#Polynomial Functions and End Behavior

Hey there, future AP Pre-Calculus master! 👋 We've explored polynomial functions, but let's nail down one crucial aspect: end behavior. This is super important, and you'll see it pop up everywhere, so let's make sure you're totally comfortable with it.

#👋🏼 Introducing End Behaviors

End behavior describes what happens to a function's output values ($y$-values) as the input values ($x$-values) approach positive or negative infinity. Think of it as the long-term trend of the function. ⏱️

# 🔚 End Behaviors in Polynomial Functions

For polynomial functions, as $x$ heads towards $\pm \infty$, the function's output will also head towards either $\infty$ or $-\infty$. Let's break down how to know which way it's going:

• Positive Leading Coefficient: If the coefficient of the term with the highest degree is positive, the function will increase without bound as $x$ approaches $\infty$ or $-\infty$. This is written as $lim_{x \to \pm \infty} p(x) = \infty$ ⬆️.
• Negative Leading Coefficient: If the coefficient of the term with the highest degree is negative, the function will decrease without bound as $x$ approaches $\infty$ or $-\infty$. This is written as $lim_{x \to \pm \infty} p(x) = -\infty$ ⬇️.

This happens because, as $x$ gets really big (positive or negative), the term with the highest degree dominates all the other terms. So, the end behavior is all about that leading term!

Image Courtesy of MilanVasicMath on YouTube

# 🤓 Examples of Determining End Behavior

Let's look at some examples to see this in action:

#Determining End Behavior: Example 1

Consider the polynomial function $f(x) = 3x^4 - 2x^3 + 5x^2 - x + 4$.

The leading term here is $3x^4$. The leading coefficient (3) is positive, and the degree (4) is even. This means that as $x$ approaches $\pm \infty$, $f(x)$ approaches $\infty$. So, $lim_{x \to \pm \infty} f(x) = \infty$.

Graph displaying the function $3x^4 - 2x^3 + 5x^2 - x + 4$

Created by Jed Q on Desmos

#Determining End Behavior: Example 2

Now, let's look at $g(x) = -2x^5 + x^3 - 6x^2 + 3x - 1$.

The leading term is $-2x^5$. The leading coefficient (-2) is negative, and the degree (5) is odd. This means that as $x$ approaches $\infty$, $g(x)$ approaches $-\infty$, and as $x$ approaches $-\infty$, $g(x)$ approaches $\infty$. So, $lim_{x \to \infty} g(x) = -\infty$ and $lim_{x \to -\infty} g(x) = \infty$.

Graph displaying the function $-2x^5 + x^3 - 6x^2 + 3x - 1$

Created by Jed Q on Desmos

#Determining End Behavior: Example 3

Let's examine $h(x) = 4x^3 - 2x^2 + 7x - 1$.

The leading term is $4x^3$. The leading coefficient (4) is positive, and the degree (3) is odd. This means that as $x$ approaches $\infty$, $h(x)$ approaches $\infty$, and as $x$ approaches $-\infty$, $h(x)$ approaches $-\infty$. So, $lim_{x \to \infty} h(x) = \infty$ and $lim_{x \to -\infty} h(x) = -\infty$.

Graph displaying the function $4x^3 - 2x^2 + 7x - 1$

Created by Jed Q on Desmos

#Determining End Behavior: Example 4

Finally, let's consider $p(x) = -x^4 + 3x^3 - 2x^2 + 5x - 7$.

The leading term is $-x^4$. The leading coefficient (-1) is negative, and the degree (4) is even. This means that as $x$ approaches $\pm \infty$, $p(x)$ approaches $-\infty$. So, $lim_{x \to \pm \infty} p(x) = -\infty$.

Graph displaying the function $-x^4 + 3x^3 - 2x^2 + 5x - 7$

Created by Jed Q on Desmos

# 🎯 Final Exam Focus

• High-Priority Topics: End behavior is a fundamental concept that links to other topics such as limits, asymptotes, and graph analysis.
• Common Question Types:
  • Multiple-choice questions asking for the end behavior of a given polynomial.
  • Free-response questions where you need to sketch a polynomial graph, paying attention to its end behavior.
  • Questions that combine end behavior with other concepts, such as finding limits at infinity.

# 📝 Practice Questions

Practice Question

Multiple Choice Questions

1. What is the end behavior of the function $f(x) = -5x^3 + 2x^2 - 7x + 1$ as $x$ approaches $\infty$? (A) $f(x) \to \infty$ (B) $f(x) \to -\infty$ (C) $f(x) \to 0$ (D) $f(x)$ oscillates

2. Which of the following polynomial functions has an end behavior of $f(x) \to -\infty$ as $x \to \infty$ and $f(x) \to \infty$ as $x \to -\infty$? (A) $f(x) = 2x^4 - 3x^2 + 1$ (B) $f(x) = -x^3 + 4x - 2$ (C) $f(x) = x^5 - 2x^3 + x$ (D) $f(x) = -3x^6 + x^4 - 2$

Free Response Question

Consider the polynomial function $p(x) = 2x^5 - 3x^4 + x^2 - 5x + 2$.

(a) Determine the end behavior of $p(x)$ as $x$ approaches $\infty$ and $-\infty$. (2 points)

(b) Sketch a general graph of $p(x)$, paying attention to its end behavior. (3 points)

(c) Explain how the leading term of $p(x)$ determines its end behavior. (2 points)

Answer Key

Multiple Choice

1. (B) $f(x) \to -\infty$
2. (B) $f(x) = -x^3 + 4x - 2$

Free Response

(a) As $x \to \infty$, $p(x) \to \infty$, and as $x \to -\infty$, $p(x) \to -\infty$. (2 points: 1 point for each correct limit)

(b) A sketch showing a graph that rises to the right and falls to the left is expected. The graph should have a general shape that is consistent with a 5th degree polynomial. (3 points: 1 point for correct end behavior, 1 point for a general shape, 1 point for showing some oscillations)

(c) The leading term of $p(x)$ is $2x^5$. The positive coefficient (2) means that as $x$ approaches $\infty$, $p(x)$ approaches $\infty$. The odd degree (5) means that as $x$ approaches $-\infty$, $p(x)$ approaches $-\infty$. (2 points: 1 point for identifying the leading term and coefficient, 1 point for explaining the impact of the odd degree).

You've got this! With a clear understanding of leading coefficients and degrees, you're well-equipped to tackle any end behavior question. Keep practicing, and you'll be an AP Pre-Calculus pro in no time! 🚀