End Behavior Part 1

Today, we started exploring the fascinating concept of end behavior in polynomial functions. End behavior describes what happens to the function's value, $f(x)$, as $x$ approaches positive infinity ($x \rightarrow \infty$) and negative infinity ($x \rightarrow -\infty$). In other words, we're looking at what the graph does way out on the far right and far left!

Key Concepts

  • Polynomial Functions: Remember that a polynomial function is a continuous function that can be described by a polynomial equation in one variable. A general form is: $$f(x) = a_n x^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0$$ where $n$ is a non-negative integer and $a_n, a_{n-1}, ..., a_0$ are real numbers with $a_n \neq 0$.
  • Degree: The degree of a polynomial is the highest power of $x$ in the function (the value of $n$ in the general form above). The degree plays a crucial role in determining end behavior.
  • Leading Coefficient: The leading coefficient is the coefficient of the term with the highest power of $x$ (the value of $a_n$ in the general form above). The sign of the leading coefficient also influences the end behavior.

How to Determine End Behavior

To determine the end behavior of a polynomial function, consider the following:

  1. The Degree of the Polynomial: Is it even or odd?
  2. The Sign of the Leading Coefficient: Is it positive or negative?

Here's a quick summary:

  • Even Degree:
    • Positive Leading Coefficient: Both ends go up ($x \rightarrow -\infty, f(x) \rightarrow \infty$ and $x \rightarrow \infty, f(x) \rightarrow \infty$)
    • Negative Leading Coefficient: Both ends go down ($x \rightarrow -\infty, f(x) \rightarrow -\infty$ and $x \rightarrow \infty, f(x) \rightarrow -\infty$)
  • Odd Degree:
    • Positive Leading Coefficient: Left end goes down, right end goes up ($x \rightarrow -\infty, f(x) \rightarrow -\infty$ and $x \rightarrow \infty, f(x) \rightarrow \infty$)
    • Negative Leading Coefficient: Left end goes up, right end goes down ($x \rightarrow -\infty, f(x) \rightarrow \infty$ and $x \rightarrow \infty, f(x) \rightarrow -\infty$)

For example, consider the function $f(x) = x^3 + 3x^2 - 1$. The degree is 3 (odd) and the leading coefficient is 1 (positive). Therefore, as $x$ approaches negative infinity, $f(x)$ approaches negative infinity, and as $x$ approaches positive infinity, $f(x)$ approaches positive infinity.

Another example: $f(x) = -2x^5 + 4x^3$. The degree is 5 (odd) and the leading coefficient is -2 (negative). Thus, as $x$ approaches negative infinity, $f(x)$ approaches positive infinity, and as $x$ approaches positive infinity, $f(x)$ approaches negative infinity.

Remember to practice identifying the degree and leading coefficient to confidently determine end behavior!

Homework

Please complete the worksheet to practice identifying end behavior. Keep up the great work!