Question

f(x) = -x³ - 21x² - 145x - 335; g(x) = -3x⁴ - 15x³ - 18x²; k(x) = 2x⁴ + 8x⁵ + 10x² + 8x + 8; h(x) = x³ + 21x² + 147x + 343

Explanation:

Step1: Analyze the degree and leading coefficient of \( h(x) \)

The function \( h(x)=x^{3}+21x^{2}+147x + 343 \) is a cubic function (degree \( n = 3 \), odd) with a leading coefficient \( a=1>0 \). For a cubic function with odd degree and positive leading coefficient, as \( x
ightarrow+\infty \), \( y
ightarrow+\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow-\infty \). But let's also check for factoring. Notice that \( h(x)=(x + 7)^{3}\) (since \( (x + 7)^{3}=x^{3}+21x^{2}+147x + 343 \)), so it has a triple root at \( x=-7 \). The graph of a function with a triple root at \( x = - 7 \) should touch and turn around? Wait, no, a triple root for a cubic: the derivative \( h^\prime(x)=3(x + 7)^{2}\), which is non - negative (since it's a square times 3), and zero only at \( x=-7 \). So the function is increasing everywhere, with a point of inflection (but horizontal tangent) at \( x=-7 \).

Step2: Analyze the other functions

• For \( f(x)=-x^{3}-21x^{2}-145x - 335 \), degree 3, leading coefficient - 1 (negative). So as \( x

ightarrow+\infty \), \( y
ightarrow-\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow+\infty \), which is the opposite of \( h(x) \)'s end - behavior (since \( h(x) \) has leading coefficient positive).

• For \( g(x)=-3x^{4}-15x^{3}-18x^{2}\), degree 4 (even), leading coefficient - 3 (negative). So as \( x

ightarrow\pm\infty \), \( y
ightarrow-\infty \), and it's a quartic, not a cubic.

• For \( k(x)=2x^{4}+8x^{5}+10x^{2}+8x + 8 \), degree 5 (odd), but let's check the leading term \( 8x^{5} \), leading coefficient 8 (positive). But the function \( h(x) \) is a cubic, while \( k(x) \) is a quintic. Also, the graph of \( h(x)=(x + 7)^{3} \) should be a smooth curve increasing through \( x=-7 \), with a horizontal tangent at \( x=-7 \). Looking at the graphs, the bottom - left graph (the third graph, the one below \( f(x) \)'s graph) should correspond to \( h(x) \) because it has the end - behavior of a cubic with positive leading coefficient (as \( x

ightarrow+\infty \), going up, as \( x
ightarrow-\infty \), going down? Wait, no, earlier mistake: for \( y = x^{3} \), as \( x
ightarrow+\infty \), \( y
ightarrow+\infty \), \( x
ightarrow-\infty \), \( y
ightarrow-\infty \). And \( h(x)=(x + 7)^{3} \) is just a horizontal shift of \( y = x^{3} \) 7 units to the left. So the graph of \( h(x) \) should look like the graph of \( y=x^{3} \) shifted left 7 units. The bottom - left graph (the one with \( h(x) \) above it) has the shape of a cubic function increasing, with a horizontal tangent at \( x=-7 \) (around \( x=-7 \) on the x - axis, the graph has a "flat" spot before increasing again). The other graphs: the top - left graph is for \( f(x) \) (since it has opposite end - behavior to \( h(x) \)), the top - right is a quartic (even degree) graph, the bottom - right is a graph that might be for a quartic or quintic with multiple roots near the origin.

Answer:

The graph corresponding to \( h(x)=x^{3}+21x^{2}+147x + 343 \) is the bottom - left graph (the graph below the \( h(x) \) equation box).