Question

match each quadratic function to its graph.
$f(x) = -2x^2 - 26x - 80 = -2(x + 8)(x + 5)$
$g(x) = -x^2 - 2$

Explanation:

Step1: Analyze \( f(x) = -2x^2 - 26x - 80 \)

Factor form: \( f(x) = -2(x + 8)(x + 5) \). The roots are \( x = -8 \) and \( x = -5 \) (set each factor to 0). The left graph crosses the x - axis at \( x=-8 \) and \( x = -5 \), so \( f(x) \) matches the left graph.

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

For \( g(x) \), when \( x = 0 \), \( g(0)=-0 - 2=-2 \), so the y - intercept is \( (0, - 2) \). The right graph has a y - intercept at \( (0, - 2) \) and no x - intercepts (discriminant \( b^2-4ac=0 - 4( - 1)( - 2)=-8<0 \)), so \( g(x) \) matches the right graph.

Answer:

\( f(x)=-2x^{2}-26x - 80 \) matches the left graph; \( g(x)=-x^{2}-2 \) matches the right graph.