End Behavior Part 2: From Equations to Infinity!

Yesterday, we explored end behavior by looking at graphs. Today, we're diving deeper and learning how to determine end behavior directly from polynomial equations. Understanding end behavior is crucial for sketching graphs and predicting function values for very large or very small values of $x$.

Key Concepts:

  • Degree of the Polynomial: The highest power of $x$ in the polynomial. This tells us the general shape of the function.
  • Leading Coefficient: The coefficient of the term with the highest power of $x$. This tells us whether the function opens upwards or downwards.

Let's break down how to determine end behavior based on these two key pieces of information:

1. Even Degree Polynomials:

  • Positive Leading Coefficient: The graph opens upwards. As $x$ approaches positive or negative infinity, $f(x)$ approaches positive infinity. We write this as:
    • As $x \to \infty$, $f(x) \to \infty$
    • As $x \to -\infty$, $f(x) \to \infty$
  • Negative Leading Coefficient: The graph opens downwards. As $x$ approaches positive or negative infinity, $f(x)$ approaches negative infinity. We write this as:
    • As $x \to \infty$, $f(x) \to -\infty$
    • As $x \to -\infty$, $f(x) \to -\infty$

Example: $f(x) = x^2 + 3x - 4$. The degree is 2 (even), and the leading coefficient is 1 (positive). Therefore, as $x$ approaches positive or negative infinity, $f(x)$ approaches positive infinity.

2. Odd Degree Polynomials:

  • Positive Leading Coefficient: As $x$ approaches positive infinity, $f(x)$ approaches positive infinity. As $x$ approaches negative infinity, $f(x)$ approaches negative infinity. We write this as:
    • As $x \to \infty$, $f(x) \to \infty$
    • As $x \to -\infty$, $f(x) \to -\infty$
  • Negative Leading Coefficient: As $x$ approaches positive infinity, $f(x)$ approaches negative infinity. As $x$ approaches negative infinity, $f(x)$ approaches positive infinity. We write this as:
    • As $x \to \infty$, $f(x) \to -\infty$
    • As $x \to -\infty$, $f(x) \to \infty$

Example: $f(x) = -x^3 + 4x^2 - 4x + 1$. The degree is 3 (odd), and the leading coefficient is -1 (negative). Therefore, as $x$ approaches positive infinity, $f(x)$ approaches negative infinity, and as $x$ approaches negative infinity, $f(x)$ approaches positive infinity.

Homework

Complete the worksheet provided in class to practice determining end behavior from equations. Remember to identify the degree and leading coefficient first! You can do it!