To graph the equation [tex]\( y = \frac{2}{x} \)[/tex] and find the corresponding [tex]\( y \)[/tex]-values for given [tex]\( x \)[/tex]-values, we will follow these steps:
1. Insert each [tex]\( x \)[/tex]-value into the equation to find the [tex]\( y \)[/tex]-value.
2. Record each calculated [tex]\( y \)[/tex]-value in the table.
Given [tex]\( x \)[/tex]-values: [tex]\( -2, -1, -\frac{1}{2}, \frac{1}{2}, 1, 2 \)[/tex].
### Step-by-step Calculation:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = \frac{2}{-2} = -1
\][/tex]
So, [tex]\( y = -1 \)[/tex].
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = \frac{2}{-1} = -2
\][/tex]
So, [tex]\( y = -2 \)[/tex].
3. For [tex]\( x = -\frac{1}{2} \)[/tex]:
[tex]\[
y = \frac{2}{-\frac{1}{2}} = -4
\][/tex]
So, [tex]\( y = -4 \)[/tex].
4. For [tex]\( x = \frac{1}{2} \)[/tex]:
[tex]\[
y = \frac{2}{\frac{1}{2}} = 4
\][/tex]
So, [tex]\( y = 4 \)[/tex].
5. For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = \frac{2}{1} = 2
\][/tex]
So, [tex]\( y = 2 \)[/tex].
6. For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = \frac{2}{2} = 1
\][/tex]
So, [tex]\( y = 1 \)[/tex].
Now, let's fill in the table with these values:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-2 & -1 \\
\hline
-1 & -2 \\
\hline
$-\frac{1}{2}$ & -4 \\
\hline
$\frac{1}{2}$ & 4 \\
\hline
1 & 2 \\
\hline
2 & 1 \\
\hline
\end{tabular}
\][/tex]
By plotting these points [tex]\((-2, -1)\)[/tex], [tex]\((-1, -2)\)[/tex], [tex]\(\left(-\frac{1}{2}, -4\right)\)[/tex], [tex]\(\left(\frac{1}{2}, 4\right)\)[/tex], [tex]\((1, 2)\)[/tex], and [tex]\((2, 1)\)[/tex] on a graph, and drawing the curve through them, we will see the shape of the hyperbola representing the equation [tex]\( y = \frac{2}{x} \)[/tex]. This graph will have two branches, one in the first and third quadrants, and the other in the second and fourth quadrants, asymptotically approaching the x-axis and y-axis but never touching them.
