Question

graph the function with a graphing calculator using a standard viewing window. state whether the graph has a turning point in this window.

$y = \frac{6}{x^2 + 2}$

choose the correct graph below.

\\(\bigcirc\\) a.
\\(\bigcirc\\) b.
\\(\bigcirc\\) c.
\\(\bigcirc\\) d.
(images of graphs a, b, c, d are shown with magnifying and other icons)

Explanation:

Step1: Analyze function behavior at x=0

Substitute $x=0$ into $y=\frac{6}{x^2 + 2}$:
$y=\frac{6}{0^2 + 2}=3$
So the graph passes through $(0, 3)$.

Step2: Analyze end behavior

As $x\to\pm\infty$, $x^2\to\infty$, so $x^2+2\to\infty$. Thus:
$\lim_{x\to\pm\infty} \frac{6}{x^2 + 2}=0$
The graph approaches $y=0$ as $x$ goes to $\pm\infty$.

Step3: Check for turning points

Find the derivative using the quotient rule: $y'=\frac{-12x}{(x^2 + 2)^2}$. Set $y'=0$:
$\frac{-12x}{(x^2 + 2)^2}=0 \implies x=0$
The only critical point is at $x=0$, which is a maximum (since the function decreases as $x$ moves away from 0 in either direction, and the value at $x=0$ is the highest point).

Step4: Match to graph options

Only option D has a maximum at $(0, 3)$, approaches $y=0$ as $x\to\pm\infty$, and has a turning point at $x=0$.

Answer:

D. <The graph with a maximum at the y-axis, approaching the x-axis on both ends>
The graph has a turning point (a maximum) in the standard viewing window at $(0, 3)$.