Question

match each polynomial function to its graph. \\( f(x) = -x^3 - 2x^2 - x - 3 \\) \\( g(x) = 3x^4 - 3x^3 - 9x^2 + 6x + 3 \\) \\( h(x) = x^3 - 2x^2 + x - 2 \\) \\( k(x) = -3x^4 - 12x^3 - 9x^2 + 3x - 3 \\)

Explanation:

Step1: Analyze the degree and leading coefficient of each function

• For \( f(x)=-x^{3}-2x^{2}-x - 3 \): Degree is 3 (odd), leading coefficient is - 1 (negative). So as \( x

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

• For \( g(x)=3x^{4}-3x^{3}-9x^{2}+6x + 3 \): Degree is 4 (even), leading coefficient is 3 (positive). So as \( x

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

• For \( h(x)=x^{3}-2x^{2}+x - 2 \): Degree is 3 (odd), leading coefficient is 1 (positive). So as \( x

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

• For \( k(x)=-3x^{4}-12x^{3}-9x^{2}+3x - 3 \): Degree is 4 (even), leading coefficient is - 3 (negative). So as \( x

ightarrow\pm\infty \), \( k(x)
ightarrow-\infty \).

Step2: Analyze the y - intercept (value of function at \( x = 0 \))

• \( f(0)=-0 - 0 - 0 - 3=-3 \)
• \( g(0)=0 - 0 - 0+0 + 3 = 3 \)
• \( h(0)=0 - 0+0 - 2=-2 \)
• \( k(0)=0 - 0 - 0+0 - 3=-3 \)

Step3: Match with graphs

• The first graph (left - top): It is a cubic function (degree 3) with the end - behavior of \( x

ightarrow+\infty,f(x)
ightarrow+\infty \) and \( x
ightarrow-\infty,f(x)
ightarrow-\infty \) (leading coefficient positive). So it matches \( h(x)=x^{3}-2x^{2}+x - 2 \) (since \( h(x) \) is cubic with positive leading coefficient and \( h(0)=-2 \), which is close to the y - intercept of this graph).

• The second graph (right - top): It is a quartic function (degree 4) with end - behavior \( x

ightarrow\pm\infty,f(x)
ightarrow+\infty \) (leading coefficient positive). So it matches \( g(x)=3x^{4}-3x^{3}-9x^{2}+6x + 3 \) (since \( g(x) \) is quartic with positive leading coefficient and \( g(0)=3 \), which matches the y - intercept of this graph).

• The third graph (left - bottom): It is a quartic function (degree 4) with end - behavior \( x

ightarrow\pm\infty,f(x)
ightarrow-\infty \) (leading coefficient negative). So it matches \( k(x)=-3x^{4}-12x^{3}-9x^{2}+3x - 3 \) (since \( k(x) \) is quartic with negative leading coefficient and \( k(0)=-3 \), which matches the y - intercept of this graph).

• The fourth graph (right - bottom): It is a cubic function (degree 3) with end - behavior \( x

ightarrow+\infty,f(x)
ightarrow-\infty \) and \( x
ightarrow-\infty,f(x)
ightarrow+\infty \) (leading coefficient negative). So it matches \( f(x)=-x^{3}-2x^{2}-x - 3 \) (since \( f(x) \) is cubic with negative leading coefficient and \( f(0)=-3 \), which matches the y - intercept of this graph).

Answer:

• \( f(x)=-x^{3}-2x^{2}-x - 3 \) matches the fourth graph (right - bottom).
• \( g(x)=3x^{4}-3x^{3}-9x^{2}+6x + 3 \) matches the second graph (right - top).
• \( h(x)=x^{3}-2x^{2}+x - 2 \) matches the first graph (left - top).
• \( k(x)=-3x^{4}-12x^{3}-9x^{2}+3x - 3 \) matches the third graph (left - bottom).