Graph of a Polynomial

A polynomial is a mathematical expression composed of several terms, where each term includes a constant, called a coefficient, and a variable raised to a non-negative integer power. The general form of a polynomial of degree 'n' in one variable 'x' is:

p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₂x² + a₁x + a₀

where aₙ is the leading coefficient and a₀ is the constant term.

PROPERTIES OF THE GRAPH OF A POLYNOMIAL

• ZEROS OF THE POLYNOMIAL (X-INTERCEPTS): Points where the graph of the polynomial intersects the x-axis. These correspond to the solutions of the equation p(x) = 0.
• END BEHAVIOR (BEHAVIOR TOWARDS INFINITY): Depends on the degree of the polynomial and the sign of the leading coefficient. For higher powers and a positive leading coefficient, the curve generally rises towards infinity on the right side (and its behavior on the left depends on whether the degree is even or odd).
• EXTREMA (LOCAL MAXIMA AND MINIMA): Points where the graph reaches a local minimum or maximum. These are determined using the first and second derivatives of the function.
• SYMMETRY: Some polynomials, such as even or odd functions, exhibit symmetry with respect to the y-axis or the origin.
• • An even function (p(-x) = p(x)) is symmetric about the y-axis.
  • An odd function (p(-x) = -p(x)) is symmetric about the origin.

DRAWING THE GRAPH OF A POLYNOMIAL

Drawing the graph of a polynomial requires knowledge of its key properties. The process includes:

1. FIND THE ZEROS: Solve the equation p(x) = 0 for x.
2. DETERMINE THE END BEHAVIOR: Check how the graph behaves for very large positive or very large negative values of x (as x → ∞ and x → -∞).
3. CALCULATE EXTREMA: Use derivatives to find points where the tangent line is horizontal (f'(x) = 0).
4. CHECK FOR SYMMETRY: Determine if the graph exhibits odd or even symmetry.
5. DRAW THE GRAPH: Use the obtained information (zeros, end behavior, extrema, y-intercept (p(0)), and any other calculated points) to sketch the graph.