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) = x^{2} - 3x + 2 | ||
3. h(x) = x(x + 1)(x - 2) = x^{3} - x^{2} - 2x |
Based on your entries in the table above,
Applying Your Knowledge
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) = x^{2} - 1 | ||
8. g(x) = x(x + 1)(x - 1) = x^{3} - x | ||
9. h(x) = x^{2} + 3x + 2 | ||
10. l(x) = x^{2} + 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.
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.