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Consider the polynomial function [tex]p(x)=6x^9 + 4x^6 + 2x^4 - 200[/tex].

What is the end behavior of the graph of [tex]p[/tex]?

Choose one answer:

A. As [tex]x \rightarrow \infty, p(x) \rightarrow \infty[/tex], and as [tex]x \rightarrow -\infty, p(x) \rightarrow \infty[/tex].

B. As [tex]x \rightarrow \infty, p(x) \rightarrow -\infty[/tex], and as [tex]x \rightarrow -\infty, p(x) \rightarrow \infty[/tex].

C. As [tex]x \rightarrow \infty, p(x) \rightarrow -\infty[/tex], and as [tex]x \rightarrow -\infty, p(x) \rightarrow -\infty[/tex].

D. As [tex]x \rightarrow \infty, p(x) \rightarrow \infty[/tex], and as [tex]x \rightarrow -\infty, p(x) \rightarrow -\infty[/tex].



Answer :

GinnyAnswer
Let's analyze the end behavior of the polynomial function [tex]\( p(x) = 6x^9 + 4x^6 + 2x^4 - 200 \)[/tex].

To determine the end behavior of the polynomial, we should focus on the term with the highest degree because it will dominate the behavior of the polynomial as [tex]\( x \)[/tex] approaches positive or negative infinity.

The highest degree term in this polynomial is [tex]\( 6x^9 \)[/tex].

- As [tex]\( x \rightarrow \infty \)[/tex]:
The term [tex]\( 6x^9 \)[/tex] will dominate, and since the coefficient (6) is positive and the exponent (9) is odd, [tex]\( x^9 \)[/tex] also grows without bound as [tex]\( x \)[/tex] increases. Therefore, [tex]\( 6x^9 \rightarrow \infty \)[/tex] as [tex]\( x \rightarrow \infty \)[/tex]. Hence, [tex]\( p(x) \rightarrow \infty \)[/tex].

- As [tex]\( x \rightarrow -\infty \)[/tex]:
Again, the term [tex]\( 6x^9 \)[/tex] will dominate. Since the exponent (9) is odd, [tex]\( (-x)^9 = -x^9 \)[/tex]; thus [tex]\( 6x^9 \rightarrow -\infty \)[/tex] as [tex]\( x \rightarrow -\infty \)[/tex]. Hence, [tex]\( p(x) \rightarrow -\infty \)[/tex].

Thus, the end behavior of the polynomial [tex]\( p(x) = 6x^9 + 4x^6 + 2x^4 - 200 \)[/tex] is:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( p(x) \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( p(x) \rightarrow -\infty \)[/tex].

Therefore, the correct answer is:
(D) As [tex]\( x \rightarrow \infty, p(x) \rightarrow \infty \)[/tex], and as [tex]\( x \rightarrow -\infty, p(x) \rightarrow -\infty \)[/tex].

Answer: D

Step-by-step explanation:

To find the end behavior of a graph, you must follow these steps:

  1. Look at the leading term of the function, which is the term with the highest exponent. In this equation, the leading term is 6x⁹
  2. Determine if the leading term has a positive or negative coefficient and an even or odd exponent. 6x⁹ has a positive coefficient and an odd exponent
  3. Next, use the rules for end behaviors to figure out what combination of coefficients and exponents generates what end behavior.The rules for end behavior of functions can be found here: https://brainly.com/question/40841761

Since the leading term has a positive coefficient and an odd exponent, the end behavior will be: As [tex]x \rightarrow \infty, p(x) \rightarrow \infty[/tex] and as [tex]x \rightarrow -\infty, p(x) \rightarrow -\infty[/tex]

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