Graphs of Polynomials - End behavior

A well-defined collection of distinct objects called elements or members.

https://youtu.be/cggv0rx2yBo

Learning Outcomes

 Visualize the end behavior of a polynomial from its algebraic expression depending on the degree and leading coefficient of the polynomial.  Compare the end behavior of the graph of a polynomial with the end behavior of its algebraic expression.

Exercise Questions 🤯

Good evening! Here in India on this Sunday, let’s explore these questions about the end behavior of polynomials. This is a concept that tells us what the graph of a function does as it goes off to the far left and far right of the chart.

Core Concepts: The Leading Coefficient Test

The end behavior of any polynomial is determined only by its leading term—the term with the highest power of $x$. We only need to look at two things from this term, $a_nx^n$:

1. The degree of the polynomial ($n$): Is it even or odd?
2. The sign of the leading coefficient ($a_n$): Is it positive or negative?

These two pieces of information tell us everything. Here’s a summary table:

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Question 1: What Determines End Behavior? (from file image_d0aa77.png)

The Question: If $p(x) = a_nx^n + a_{n-1}x^{n-1} + ….. + a_1x + a_0$ be a polynomial then end behavior of the polynomial depends __________.

Detailed Solution: As explained in the core concepts, the end behavior is completely dominated by the term with the highest power, $a_nx^n$. All the lower-power terms become insignificant as $x$ approaches $\infty$ or $-\infty$. Therefore, the behavior depends entirely on the leading coefficient, $a_n$, and the degree, $n$.

Final Answer: Only on the coefficient $a_n$ and the degree of the polynomial. {{< /border >}}

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Question 2: Even Degree, Positive Leading Coefficient (from file image_d0aa77.png)

The Question: If $a_n > 0$ and $n$ is even, then which of the following is(are) true?

Detailed Solution: This is the “Up-Up” case, which behaves like a standard parabola, $y=x^2$.

• Degree is even: Both ends of the graph point in the same direction.
• Leading coefficient is positive: That direction is up.
• Therefore:
  • As $x$ goes to the far right ($x \to \infty$), the graph goes up ($p(x) \to \infty$).
  • As $x$ goes to the far left ($x \to -\infty$), the graph also goes up ($p(x) \to \infty$).

Final Answer: The correct statements are:

• $x \to \infty \implies p(x) \to \infty$
• $x \to -\infty \implies p(x) \to \infty$ {{< /border >}}

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Question 3: Odd Degree, Positive Leading Coefficient (from file image_d0aa1e.png)

The Question: If $a_n > 0$ and $n$ is odd, then which of the following is(are) true?

Detailed Solution: This is the “Down-Up” case, which behaves like a standard cubic function, $y=x^3$.

• Degree is odd: The ends of the graph point in opposite directions.
• Leading coefficient is positive: The graph goes down on the left and up on the right.
• Therefore:
  • As $x$ goes to the far right ($x \to \infty$), the graph goes up ($p(x) \to \infty$).
  • As $x$ goes to the far left ($x \to -\infty$), the graph goes down ($p(x) \to -\infty$).

Final Answer: The correct statements are:

• $x \to \infty \implies p(x) \to \infty$
• $x \to -\infty \implies p(x) \to -\infty$ {{< /border >}}

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Question 4: Odd Degree, Negative Leading Coefficient (from file image_d0aa1e.png)

The Question: If $a_n < 0$ and $n$ is odd, then which of the following is(are) true?

Detailed Solution: This is the “Up-Down” case, which behaves like $y=-x^3$.

• Degree is odd: The ends of the graph point in opposite directions.
• Leading coefficient is negative: The graph goes up on the left and down on the right.
• Therefore:
  • As $x$ goes to the far right ($x \to \infty$), the graph goes down ($p(x) \to -\infty$).
  • As $x$ goes to the far left ($x \to -\infty$), the graph goes up ($p(x) \to \infty$).

Final Answer: The correct statements are:

• $x \to \infty \implies p(x) \to -\infty$
• $x \to -\infty \implies p(x) \to \infty$ {{< /border >}}

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Question 5: Even Degree, Negative Leading Coefficient (from file image_d0aa1e.png)

The Question: If $a_n < 0$ and $n$ is even, then which of the following is(are) true?

Detailed Solution: This is the “Down-Down” case, which behaves like an upside-down parabola, $y=-x^2$.

• Degree is even: Both ends of the graph point in the same direction.
• Leading coefficient is negative: That direction is down.
• Therefore:
  • As $x$ goes to the far right ($x \to \infty$), the graph goes down ($p(x) \to -\infty$).
  • As $x$ goes to the far left ($x \to -\infty$), the graph also goes down ($p(x) \to -\infty$).

Final Answer: The correct statements are:

• $x \to \infty \implies p(x) \to -\infty$
• $x \to -\infty \implies p(x) \to -\infty$ {{< /border >}}

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Question 6: End Behavior of a Specific Polynomial (from file image_d0a9bc.png)

The Question: If $p(x) = (-5)^3x^3 + 2x^2 + x + 1$ be a polynomial, then which of the following is(are) true?

Detailed Solution:

1. Find the leading term: We must first simplify the leading coefficient.
   • The leading term is $(-5)^3x^3 = -125x^3$.
2. Identify its properties:
   • Degree ($n$): The degree is 3, which is odd.
   • Leading Coefficient ($a_n$): The coefficient is -125, which is negative.
3. Determine the end behavior: This is the case of an odd degree and a negative leading coefficient (“Up-Down”).
   • As $x$ goes right, the graph goes down: $x \to \infty \implies p(x) \to -\infty$.
   • As $x$ goes left, the graph goes up: $x \to -\infty \implies p(x) \to \infty$.

Final Answer: The correct statements are:

• $x \to \infty \implies p(x) \to -\infty$
• $x \to -\infty \implies p(x) \to \infty$ {{< /border >}}

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Question 7: Reverse Logic (from file image_d0a9bc.png)

The Question: If $p(x) = 7x^n + 9x^5 + 1$ be a polynomial and $x \to \infty \implies p(x) \to -\infty$, then which of the following is(are) true?

Detailed Solution: This question contains a logical contradiction, meaning none of the options can be correct based on the premise. Here is the reasoning:

1. Analyze the Required End Behavior: The condition $x \to \infty \implies p(x) \to -\infty$ means the graph must go down on the far right. According to our rules, this can only happen if the leading coefficient is negative.

2. Analyze the Given Polynomial: $p(x) = 7x^n + 9x^5 + 1$.

   • Case 1: $n > 5$. The leading term is $7x^n$. The leading coefficient is 7 (positive).
   • Case 2: $n = 5$. The leading term is $7x^5 + 9x^5 = 16x^5$. The leading coefficient is 16 (positive).
   • Case 3: $n < 5$. The leading term is $9x^5$. The leading coefficient is 9 (positive).

3. The Contradiction: In all possible cases, the leading coefficient of this polynomial is positive. A polynomial with a positive leading coefficient must go to $+\infty$ as $x \to \infty$. This directly contradicts the condition given in the question.

Final Answer: The premise of the question is flawed, as a polynomial with this structure can never go to $-\infty$ as $x \to \infty$. There is no correct option among the choices. {{< /border >}}