1. 4.2 Properties of Polynomial Graphs
For a polynomial function of degree n, the following apply:
The domain is the set of all real numbers.
The graph has at most n-1 turning points.
The graph has at most n x-intercepts.

2. Graph the following in your graphing calculator and determine
Degree and leading coefficient
Number of turning points,
Number of x-intercepts
Number of relative maximum and minimums, Number of absolute maximum and minimums.
End behavior

3. Graph the following in your graphing calculator and determine
Degree and leading coefficient
Number of turning points,
Number of x-intercepts
Number of relative maximum and minimums, Number of absolute maximum and minimums.
End behavior

4. Given the graph, determine the number of turning points,
Estimate the x-intercepts
Positive or negative leading coefficient of the corresponding function
Graph of a cubic or quartic function.
Equation of function:

5. Sketching the Graph of a Polynomial
Find and plot the y-intercept
Determine the end behavior according to the degree of the polynomial and the sign of the leading coefficient.
Find all the real zeros and their multiplicities.
Find and locate a test point between each real zero and connect the points with a smooth curve.
(Note: Without applying calculus techniques, we will only approximate the turning points.)

6. Sketch the graph of

7. Find a polynomial function that could be represented by the graph.

8. Write the equation of the function in factored form & polynomial form.
Use the graph of f to factor f(x) = 2x3 – 4x2 – 10x + 12.

9. The Intermediate Value Theorem
For any polynomial function P(x) with real coefficients, suppose that for a ≠ b, P(a) and P(b) are of opposite signs. Then the function has at least one real zero between a and b. The intermediate value theorem cannot be used to determine whether there is a real zero between a and b when P(a) and P(b) have the same sign.

10. Using the intermediate value theorem, determine, if possible, whether the function has at least one real zero between a and b.

11. Example Sketch the graph of f (x) = 2x3 + x2  8x  4. y-intercept
End behavior
Real zeros and their multiplicities.
Test point between each real zero and connect the points with a smooth curve.