### MATH100: Graphs of Polynomial Functions

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The purpose of this exercise is to explore how the equations of polynomial functions are related to the zeros (x-intercepts) of those functions and the shapes of their graphs. Please work in pairs; each partner should contribute to the final answers given.

Exploring Examples

In this exercise you will graph several functions and observe their zeros and the shapes of their graphs. For each of the following functions, graph the function, count the x-intercepts of its graph, and determine which inputs x produce the output 0.

 Function Number of x-intercepts Zeros (location) 1. f(x) = x - 3 2. g(x) = (x - 1)(x - 2) = x2 - 3x + 2 3. h(x) = x(x + 1)(x - 2) = x3 - x2 - 2x

Based on your entries in the table above,

1. What relationship do you observe between the equation of a polynomial function and the number of x-intercepts it has?

2. What relationship do you observe between the equation of a polynomial function and the location of its x-intercepts?

3. Graph the functions f(x) = x2 and g(x) = x2 + 1. How do these polynomial functions support the conjectures you made in the last two questions? How do they contradict your conjectures?

Use the observations you made in the previous section to predict how many times the graph of each function will cross the x-axis and what the zeros of each function will be. Check your work by graphing.

 Function Number of x-intercepts Zeros 7. f(x) = (x + 1)(x - 1) = x2 - 1 8. g(x) = x(x + 1)(x - 1) = x3 - x 9. h(x) = x2 + 3x + 2 10. l(x) = x2 + 4

For each description below, find the equation of a function that fits that description. (There may be more than one correct answer.) Check your work by graphing.

1. The graph of the function crosses the x-axis twice, at x = 1 and at x = 2.

2. The function is a degree three polynomial function and its graph crosses the x-axis at x = 0, x = 1 and x = 5.

3. The function is a degree three polynomial function but its graph crosses the x-axis only once.

4. The function is quadradic (degree 2) and its graph touches the x-axis only once.

5. The function is quadradic and has no zeros.

6. The graph of the function looks like the one shown on the right.

Further Exploration

Optional: Invent a polynomial function whose graph crosses the x-axis several times. Show your graph (but not your equation) to your partner. Try to guess your partner's equation while she or he tries to guess yours.
Bonus: (5 points) Do problem 72 on page 299 on a separate sheet of paper and turn it in.