The document discusses curve sketching of polynomial functions. It explains that the appearance of a polynomial graph depends on whether the exponent is odd or even. It also notes that the maximum number of roots is equal to the degree of the polynomial function. The steps for sketching a polynomial graph are outlined as: 1) Find the y-intercept, 2) Find all roots, 3) Determine the sign over intervals defined by roots, and 4) Sketch the graph.

1. Curve Sketching
A treasured sketch !! :) by flickr user atinirdosh

2. Solve by factoring each polynomial completely.
3
(a) x + 5x 2 + 2x - 8 = 0

3. Solve by factoring each polynomial completely.
3
(a) x + 5x 2 + 2x - 8 = 0

4. Solve by factoring each polynomial completely.
(d) 2x3 + 7x 2 + x - 10 = 0

5. Graphing Polynomial Functions
ƒ(x) = x n
Appearance
Where n is odd, the
Where n is even, the
graph looks like this:
graph looks like this:

6. Graphing Polynomial Functions
Roots
The maximum number of roots for any polynomial function is
equal to the degree of the function.
Examples: Cubic Quartic Quintic
3
ƒ(x) = x 4 ƒ(x) = x 5
ƒ(x) = x
max. # of roots 3 4 5

7. Graphing Polynomial Functions
Sketching
Step 1: Find the y-intercept (let x = 0)
Step 2: Find all roots. (Use rational roots theorem if necessary.)
Step 3: Determine the sign of the function over the intervals
defined by the roots.
Step 4: Sketch the graph

8. Solve by factoring each polynomial completely.
3
(a) x + 5x 2 + 2x - 8 = 0