Quadratic and Polynomial Functions
Polynomial Functions and Their Graphs
Degree, leading coefficient and what they determine, the family of power functions y = xn, and how the special cases fit together.
What this page covers
- Identify the degree and coefficients of a polynomial function
- Name the special cases up to degree three
- Describe the family of curves y = xn
- Predict end behaviour from the leading term
The general form
Reading off the coefficients — the source's example
For y = 2x4 - 3x3 + x:
| Coefficient | a4 | a3 | a2 | a1 | a0 |
|---|---|---|---|---|---|
| Value | 2 | −3 | 0 | 1 | 0 |
Source, Week 6, page 5. The zeros matter: recording a2 = 0 and a0 = 0 explicitly is what makes long division and synthetic methods work, since both need a placeholder for every missing power.
This is a polynomial of degree 4, because 4 is the largest exponent with a non-zero coefficient.
The special cases
| Degree | Name | General form | Source example | Graph |
|---|---|---|---|---|
| 0 | Constant | y = a0 | y = 5 | Horizontal line |
| 1 | Linear | y = a1x + a0 | y = 2x - 7 | Straight line |
| 2 | Quadratic | y = a2x2 + a1x + a0 | y = 2x2 + 3x + 5 | Parabola |
| 3 | Cubic | y = a3x3 + a2x2 + a1x + a0 | y = -2x3 + 5x2 - 2x + 7 | One or two turning points |
Beyond degree three the names run out and the degree is quoted directly — quartic and quintic exist but are rarely used. What matters from there on is the degree itself.
The power functions y = xn
The source sketches the family y = x2, x4, x6 on one set of axes and y = x, x3, x5 on another. The two pictures are quite different, and the difference is entirely the parity of n.
| Feature | Even n | Odd n |
|---|---|---|
| Symmetry | About the y-axis (even function) | About the origin (odd function) |
| Sign of y | Never negative | Same sign as x |
| Shape near 0 | A flat-bottomed valley | A flattening then a rise |
| Both ends | Rise together | Go in opposite directions |
| Passes through | (0,0), (1,1), (-1,1) | (0,0), (1,1), (-1,-1) |
| Turning point at 0 | Yes, a minimum | No |
Within each family, larger n means flatter near the origin and steeper outside |x| = 1. Between -1 and 1, raising to a higher power makes the value smaller; outside, larger. All the curves cross at x = ±1, which is why the source's sketches show them fanning out from those points.
| x | x2 | x4 | x6 |
|---|---|---|---|
| 0.5 | 0.25 | 0.0625 | 0.0156 |
| 1 | 1 | 1 | 1 |
| 2 | 4 | 16 | 64 |
End behaviour
For large |x| the leading term dominates every other, because it grows fastest. So the two ends of any polynomial graph behave exactly like those of y = anxn.
| Degree | an | As x → -∞ | As x → +∞ |
|---|---|---|---|
| Even | Positive | y → +∞ | y → +∞ |
| Even | Negative | y → -∞ | y → -∞ |
| Odd | Positive | y → -∞ | y → +∞ |
| Odd | Negative | y → +∞ | y → -∞ |
The third row has an important consequence. A polynomial of odd degree runs from -∞ to +∞, so it must cross the x-axis at least once. Every odd-degree polynomial with real coefficients has a real root.
y = x2 + 1 never reaches the axis. Both ends rise, and the curve stays above it throughout.
Turning points and roots
| Degree n | Roots (with multiplicity, over ℂ) | Turning points, at most |
|---|---|---|
| 1 | 1 | 0 |
| 2 | 2 | 1 |
| 3 | 3 | 2 |
| 4 | 4 | 3 |
| n | n | n - 1 |
The root count is exact over the complex numbers; over the reals it is an upper bound, since complex roots come in conjugate pairs. The turning-point count is always an upper bound — y = x3 is a cubic with none at all.
Combining the tools already available gives a usable sketching procedure for a polynomial: find the roots using The Remainder and Factor Theorems, determine the sign between consecutive roots with a sign diagram, and fix the two ends from the leading term.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Omitting zero coefficients | Record every ai | Long division needs the placeholders |
| Reading the degree from the first term written | It is the largest exponent | Put it in standard form first |
| Expecting n turning points | At most n - 1 | y = x3 has none |
| Assuming every polynomial crosses the axis | Only odd degree must | x2 + 1 never does |
| Using a middle term to judge end behaviour | The leading term dominates | Compare growth rates |
| Thinking higher powers are always larger | Only outside |x| = 1 | 0.54 < 0.52 |
Frequently asked questions
What decides the overall shape?
The leading term. For large |x| it dominates every other term, so the far-field behaviour of y = 2x4 - 3x3 + x matches that of y = 2x4.
Why do even and odd powers behave differently?
An even power sends both large positive and large negative x to large positive y; an odd power preserves the sign, so the two ends go in opposite directions.
How many turning points can a polynomial have?
At most one fewer than its degree. A cubic has at most two, a quartic at most three. It may have fewer.
Why do all the curves y = xn pass through the same two points?
At x = 1 every power is 1, and at x = 0 every positive power is 0. So they all pass through (0, 0) and (1, 1), and the odd ones also through (-1, -1).
Source. Handwritten teaching notes, Week 6, pages 5-6.
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.