Let's find the corresponding [tex]$y$[/tex]-values for the given [tex]$x$[/tex]-values using the equation [tex]\( y = \frac{4}{x} \)[/tex].
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = \frac{4}{-2} = -2.0
\][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = \frac{4}{-1} = -4.0
\][/tex]
3. For [tex]\( x = -\frac{1}{2} \)[/tex]:
[tex]\[
y = \frac{4}{-\frac{1}{2}} = -8.0
\][/tex]
4. For [tex]\( x = \frac{1}{2} \)[/tex]:
[tex]\[
y = \frac{4}{\frac{1}{2}} = 8.0
\][/tex]
5. For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = \frac{4}{1} = 4.0
\][/tex]
6. For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = \frac{4}{2} = 2.0
\][/tex]
Now we can summarize our results in the table:
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
-2 & -2.0 \\
\hline
-1 & -4.0 \\
\hline
-0.5 & -8.0 \\
\hline
0.5 & 8.0 \\
\hline
1 & 4.0 \\
\hline
2 & 2.0 \\
\hline
\end{tabular}
To graph these points, we plot each [tex]$(x, y)$[/tex] pair on a coordinate plane.
1. [tex]$(-2, -2.0)$[/tex]
2. [tex]$(-1, -4.0)$[/tex]
3. [tex]$(-0.5, -8.0)$[/tex]
4. [tex]$(0.5, 8.0)$[/tex]
5. [tex]$(1, 4.0)$[/tex]
6. [tex]$(2, 2.0)$[/tex]
These points should help determine the shape of the graph for the equation [tex]\( y = \frac{4}{x} \)[/tex]. This type of equation represents a hyperbola with asymptotes at [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex].
We identified the [tex]$y$[/tex]-value for [tex]\( x = -2 \)[/tex] as [tex]\(-2.0\)[/tex], and similarly found values for all the given [tex]$x$[/tex]-values. Verifying these values on a graph, we should see a curve approaching the vertical and horizontal axes (the asymptotes) but never touching them.
