Question

match each polynomial function to its graph.
$f(x) = -x^3 - 18x^2 - 108x - 216 = -(x + 6)^3$
$g(x) = -x^3 + 21x^2 - 147x + 343 = -(x - 7)^3$
$f(x) = -x^3 - 18x^2 - 108x - 216$ $g(x) = -x^3 + 21x^2 - 147x + 343$

Explanation:

Step1: Analyze \( f(x) = -(x + 6)^3 \)

The function \( f(x) = -(x + 6)^3 \) has a root at \( x = -6 \) (with multiplicity 3). The leading coefficient is negative, so as \( x \to +\infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to +\infty \). At \( x = -6 \), the graph touches and turns? No, for a cubic with odd multiplicity, it crosses or has a point of inflection. Wait, \( y = - (x + 6)^3 \) is a cubic function, reflected over x - axis and shifted left 6 units. The standard \( y = x^3 \) has a point of inflection at the origin, so \( y = - (x + 6)^3 \) has a point of inflection at \( x = -6 \), and the graph should pass through (or have a point of inflection at) \( x = -6 \). Looking at the right graph: it has a point around \( x = -6 \), and as \( x \) increases from \( -6 \), the graph goes down (since leading coefficient negative). So \( f(x) \) matches the right graph.

Step2: Analyze \( g(x) = -(x - 7)^3 \)

The function \( g(x) = -(x - 7)^3 \) has a root at \( x = 7 \) (multiplicity 3). Leading coefficient negative, so as \( x \to +\infty \), \( g(x) \to -\infty \), as \( x \to -\infty \), \( g(x) \to +\infty \). The point of inflection is at \( x = 7 \). Looking at the left graph: it has a point around \( x = 7 \), and as \( x \) increases from \( 7 \), the graph goes down. So \( g(x) \) matches the left graph.

Answer:

\( f(x) = -x^3 - 18x^2 - 108x - 216 \) matches the Right Graph.
\( g(x) = -x^3 + 21x^2 - 147x + 343 \) matches the Left Graph.