Polynomial - Graphs

Graphs

A polynomial function in one real variable can be represented by a graph.

• The graph of the zero polynomial

f(x) = 0

is the x-axis.

• The graph of a degree 0 polynomial

f(x) = a₀, where a₀ ≠ 0,

is a horizontal line with y-intercept a₀

• The graph of a degree 1 polynomial (or linear function)

f(x) = a₀ + a₁x, where a₁ ≠ 0,

is an oblique line with y-intercept a₀ and slope a₁.

• The graph of a degree 2 polynomial

f(x) = a₀ + a₁x + a₂x2, where a₂ ≠ 0

is a parabola.

• The graph of a degree 3 polynomial

f(x) = a₀ + a₁x + a₂x2, + a₃x3, where a₃ ≠ 0

is a cubic curve.

• The graph of any polynomial with degree 2 or greater

f(x) = a₀ + a₁x + a₂x2 + ... + a_nxn, where a_n ≠ 0 and n ≥ 2

is a continuous non-linear curve.

The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value).

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

The illustrations below show graphs of polynomials.

• Polynomial of degree 2:
  f(x) = x2 - x - 2 = (x+1)(x-2)

• Polynomial of degree 3:
  f(x) = x3/4 + 3x2/4 - 3x/2 - 2 = 1/4 (x+4)(x+1)(x-2)

• Polynomial of degree 4:
  f(x) = 1/14 (x+4)(x+1)(x-1)(x-3) + 0.5

• Polynomial of degree 5:
  f(x) = 1/20 (x+4)(x+2)(x+1)(x-1)(x-3) + 2

• Polynomial of degree 6:
  f(x) = 1/30 (x+3.5)(x+2)(x+1)(x-1)(x-3)(x-4) + 2

• Polynomial of degree 7:
  f(x) = (x-3)(x-2)(x-1)(x)(x+1)(x+2)(x+3)