Complete the Lesson 1 Assignment that you saved in your course folder at the beginning of the lesson. Show work to support your answers.
Save your responses in your course folder.
Go to Module 3 Project: Graphic Design Using Polynomials, and read over all project requirements to become familiar with what you will be doing and how you will be assessed.
Complete Part 1 of the Module 3 Project.
Save your responses in your course folder.
It should be noted that the end behaviour described in this lesson was generalized by looking at the graphs of many functions. This is not a mathematical proof. In later courses you will learn formal methods for proving these conclusions.
The following shows an informal proof of the end behaviour for f(x) = 2x4 + 3x3 − x2 + 5x.
Factoring out 2x4 results in the following equivalent function:
Now, think about what happens as x gets larger and larger:
As x gets larger, the fractions become smaller and smaller. A mathematician would say that as x approaches infinity, the fractions approach zero. So, as x approaches infinity, you can informally write
And this is exactly the end behaviour expected for an even function: as x approaches ∞, y also approaches ∞.
Similar reasoning can be used to show that as x approaches −∞, y approaches positive ∞—again, the end behaviour you expect for an even-degree function.