Question

match each quadratic function to its graph.
$f(x) = x^2 + 14x + 55$
$g(x) = -x^2 + 2x - 6$

Explanation:

Step1: Identify parabola direction

For $f(x)=x^2+14x+55$, the coefficient of $x^2$ is $1>0$, so it opens upward.
For $g(x)=-x^2+2x-6$, the coefficient of $x^2$ is $-1<0$, so it opens downward.

Step2: Find vertex of $f(x)$

Use vertex formula $x=-\frac{b}{2a}$ for $ax^2+bx+c$.
$x=-\frac{14}{2\times1}=-7$
Substitute $x=-7$ into $f(x)$:
$f(-7)=(-7)^2+14\times(-7)+55=49-98+55=6$
Vertex of $f(x)$ is $(-7,6)$.

Step3: Find vertex of $g(x)$

Use vertex formula $x=-\frac{b}{2a}$:
$x=-\frac{2}{2\times(-1)}=1$
Substitute $x=1$ into $g(x)$:
$g(1)=-(1)^2+2\times1-6=-1+2-6=-5$
Vertex of $g(x)$ is $(1,-5)$.

Step4: Match to graphs

The bottom graph opens upward, matching $f(x)$'s direction, and its vertex is in the positive x,y region, consistent with $(-7,6)$ (left upper area).
The top graph opens downward, matching $g(x)$'s direction, and its vertex is in the negative x,y region, consistent with $(1,-5)$ (right lower area).

Answer:

1. $f(x) = x^2 + 14x + 55$ matches the bottom (right upper) graph.
2. $g(x) = -x^2 + 2x - 6$ matches the top (left lower) graph.