Polynomial Function Visualization

Definition

A polynomial function is a type of mathematical function that involves only powers of the variable that are non-negative integers, combined using addition, subtraction, and multiplication.

A polynomial function has only non‑negative integer powers of x:

\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \] Where,

\(x\) is a variable
\(a_n, a_{n-1}, a{n-2} ..., a_1, a_0\) are real number coefficients,
\(n\) is non-negative integer, i.e, \((n = 0, 1, 2, 3, ...) \)
\(a_n \ne 0\) is the leading coefficient
\(n\) is the degree of the polynomial

Examples of the polynomial function

Polynomial	degree	description
\(f(x) = 9\)	0	Constant polynomial
\(g(x) = 7x + 3\)	1	Linear polynomial
\(h(x)= x^2 - 5x + 6\)	2	Quadratic polynomial
\(p(x) = 4x^3 + 2x^2 -x +7\)	3	Cubic polynomial

what is NOT a polynomial function

A function is not a polynomial if it includes:

Negative exponents:
Example: \[f(x) = \frac{1}{x}\]
Variables in the denominator
Examples:
1. \[f(x) = \frac{1}{x + 2} \]
2. \[g(x)= x^{-3} + 2x \]
Square roots or fractional exponents
Examples
1. \[f(x) = \sqrt(x)\]
2. \[g(x) = x^\frac{4}{3} \]
Absolute values
Example \[f(x) = |x|\]

How to Identify a Polynomial Function

To identify if a function is a polynomial, follow this checklist:
✅ Yes, it's a polynomial if:

The function has terms that are a product of real numbers and non-negative integer powers of the variable.
The exponents of the variable are whole numbers: 0, 1, 2, 3, ...
he variables do not appear in denominators or under radicals.

🚫 No, it's not a polynomial if:

There's a variable in the denominator or under a square root.
Any exponent of the variable is fractional or negative.
The function contains absolute values, trigonometric, logarithmic, or exponential functions.

Real-Life Application Example

Example: Cost function
A company models its cost to produce 𝑥 items as: \[C(x) = 5x^2 + 20x + 100\] Here,

\(x\): number of items,
\(C(x)\): total cost,
This is a polynomial of degree 2 (a quadratic polynomial),
The graph is a parabola opening upwards.

Graphical Characteristics of Polynomial Functions

The graph is continuous and smooth (no breaks or sharp corners).
The degree determines the shape and end behavior:
- Degree 1: Straight line
- Degree 2: Parabola
- Degree 3+: Curves with turning points
- and so on ...

Step-by-Step process for Graphing Process:

Example:
Let's graph \[f(x) = x^3 - 4x\] This is a cubic polynomial, so expect some curve changes (turning points).

Step 1: Identify Key Features
1. Degree:
  - egree is 3 → This is a cubic polynomial.
  - Graph has a “S” shape (one bend).
2. Leading Coefficient:
  - Coefficient of \(x^3\) is positive → Graph rises to the right and falls to the left.
Step 2: Find the x-intercepts (Roots)
set \(f(x) = 0\) \[x^3 - 4x = 0\] \[x(x^2 - 4) = 0\] \[x(x -2)(x + 2) = 0\] So the x-intercepts are: \[x = 0, \quad x = - 2, \quad x = 2\]
Step 3: Find the y-intercept let \( x = 0,\) \[f(0) = 0^3 - 4(0) = 0\] So the y-intercept is also \((0,0)\)

Step 4: Plot Additional Points Let’s pick a few values around the x-intercepts:

\(x\)	\(f(x) = x^3 - 4x\)
-3	- 27 + 12 = - 15
-1	-1 + 4 = 3
1	1 - 4 = - 3
3	27 - 12 = 15

Step 5: Sketch the Graph
- Plot the x-intercepts: (-2, 0), (0, 0), (2, 0)
- Plot additional points: (-3, -15), (-1, 3), (1, -3), (3, 15)
- Connect the points smoothly to form a continuous curve with an S-shape

Graph (Visual Idea)

Here's a rough sketch of the graph’s behavior:

Interactive Graph

Click “Next Plot” to generate a new random polynomial:

\( f(x) = x^3 - 4x \)