1. Enter points by typing, for example, “(3,0)” and ENTER in the Input: bar at the bottom. You should see this appear as "A=(3,0)" in the left column.[br]2. Type these additional points and ENTER after each one: (-1,0), (1,0), and (0,-3)[br]3. Type “Polynomial[A,B,C,D]” to see the interpolating polynomial for the points.
What is the equation of the resulting polynomial?
[math]-x^3+3x^2+x-3[/math]
Create a polynomial using the process in Task 1 that has the following characteristics:[br][list][*]negative y-intercept[/*][*]at least 3 x-intercepts[/*][*]increasing over the interval (0,4)[/*][/list]
What is the equation of the polynomial you created?
Move the blue points and observe how the equation changes. Then note your observations in the answer box below.
How are the zeros related to the equation of the polynomial? Answer in 1-2 sentences.
The zeros appear in each factor of the equation with an opposite sign.
What happens to the curve when [b]two [/b]blue dots overlap (are the same zero)? What happens to the corresponding factor in the equation? Answer in 2-3 sentences.
What happens to the curve when [b]three [/b]blue dots overlap (are the same zero)? What happens to the corresponding factor in the equation? Answer in 2-3 sentences.
Make an observation between the relationship of each factor, its exponent, and what that means on the graph of the polynomial. Explain your observation below.
Notice the y-intercept. How is the y-intercept affected as the zeros change? Answer in 1-2 sentences.
Change the number of zeros. Make an observation between the number of zeros and the end behavior of the graph, and explain it in 2-3 sentences below.