Graphing Polynomial Functions

Graphing Polynomial Functions

How to graph polynomial functions by using its zeros: basic graphs, examples, and their solutions.

Graphs: Odd Power Functions

The graphs of the odd power functions (y = x, y = x^3, y = x^5, ...) pass through the x-axis at their zeros.

The graphs of odd power functions
(y = x, y = x3, y = x5, ...)
pass through the x-axis
at their zeros.

Graphs: Even Power Functions

The graphs of the even power functions (y = x^2, y = x^4, y = x^6, ...) bounce off the x-axis at their zeros.

The graphs of even power functions
(y = x2, y = x4, y = x6, ...)
bounce off the x-axis
at their zeros.

Example 1

By using the zeros of the given function, graph the function. y = x^3 - 3x^2 + 4

Use the synthetic division
to factor the right side of the function.

y = (x + 1)(x - 2)2

Factor theorem

The zeros are -1 and 2.

Mark the zeros on the x-axis: -1, 2.

The sign of the highest degree term is (+). (brown)
As x → ∞, y → ∞.
So the graph starts at quadrant I. (brown point)

Graph the function from the right to the left.

(x - 2)2 is an even power factor.
So at x = 2,
the graph bounces off the x-axis.
[↖ ↙]

(x + 1) is an odd power factor.
So at x = -1,
the graph passes through the x-axis.
[↙ ↙]

Example 2

By using the zeros of the given function, graph the function. y = x^4 - x^2

Factor the right side of the function.

y = x2(x + 1)(x - 1)

The zeros are 0, -1, and 1.

Factoring the difference of squares

Mark the zeros on the x-axis: -1, 0, 1.

The sign of the highest degree term is (+). (brown)
So the graph starts at the quadrant I. (brown point)

Graph the function from the right to the left.

(x - 1) is an odd power factor.
So at x = -1,
the graph passes through the x-axis.
[↙ ↙]

x2 is an even power factor.
So at x = 0,
the graph bounces off the x-axis.
[↙ ↖]

(x + 1) is an odd power factor.
So at x = -1,
the graph passes through the x-axis.
[↖ ↖]

Example 3

By using the zeros of the given function, graph the function. y = -x^3(x + 3)^2

The function is already factored.

And the zeros are 0 and -3.

Mark the zeros on the x-axis: -3, 0.

The sign of the highest degree term is (-). (brown)
As x → ∞, y → -∞.
So the graph starts at the quadrant IV. (brown point)

Graph the function from the right to the left.

x3 is an odd power factor.
So at x = 0,
the graph passes through the x-axis.
[↖ ↖]

(x + 3)2 is an even power factor.
So at x = -3,
the graph bounces off the x-axis.
[↖ ↙]