Question

which function has this end behavior of up and up based on leading coefficient being positive and the degree is even?
$x^4+3x^3-4x+1$
$x^5-4x^4+2x^2-1$
$-x^2-3x+1$
$-x^3+2x^2+3$

Explanation:

Step1: Recall end behavior rules

For a polynomial \( f(x)=a_nx^n + a_{n - 1}x^{n - 1}+\dots+a_0\), end behavior is determined by leading term \(a_nx^n\). If \(n\) (degree) is even:

• If \(a_n>0\), as \(x

ightarrow\pm\infty\), \(f(x)
ightarrow+\infty\).

• If \(a_n<0\), as \(x

ightarrow\pm\infty\), \(f(x)
ightarrow-\infty\).
If \(n\) is odd:

• If \(a_n>0\), as \(x

ightarrow+\infty\), \(f(x)
ightarrow+\infty\); as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\).

• If \(a_n<0\), as \(x

ightarrow+\infty\), \(f(x)
ightarrow-\infty\); as \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\).

Step2: Analyze each option

• Option 1 (\(x^4 + 3x^3-4x + 1\)):

Degree \(n = 4\) (even), leading coefficient \(a_4=1>0\). So end behavior: \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\); \(x
ightarrow+\infty\), \(f(x)
ightarrow+\infty\) (matches the required behavior).

• Option 2 (\(x^5-4x^4 + 2x^2-1\)):

Degree \(n = 5\) (odd), so end behavior will be different (e.g., as \(x
ightarrow+\infty\), \(f(x)
ightarrow+\infty\); as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\)) – does not match.

• Option 3 (\(-x^2-3x + 1\)):

Degree \(n = 2\) (even), but leading coefficient \(a_2=-1<0\). So end behavior: \(x
ightarrow\pm\infty\), \(f(x)
ightarrow-\infty\) – does not match.

• Option 4 (\(-x^3 + 2x^2+3\)):

Degree \(n = 3\) (odd), leading coefficient \(a_3=-1<0\). End behavior: \(x
ightarrow+\infty\), \(f(x)
ightarrow-\infty\); \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\) – does not match.

Answer:

The function with the given end behavior is \(x^4 + 3x^3-4x + 1\) (the first option, green box).