Understanding polynomial end behavior using the Leading Term Test

📚 Quick Study Guide

• 🔢Leading Term: The term with the highest degree in the polynomial. It determines the end behavior.
• 📈Even Degree: If the leading term's degree is even, the end behavior is the same on both sides (either both up or both down).
• 📉Odd Degree: If the leading term's degree is odd, the end behavior is opposite on each side (one up, one down).
• ➕Positive Leading Coefficient: For even degree, both ends go up. For odd degree, left goes down, right goes up.
• ➖Negative Leading Coefficient: For even degree, both ends go down. For odd degree, left goes up, right goes down.
• ✏️General Form: A polynomial is in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where $a_n$ is the leading coefficient and $n$ is the degree.
• 💡Leading Term Test Summary:
  • Even Degree, Positive Coefficient: As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$.
  • Even Degree, Negative Coefficient: As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$.
  • Odd Degree, Positive Coefficient: As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$.
  • Odd Degree, Negative Coefficient: As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$.

🧠 Practice Quiz

1. Which of the following statements is true regarding the end behavior of the polynomial $f(x) = -3x^4 + 2x^2 - x + 5$?

   1. As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$
   2. As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$
   3. As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$
   4. As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$

2. What is the end behavior of the polynomial $g(x) = 5x^3 - x^2 + 7x - 1$?

   1. As $x \rightarrow \infty$, $g(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $g(x) \rightarrow \infty$
   2. As $x \rightarrow \infty$, $g(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $g(x) \rightarrow -\infty$
   3. As $x \rightarrow \infty$, $g(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $g(x) \rightarrow -\infty$
   4. As $x \rightarrow \infty$, $g(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $g(x) \rightarrow \infty$

3. The leading term of a polynomial is $-2x^5$. What is its end behavior?

   1. As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$
   2. As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$
   3. As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$
   4. As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$

4. Consider the polynomial $h(x) = x^{6} - 4x^{4} + 2x - 9$. Which statement accurately describes its end behavior?

   1. Both ends go down.
   2. Both ends go up.
   3. The left end goes up, and the right end goes down.
   4. The left end goes down, and the right end goes up.

5. A polynomial has odd degree and a positive leading coefficient. Which of the following is true?

   1. Both ends go up.
   2. Both ends go down.
   3. The left end goes down, and the right end goes up.
   4. The left end goes up, and the right end goes down.

6. What is the end behavior of $f(x) = -x + 3x^3 - 2x^5 + 1$?

   1. As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$
   2. As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$
   3. As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$
   4. As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$

7. Which polynomial has the following end behavior: As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$?

   1. $f(x) = 2x^3 + x - 1$
   2. $f(x) = -3x^2 + 5$
   3. $f(x) = 4x^5 - 2x^2$
   4. $f(x) = x^4 + 7x$