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Lab06 Polynomial Function Graphs; Long and Synthetic Division

1) End Behavior of Polynomials

In the following applet, investigate the degree and leading coefficient of each polynomial and compare it to the graphs end behavior.

2) Zeros of a Polynomial Function

Use the slides to investigate the behavior of a zero based on the degree of the factor.

3) Intermediate Value Theorem

The Intermediate Value Theorem is an Existence Theorem. it can either confirm the existence of a value or it can not confirm. We will use this theorem to narrow down zeros of our polynomial functions. Verify the first table entry in the Lab06. Move the sliders along the axis so that N = 0, a = 1 and b = 3. Complete the table in the lab.

4) Quadratic Modeling

On pg 115 of the textbook, the path of the baseball can be modeled using the function . We have already determined the distance and maximum height of the ball (vertex) algebraically but this can also be performed on GeoGebra. In general, when you know how to determine something 100% algebraically, you can then explore using technology. Determine the x-intercepts algebraically (quadratic formula) and fill in the lab. Enter the function for the path of the ball in the applet below. A synonymous term for x-intercept is root. We can confirm our Quadratic Formula by typing the command root(f(x))  typing "f(x)" and not the entire function.  Watch for the two roots to be listed as points A and B.  *If your algebra does not match up, fix your work before submitting the snapshot.  The roots give us the practical domain [A,B] on which our function takes on practical range values ( positive values [0, MAXIMUM] for the height of a ball).  Use the x-intercepts as left and right bounds to find the maximum function value by typing the command: max(f(x), the x-value from point A,the x-value from point B). This should show up as point "C".

5) Use the GeoGebra Applet Above

Reset the applet above and use the applet to complete pg 118 # 79 (The number of fixtures produced to yield a minimum cost). This function has no roots so we need to create a really wide range for the minimum. Type the min command and use the x-values 0 and 1000000. No algebraic work is necessary, all work should be done on GeoGebra.