (half-turn symmetry about the origin). For polynomials, check the exponents after expanding: all even → even; all odd → odd; mixed → neither.

How to help: Ask your teen to show you the fold test and the spin test on a graph they drew. Then give a quick one, such as x⁴ − x², and ask for the reason, not just the answer.

Common difficulties:

  • Thinking even degree means even function.
  • Forgetting that a constant term counts as an even power (so x³ + 1 is not odd).
  • Not expanding a factored form before using the exponent shortcut.
  • Writing (−x)² as −x².