Question

match each quadratic function to its graph.
$f(x) = x^2 + 5$
$g(x) = -2x^2$
$h(x) = 2x^2 - 8 = 2(x + 2)(x - 2)$
$k(x) = -x^2 - x + 6 = -(x + 3)(x - 2)$
$f(x)=x^2 + 5$
$g(x)=-2x^2$
$h(x)=2x^2 - 8$
$k(x)=-x^2 - x + 6$

Explanation:

Step1: Analyze \( f(x) = x^2 + 5 \)

The function is a quadratic with \( a = 1>0 \), so it opens upward. The vertex form is \( f(x)=x^2 + 5 \), so the vertex is at \( (0, 5) \). Look for the upward - opening parabola with vertex on the y - axis at \( y = 5 \). The top - right graph (second graph in the first row) has a vertex at \( (0,5) \) and opens upward, so \( f(x)=x^2 + 5 \) matches this graph.

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

The function is a quadratic with \( a=-2 < 0 \), so it opens downward. The vertex form is \( g(x)=-2x^2 \), so the vertex is at \( (0,0) \). Look for the downward - opening parabola with vertex at the origin. The bottom - left graph (second graph in the second row) has a vertex at \( (0,0) \) and opens downward, so \( g(x) = - 2x^2\) matches this graph.

Step3: Analyze \( h(x)=2x^2 - 8=2(x + 2)(x - 2) \)

The function is a quadratic with \( a = 2>0 \), so it opens upward. The roots are found by setting \( h(x)=0 \), \( 2(x + 2)(x - 2)=0\), so \( x=-2 \) or \( x = 2 \). Look for the upward - opening parabola with x - intercepts at \( x=-2 \) and \( x = 2 \). The bottom - right graph (second graph in the second row) has x - intercepts at \( x=-2 \) and \( x = 2 \) and opens upward, so \( h(x)=2x^2 - 8 \) matches this graph.

Step4: Analyze \( k(x)=-x^2 - x + 6=-(x + 3)(x - 2) \)

The function is a quadratic with \( a=-1<0 \), so it opens downward. The roots are found by setting \( k(x)=0 \), \( -(x + 3)(x - 2)=0\), so \( x=-3 \) or \( x = 2 \). Look for the downward - opening parabola with x - intercepts at \( x=-3 \) and \( x = 2 \). The top - left graph (first graph in the first row) has x - intercepts at \( x=-3 \) and \( x = 2 \) and opens downward, so \( k(x)=-x^2 - x + 6 \) matches this graph.

Answer:

• \( f(x)=x^2 + 5 \) matches the top - right graph.
• \( g(x)=-2x^2 \) matches the bottom - left graph.
• \( h(x)=2x^2 - 8 \) matches the bottom - right graph.
• \( k(x)=-x^2 - x + 6 \) matches the top - left graph.