Question

match each quadratic function to its graph.
$f(x) = -2x^{2} - 24x - 64 = -2(x + 8)(x + 4)$
$g(x) = -x^{2} + 3x - 2 = -(x - 1)(x - 2)$
$f(x) = -2x^{2} - 24x - 64$ $g(x) = -x^{2} + 3x - 2$

Explanation:

Step1: Analyze \( f(x) = -2x^2 - 24x - 64 \)

First, find the x - intercepts from the factored form \( f(x)=-2(x + 8)(x + 4) \). Set \( f(x)=0 \), then \( x+8 = 0 \) or \( x + 4=0 \), so \( x=-8 \) or \( x=-4 \). The parabola opens downward (since the coefficient of \( x^2\) is - 2, negative).

Step2: Analyze \( g(x)=-x^2 + 3x - 2 \)

From the factored form \( g(x)=-(x - 1)(x - 2) \), set \( g(x)=0 \), then \( x - 1=0 \) or \( x - 2=0 \), so \( x = 1 \) or \( x=2 \). The parabola opens downward (coefficient of \( x^2\) is - 1, negative).

Step3: Match with graphs

The left - hand graph has x - intercepts around 1 and 2 (matches \( g(x)\)'s x - intercepts 1 and 2). The right - hand graph has x - intercepts around - 8 and - 4 (matches \( f(x)\)'s x - intercepts - 8 and - 4). So \( f(x)=-2x^2 - 24x - 64 \) matches the right graph, and \( g(x)=-x^2 + 3x - 2 \) matches the left graph.

Answer:

\( f(x)=-2x^2 - 24x - 64 \) matches the Right Graph; \( g(x)=-x^2 + 3x - 2 \) matches the Left Graph.