Question

match each quadratic function to its graph.
$f(x) = 2x^{2} + 26x + 80 = 2(x + 8)(x + 5)$
$g(x) = x^{2} + 18x + 81 = (x + 9)^{2}$
$f(x) = 2x^{2} + 26x + 80$ $g(x) = x^{2} + 18x + 81$

Explanation:

Step1: Find roots of $f(x)$

From factored form $f(x)=2(x+8)(x+5)$, set to 0:
$x+8=0 \implies x=-8$; $x+5=0 \implies x=-5$

Step2: Find vertex of $f(x)$

Vertex $x$-coordinate: $\frac{-8 + (-5)}{2}=-6.5$
Substitute $x=-6.5$: $f(-6.5)=2(-6.5+8)(-6.5+5)=2(1.5)(-1.5)=-4.5$
Vertex: $(-6.5, -4.5)$

Step3: Find root/vertex of $g(x)$

From $g(x)=(x+9)^2$, set to 0: $x=-9$ (double root, vertex at $(-9,0)$)

Step4: Match to graphs

Left graph has roots at $x=-8, x=-5$ and vertex below x-axis; right graph touches x-axis at $x=-9$.

Answer:

• $f(x)=2x^2 + 26x + 80$ matches the left graph (with vertex at $(-6.5, -4.5)$ and x-intercepts at $x=-8$ and $x=-5$)
• $g(x)=x^2 + 18x + 81$ matches the right graph (with vertex and x-intercept at $(-9, 0)$)