Question

match each polynomial function to its graph.
h(x)=-x^{3}-12x^{2}-48x - 64
k(x)=x^{4}+16x^{3}+96x^{2}+256x + 255
f(x)=x^{3}+4x^{2}+3x + 1
g(x)=-x^{4}-5x^{2}-4

Explanation:

Step1: Analyze end - behavior of $h(x)=-x^{3}-12x^{2}-48x - 64$

The leading term is $-x^{3}$. For odd - degree polynomials with a negative leading coefficient, as $x\to+\infty$, $y\to-\infty$ and as $x\to-\infty$, $y\to+\infty$.

Step2: Analyze end - behavior of $k(x)=x^{4}+16x^{3}+96x^{2}+256x + 255$

The leading term is $x^{4}$. For even - degree polynomials with a positive leading coefficient, as $x\to\pm\infty$, $y\to+\infty$.

Step3: Analyze end - behavior of $f(x)=x^{3}+4x^{2}+3x + 1$

The leading term is $x^{3}$. For odd - degree polynomials with a positive leading coefficient, as $x\to+\infty$, $y\to+\infty$ and as $x\to-\infty$, $y\to-\infty$.

Step4: Analyze end - behavior of $g(x)=-x^{4}-5x^{2}-4$

The leading term is $-x^{4}$. For even - degree polynomials with a negative leading coefficient, as $x\to\pm\infty$, $y\to-\infty$.

Answer:

Match based on end - behavior and number of turning points (not shown in full analysis here but can be further investigated using derivatives). Without specific graph labels, we can only say in general:

• $h(x)=-x^{3}-12x^{2}-48x - 64$ has odd - degree and negative leading coefficient.
• $k(x)=x^{4}+16x^{3}+96x^{2}+256x + 255$ has even - degree and positive leading coefficient.
• $f(x)=x^{3}+4x^{2}+3x + 1$ has odd - degree and positive leading coefficient.
• $g(x)=-x^{4}-5x^{2}-4$ has even - degree and negative leading coefficient.