Question

match each polynomial function to its graph.
$f(x) = -2x^3 + 42x^2 - 288x + 640 = -2(x - 5)(x - 8)^2$
$g(x) = -x^3 - 24x^2 - 192x - 512 = -(x + 8)^3$
$f(x) = -2x^3 + 42x^2 - 288x + 640$ $g(x) = -x^3 - 24x^2 - 192x - 512$

Explanation:

Step1: Analyze \( f(x) \)

The factored form of \( f(x) \) is \( -2(x - 5)(x - 8)^2 \). The roots are \( x = 5 \) (with multiplicity 1) and \( x = 8 \) (with multiplicity 2). At \( x = 8 \), the graph touches the x - axis (due to even multiplicity) and turns around, and at \( x = 5 \), it crosses the x - axis (due to odd multiplicity). Also, the leading coefficient is - 2 (negative), so as \( x
ightarrow+\infty \), \( f(x)
ightarrow-\infty \) and as \( x
ightarrow-\infty \), \( f(x)
ightarrow+\infty \). Looking at the two graphs, the right - hand graph has a touch - and - turn at \( x = 8 \) and a cross at \( x = 5 \), so \( f(x) \) matches the right graph.

Step2: Analyze \( g(x) \)

The factored form of \( g(x) \) is \( -(x + 8)^3 \). The root is \( x=-8 \) with multiplicity 3 (odd). So the graph crosses the x - axis at \( x = - 8 \). The leading coefficient is - 1 (negative), so as \( x
ightarrow+\infty \), \( g(x)
ightarrow-\infty \) and as \( x
ightarrow-\infty \), \( g(x)
ightarrow+\infty \). The left - hand graph has a single root at \( x=-8 \) (crossing the x - axis), so \( g(x) \) matches the left graph.

Answer:

\( f(x)=-2x^{3}+42x^{2}-288x + 640 \) matches the right graph.
\( g(x)=-x^{3}-24x^{2}-192x - 512 \) matches the left graph.