Question

match each quadratic function to its graph.
$f(x) = 2x^{2} + 8x = 2x(x + 4)$
$g(x) = -x^{2} + 18x - 81 = -(x - 9)^{2}$
$f(x) = 2x^{2} + 8x$ $g(x) = -x^{2} + 18x - 81$
(graphs are shown with coordinate grids, parabolas; one graph has a parabola opening upwards with roots around -4 and 0, another with a parabola opening downwards with vertex around (9,0))
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Explanation:

Step1: Analyze \( f(x) = 2x^2 + 8x \)

The coefficient of \( x^2 \) is \( 2>0 \), so the parabola opens upward. The factored form is \( 2x(x + 4) \), so the roots are \( x = 0 \) and \( x=-4 \). Looking at the graphs, the left graph has a parabola opening upward with roots near \( x = 0 \) and \( x=-4 \), so \( f(x) \) matches the left graph.

Step2: Analyze \( g(x) = -x^2 + 18x - 81 \)

The coefficient of \( x^2 \) is \( -1<0 \), so the parabola opens downward. The vertex form is \( -(x - 9)^2 \), so the vertex is at \( (9, 0) \) (a repeated root). The right graph has a parabola opening downward with a vertex near \( x = 9 \), so \( g(x) \) matches the right graph.

Answer:

\( f(x) = 2x^2 + 8x \) matches the Left Graph; \( g(x) = -x^2 + 18x - 81 \) matches the Right Graph.