Students understand how to use the degree of the polynomial to identify the end behavior of a polynomial, which is how the outputs change as we look at input values farther and farther from 0. 

Students begin by noticing differences between the graphs of an even- and an odd-degree polynomial. Then they practice evaluating polynomials at specific inputs in two ways. First they evaluate polynomials of different degrees for different positive and negative inputs. They notice patterns in how the output values change as the input values get larger in the positive or negative directions depending on the degree of the polynomial (MP8). 

Next they evaluate a single polynomial at a variety of inputs to see how the leading term dictates end behavior even when other terms may have larger values at inputs nearer to 0 due to their coefficients.

Starting in this lesson, students will use language such as “larger and larger in the positive direction” and “larger and larger in the negative direction” when describing the end behavior of a function. Standard terminology and symbols like are not introduced in this course because they are statements about limits. For example, “As , ” is a shorthand way to express a complicated idea: for any real number , there is some value such that if , then .  Describing this idea with phrases like “as approaches infinity” can leave students with the mistaken impression that infinity is a number and is approaching it. The language used to describe end behavior in this course was chosen to be student-friendly while also making the point that when we talk about the end behavior of a polynomial function , we are considering how the value of the outputs are changing in relation to inputs of increasing magnitude. The word “greater” is avoided when describing end behavior in order to prevent any confusion on the part of the students with the language of inequalities. This level of understanding is appropriate for Math 3, and is one that can be built on without introducing possible misconceptions in later mathematics courses in which limits are a focus.

If students are familiar with the term “magnitude,“ use it during this lesson to help describe end behavior. For example, “as  gets larger and larger in magnitude in the negative direction.” As students gain confidence in thinking and talking about end behavior, extra words like “magnitude” can be phased out.