Question

match each quadratic function to its graph. \\( f(x) = x^2 + 14x + 44 \\) \\( g(x) = -x^2 + 2x + 5 \\)

Explanation:

Step1: Analyze \( f(x) = x^2 + 14x + 44 \)

The coefficient of \( x^2 \) is positive (\( 1>0 \)), so the parabola opens upward. Let's find the vertex. The x - coordinate of the vertex of a quadratic \( ax^2+bx + c \) is \( x=-\frac{b}{2a} \). For \( f(x) \), \( a = 1 \), \( b=14 \), so \( x=-\frac{14}{2\times1}=-7 \). Substitute \( x = - 7 \) into \( f(x) \): \( f(-7)=(-7)^2+14\times(-7)+44=49 - 98 + 44=-5 \). The vertex is \( (-7,-5) \), and the parabola opens upward, so it should match the left - hand graph.

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

The coefficient of \( x^2 \) is negative (\( - 1<0 \)), so the parabola opens downward. The x - coordinate of the vertex is \( x =-\frac{b}{2a}=-\frac{2}{2\times(-1)} = 1 \). Substitute \( x = 1 \) into \( g(x) \): \( g(1)=-(1)^2+2\times1 + 5=-1 + 2+5 = 6 \). The vertex is \( (1,6) \), and the parabola opens downward, so it should match the right - hand graph.

Answer:

\( f(x)=x^2 + 14x + 44 \) matches the left - hand graph (with vertex at \( (-7,-5) \) and opening upward). \( g(x)=-x^2 + 2x + 5 \) matches the right - hand graph (with vertex at \( (1,6) \) and opening downward).