Question

what is the end behavior of the graph of the polynomial function $f(x) = 2x^3 - 26x - 24$?
as $x \
ightarrow -\infty$, $y \
ightarrow \infty$ and as $x \
ightarrow \infty$, $y \
ightarrow \infty$.
as $x \
ightarrow -\infty$, $y \
ightarrow -\infty$ and as $x \
ightarrow \infty$, $y \
ightarrow \infty$.
as $x \
ightarrow -\infty$, $y \
ightarrow -\infty$ and as $x \
ightarrow \infty$, $y \
ightarrow -\infty$.
as $x \
ightarrow -\infty$, $y \
ightarrow \infty$ and as $x \
ightarrow \infty$, $y \
ightarrow -\infty$.

Explanation:

Step1: Identify the leading term

For the polynomial \( f(x)=2x^{3}-26x - 24 \), the leading term is \( 2x^{3} \) (the term with the highest power of \( x \)).

Step2: Analyze the degree and leading coefficient

The degree of the polynomial (the power of \( x \) in the leading term) is \( n = 3 \), which is odd. The leading coefficient \( a=2 \), which is positive.

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Answer:

The fourth option: As \( x \to -\infty \), \( y \to \infty \) and as \( x \to \infty \), \( y \to -\infty \) (Wait, correction: Wait, for a cubic function \( f(x)=ax^3+bx^2+cx + d \), the leading term is \( ax^3 \). Here \( a = 2>0 \). The end - behavior of a cubic function \( y = ax^3\) is: when \( a>0 \), as \( x\to-\infty \), \( y\to-\infty \) and as \( x\to\infty \), \( y\to\infty \)? Wait, no, wait: Let's re - calculate.

Wait, the function is \( f(x)=2x^{3}-26x - 24 \). The leading term is \( 2x^{3}\). For the end - behavior of a polynomial, we only need to consider the leading term.

For a power function \( y = ax^{n}\), when \( n \) is odd:

• If \( a>0 \), as \( x\to-\infty \), \( y\to-\infty \) (because \( (-\infty)^{3}=-\infty \), and multiplying by \( 2>0 \) gives \( -\infty \)) and as \( x\to\infty \), \( y\to\infty \) (because \( (\infty)^{3}=\infty \), multiplying by \( 2>0 \) gives \( \infty \)). Wait, but the options:

Wait the first option: As \( x\to-\infty \), \( y\to\infty \) and as \( x\to\infty \), \( y\to\infty \) (wrong)

Second option: As \( x\to-\infty \), \( y\to-\infty \) and as \( x\to\infty \), \( y\to\infty \) (this is correct for \( a = 2>0 \) and \( n = 3 \) (odd))

Wait I made a mistake earlier. Let's re - derive:

The leading term is \( 2x^{3}\). When \( x\to-\infty \), \( x^{3}\to-\infty \), and \( 2\times(-\infty)=-\infty \), so \( y\to-\infty \). When \( x\to\infty \), \( x^{3}\to\infty \), and \( 2\times\infty=\infty \), so \( y\to\infty \). So the second option: "As \( x \to -\infty \), \( y \to -\infty \) and as \( x \to \infty \), \( y \to \infty \)" is correct.