Question

match each polynomial function to its graph.
$f(x) = x^4 + 4x^2 - x + 3$
$g(x) = -x^4 - 4x^2 + x - 4$
two graphs on a coordinate grid are shown, one with a upward - opening curve and one with a downward - opening curve, each with x - axis from - 10 to 10 and y - axis from - 10 to 10, with grid lines.

Explanation:

Step1: Analyze the leading term of \( f(x) \)

The leading term of \( f(x)=x^{4}+4x^{2}-x + 3 \) is \( x^{4} \) with a positive coefficient (\( 1>0 \)). For even - degree polynomials, when the leading coefficient is positive, as \( x
ightarrow\pm\infty \), \( f(x)
ightarrow+\infty \). So the graph of \( f(x) \) should open upwards (both ends go to positive infinity). Looking at the two graphs, the first graph (with the curve opening upwards) matches the end - behavior of \( f(x) \).

Step2: Analyze the leading term of \( g(x) \)

The leading term of \( g(x)=-x^{4}-4x^{2}+x - 4 \) is \( -x^{4} \) with a negative coefficient (\( - 1<0 \)). For even - degree polynomials, when the leading coefficient is negative, as \( x
ightarrow\pm\infty \), \( g(x)
ightarrow-\infty \). So the graph of \( g(x) \) should open downwards (both ends go to negative infinity). Looking at the two graphs, the second graph (with the curve opening downwards) matches the end - behavior of \( g(x) \).

Answer:

\( f(x)=x^{4}+4x^{2}-x + 3 \) matches the first graph (opening upwards), \( g(x)=-x^{4}-4x^{2}+x - 4 \) matches the second graph (opening downwards)