Polynomial Curve Sketching

Warm up

(a) Determine the value of b.

When the polynomial 2x + bx - 5 is divided by x - 3, the remainder is 7.2

Warm up

(a) x + 5x + 2x - 8 = 03 2

Solve by factoring the polynomial completely.

Use your TI 83 to find the roots of the polynomial. x3 - 2x2 - 5x + 6

Answer x = -2, 1, 3

Degree n of a polynomial is odd

The function has opposite behaviour

If the leading coefficient is >0

The graph rises to the right and falls to the left

If the leading coefficient is <0

The graph rises to the left and falls to the right

x3+2x

2+1

-2x3+2x

2+1

When the degree n of a polynomial is even, then the graph has similar behaviour on the left as on the right

If the leading coefficient >0 the graph rises on the left and rises on the rightx

4 + x

3 - 2x

2 + x + 1

If the leading coefficient <0 the graph falls on the left and falls on the right

-x4 - 2x3 + 2x2 + x + 1

Graphing Polynomial Functions

Appearance

Where n is even, the graph looks like this:

ƒ(x) = xn

Where n is odd, the graph looks like this:

Graphing Polynomial Functions

The maximum number of roots for any polynomial function is equal to the degree of the function.

Roots

Examples:

max. # of roots 3 4 5

ƒ(x) = x ƒ(x) = x ƒ(x) = x 3 4 5

Cubic Quartic Quintic

Graphing Polynomial Functions

Sketching

Step 1: Find the y-intercept (let x = 0)

Step 4: Sketch the graph

Step 3: Determine the sign of the function over the intervals defined by the roots.

Step 2: Find all roots. (Use rational roots theorem if necessary.)

Factor the polynomial completely. Sketch the graph.ƒ(x) = x + 5x + 2x - 8 3 2

Sketch the graph of