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 any polynomial with degree 2 or greater
f(x) = a0 + a1x + a2x2 + . . . + anxn , where an ≠ 0 and n ≥ 2
is a continuous non-linear curve.

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) = x3/5 + 4x2/5 - 7x/5 - 2= 1/5 (x+5)(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