Question

match each polynomial function to its graph.
$f(x) = -x^2 - 9x - 14 = -(x + 7)(x + 2)$
$g(x) = -x^2 - 5$
$f(x) = -x^2 - 9x - 14$ $g(x) = -x^2 - 5$

Explanation:

Step1: Analyze \( f(x) = -x^2 - 9x - 14 = -(x + 7)(x + 2) \)

The roots of \( f(x) \) are found by setting \( f(x) = 0 \), so \( -(x + 7)(x + 2)=0 \). This gives \( x = -7 \) and \( x = -2 \). So the graph of \( f(x) \) should intersect the x - axis at \( x=-7 \) and \( x = -2 \). Looking at the two graphs, the right - hand graph (the one with the vertex more to the left and intersecting the x - axis) has x - intercepts around \( x=-7 \) and \( x=-2 \), so \( f(x) \) matches the right - hand graph.

Step2: Analyze \( g(x)=-x^2 - 5 \)

For \( g(x)=-x^2 - 5 \), set \( g(x) = 0 \), then \( -x^2-5 = 0\Rightarrow x^2=-5 \), which has no real solutions. So the graph of \( g(x) \) does not intersect the x - axis. The left - hand graph does not intersect the x - axis (it is a downward - opening parabola with vertex below the x - axis and no x - intercepts), so \( g(x) \) matches the left - hand graph.

Answer:

\( f(x)=-x^2 - 9x - 14 \) matches the right - hand graph (the one with x - intercepts at \( x\approx - 7 \) and \( x\approx - 2 \)) and \( g(x)=-x^2 - 5 \) matches the left - hand graph (the one with no x - intercepts).