Graphs of polynomial functions

1. GRAPHS of polynomial functions

2. THE GRAPH OF POLYNOMIAL FUNCTIONS The graph of a polynomial function has the following characteristics SMOOTH CURVE - the turning points are not sharp CONTINUOUS CURVE – if you traced the graph with a pen, you would never have to lift the pen The DOMAIN is the set of real numbers The X – INTERCEPT is the abscissa of the point where the graph touches the x – axis. ABSOLUTE MAXIMUM/MINIMUM is the highest or lowest point (respectively) of the graph of a polynomial function. RELATIVE MAXIMUM/MINIMUM are the turning points of the graph of a polynomial function.

3. Let’s explore the graphs of the following functions

4. Let’s explore the graphs of the following functions

5. P(x) Degree (Odd/Even) Value of leading coefficient Rational zeros Number of x-intercepts Number of turning points 1 Odd a > 0 1 2 Odd a > 0 2, 4, 6 3 2 3 Odd a < 0 4 1 2 4 Even a > 0 1 1 5 Even a > 0 1, -1, 2, -2 4 3 6 Even a < 0 none 1

6. What conclusions can we draw from the table? How would you relate number of turning points with the degree of each function? What can be said about the number of zeros that each graph has and its relationship with the degree of its respective function? What seems to be true with the graph’s behavior and its degree? the value of its leading coefficient?

7. Maximum Number of Turning Points and X-intercepts A polynomial function of degree n has a maximum number of n-1 turning points at most n x-intercepts

8. BEHAVIOR/TREND OF THE GRAPHS Leading coefficient Degree (Odd/Even) Description of the Graph a > 0 Even Comes down from the left, goes up to the right a > 0 Odd Comes up from the left, goes up to the right a < 0 Even Comes up from the left, goes down to the right a < 0 Odd Comes down from the left, goes down to the right

9. Let’s Practice Describe the behavior of the following polynomial functions and identify the number of maximum zeros and turning points. 1. 2. 3.