Question

match each quadratic function to its graph.
$f(x) = x^2 - 18x + 81 = (x - 9)^2$
$g(x) = 2x^2 + 28x + 90 = 2(x + 9)(x + 5)$
$f(x) = x^2 - 18x + 81$ $g(x) = 2x^2 + 28x + 90$

Explanation:

Step1: Analyze \( f(x) = (x - 9)^2 \)

The vertex form of a quadratic is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For \( f(x)=(x - 9)^2 \), \( a = 1 \) (opens upward), vertex at \((9, 0)\). So its graph should have the vertex on the x - axis at \( x = 9 \), which matches the right - hand graph (since the right graph has its vertex near \( x = 9,y = 0 \)).

Step2: Analyze \( g(x)=2(x + 9)(x + 5) \)

First, find the roots by setting \( g(x)=0 \). \( 2(x + 9)(x + 5)=0\Rightarrow x=-9 \) or \( x=-5 \). The parabola opens upward (since \( a = 2>0 \)) and has roots at \( x=-9 \) and \( x=-5 \), so its vertex is between \( x=-9 \) and \( x=-5 \) (midpoint \( x=\frac{-9 + (-5)}{2}=-7 \)), and it should match the left - hand graph (which has roots around \( x=-9 \) and \( x=-5 \)).

Answer:

\( f(x)=x^{2}-18x + 81=(x - 9)^{2} \) matches the right - hand graph; \( g(x)=2x^{2}+28x + 90 = 2(x + 9)(x + 5) \) matches the left - hand graph.