Question

f(x) = 2x² - 16x + 32 = 2(x - 4)²
g(x) = -2x² - 8x - 8 = -2(x + 2)²
h(x) = -x² - 5
k(x) = -x² - 11x - 28 = -(x + 7)(x + 4)
f(x) = 2x² - 16x + 32 g(x) = -2x² - 8x - 8
h(x) = -x² - 5 k(x) = -x² - 11x - 28

Explanation:

Step1: Analyze \( f(x) = 2x^2 - 16x + 32 \)

The coefficient of \( x^2 \) is \( 2>0 \), so the parabola opens upward. The vertex form is \( 2(x - 4)^2 \), so the vertex is at \( (4, 0) \).

Step2: Analyze \( g(x)=-2x^2 - 8x - 8 \)

The coefficient of \( x^2 \) is \( -2<0 \), so the parabola opens downward. The vertex form is \( -2(x + 2)^2 \), so the vertex is at \( (-2, 0) \).

Step3: Analyze \( h(x)=-x^2 - 5 \)

The coefficient of \( x^2 \) is \( -1<0 \), so the parabola opens downward. The vertex is at \( (0, -5) \) (since there is no \( x \)-term, the axis of symmetry is \( x = 0 \)).

Step4: Analyze \( k(x)=-x^2 - 11x - 28 \)

The coefficient of \( x^2 \) is \( -1<0 \), so the parabola opens downward. Factored form is \( -(x + 7)(x + 4) \), so the roots are at \( x=-7 \) and \( x = -4 \), and the vertex is at \( x=\frac{-7-4}{2}=-\frac{11}{2}=-5.5 \), \( y=-(-5.5)^2-11(-5.5)-28 = 4.25 \) (vertex formula \( x=-\frac{b}{2a} \), \( y = k(-\frac{b}{2a}) \)).

Now, match each function to its graph:

• \( f(x) \) opens upward, vertex at \( (4,0) \): corresponds to the bottom - left graph (the upward - opening parabola with vertex around \( x = 4 \)).
• \( g(x) \) opens downward, vertex at \( (-2,0) \): corresponds to the top - right graph (the downward - opening parabola with vertex at \( x=-2 \), \( y = 0 \)).
• \( h(x) \) opens downward, vertex at \( (0,-5) \): corresponds to the bottom - right graph (the downward - opening parabola with vertex at \( (0,-5) \)).
• \( k(x) \) opens downward, roots at \( x=-7,-4 \), vertex at \( x=-5.5,y = 4.25 \): corresponds to the top - left graph (the downward - opening parabola with roots around \( x=-7,-4 \)).

Answer:

• \( f(x)=2x^2 - 16x + 32 \): The bottom - left graph.
• \( g(x)=-2x^2 - 8x - 8 \): The top - right graph.
• \( h(x)=-x^2 - 5 \): The bottom - right graph.
• \( k(x)=-x^2 - 11x - 28 \): The top - left graph.