What is a polynomial function?
A polynomial function has the form
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
with whole-number exponents. The degree n and the leading coefficient aₙ shape the graph.
End behaviour
End behaviour describes what happens at the far left and far right.
| Degree | Leading coefficient | Left end | Right end | Like |
|---|---|---|---|---|
| even | + | up | up | y = x² |
| even | − | down | down | y = −x² |
| odd | + | down | up | y = x³ |
| odd | − | up | down | y = −x³ |
Turning points
A turning point is where the graph switches direction (a local max or min). A degree-n polynomial has at most n − 1 turning points.
- Quadratic: 1
- Cubic: up to 2
- Quartic: up to 3
Zeros from factored form
f(x) = (x + 2)(x − 1)(x − 4) has zeros −2, 1 and 4.
y-intercept: f(0) = (2)(−1)(−4) = 8.
Multiplicity
The multiplicity of a zero is how many times its factor appears.
| Multiplicity | Graph at the zero |
|---|---|
| 1 | crosses straight through |
| 2 | touches and turns back (bounces) |
| 3 | crosses, flattening like an S |
Example: f(x) = (x − 1)²(x + 3)
- x = 1: multiplicity 2 → bounces
- x = −3: multiplicity 1 → crosses
- Degree 3, positive → down on the left, up on the right
Sketching a polynomial
- Find the degree and leading coefficient → end behaviour.
- Find the zeros and their multiplicities.
- Find the y-intercept.
- Connect smoothly, crossing or bouncing at each zero.
Example: f(x) = −x(x − 2)(x + 3)
- Degree 3, leading coefficient −1 → up on the left, down on the right
- Zeros: 0, 2, −3 (all cross)
- Sketch: from top left, cross at −3, turn, cross at 0, turn, cross at 2, head down.
Even and odd symmetry
- Even functions: f(−x) = f(x); symmetric about the y-axis (e.g., x⁴ − 3x²).
- Odd functions: f(−x) = −f(x); symmetric about the origin (e.g., x³ − x).
Applications
Polynomials model volumes of boxes (V = x(20 − 2x)(30 − 2x)), roller coaster tracks, and data trends over limited ranges.
Worked examples
Example 1: End behaviour of f(x) = 2x⁵ − x? Odd, positive → down left, up right.
Example 2: Max turning points for a degree-4 polynomial? 3.
Example 3: Zeros and behaviour of f(x) = x²(x − 5)? 0 (bounce), 5 (cross).
Common mistakes
- Using the first term before writing in standard form to find the leading coefficient.
- Assuming exactly n − 1 turning points (it’s at most).
- Ignoring multiplicity.
Try it yourself
- End behaviour of f(x) = −3x⁴ + x?
- Zeros of f(x) = (x − 2)(x + 1)²?
- Behaviour at each zero in question 2?
- y-intercept of f(x) = (x + 1)(x − 3)(x − 2)?
Answers: 1) down on both ends 2) 2 and −1 3) crosses at 2; bounces at −1 4) 6
Quick recap
- The degree and leading coefficient give end behaviour.
- At most n − 1 turning points.
- Zeros and their multiplicity shape the graph.