The goal of this activity is for students to understand how to identify the end behavior of a polynomial using the degree. End behavior is defined as how the outputs change as we look at input values farther and farther from 0. A focus of the lesson is using the structure of the expressions to understand how the term with the highest exponent dictates end behavior even when other terms may have larger values at inputs nearer to zero due to coefficients (MP7). In the process of finding out how the leading term affects the shape of the graph, students also practice evaluating polynomials at specific inputs. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5). Once students have seen that the leading term determines the end behavior, they apply this knowledge to an equation in factored form by rewriting it in standard form. In this lesson, the coefficients of all leading terms are positive and students will investigate the effects of a negative leading coefficient in the next lesson.
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 \(x \rightarrow +\infty\) are not introduced in this course because they are statements about limits. For example, “As \(x \rightarrow +\infty\), \(f(x) \rightarrow +\infty\)” is a shorthand way to express a complicated idea: for any real number \(r\), there is some value \(x_0\) such that if \(x>x_0\), then \(f(x) > r\). Describing this idea with phrases like “as \(x\) approaches infinity” can leave students with the mistaken impression that infinity is a number and \(x\) 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 \(f\), we are considering how the value of the outputs \(f(x)\) are changing in relation to inputs \(x\) 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 Algebra 2, 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 \(x\) 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.
- Compare and contrast (orally) the end behavior of polynomials of even and odd degree.
- Generalize about end behavior based on specific functions and calculations.
- Identify end behavior of polynomial functions from equations.
- Let’s investigate the shape of polynomials.
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
- I understand why a function's end behavior is determined by its leading term.
How the outputs of a function change as we look at input values further and further from 0.
This function shows different end behavior in the positive and negative directions. In the positive direction the values get larger and larger. In the negative direction the values get closer and closer to -3.