How can you make sure that all important aspects of a polynomial function are contained in the graph?
See solution.
Practice makes perfect
Recall that a polynomial function has a specific end behavior depending on its degree and leading coefficient. This can help us to determine if all important aspects of a polynomial function are contained in the graph. To get a better idea of this, let's first recall the possible end behaviors for an even degree polynomial function.
With this in mind, we will now analyze two different graphs for the quartic function f(x) = x^4 - x^3 - 9x^2 + 9x + 5.
Take a look at the graph shown above. This graph can be misleading as it gives the impression that f(x)→ + ∞ as x →+ ∞ and f(x)→ - ∞ as x → - ∞. Nevertheless, this is not possible for a even degree polynomial function. Let's try with a different window.
As we can see, we were missing a zero if we used the previous view. Furthermore, we can now see the expected end behavior. Therefore, to be sure that the window of the graph is the appropriate one, we need to be able to see the end behavior of the function.
