Certainly! Let's start by completing the table for the function [tex]\( y = 0.1^x \)[/tex].
First, we will calculate [tex]\( y \)[/tex] for each given value of [tex]\( x \)[/tex].
1. When [tex]\( x = -2 \)[/tex]:
[tex]\[
y = 0.1^{-2}
\][/tex]
Recall that [tex]\( 0.1^{-2} = (0.1)^{-2} = \left(\frac{1}{10}\right)^{-2} = 10^2 = 100 \)[/tex].
Therefore, [tex]\( y = 100 \)[/tex].
2. When [tex]\( x = -1 \)[/tex]:
[tex]\[
y = 0.1^{-1}
\][/tex]
Recall that [tex]\( 0.1^{-1} = (0.1)^{-1} = \left(\frac{1}{10}\right)^{-1} = 10 \)[/tex].
Therefore, [tex]\( y = 10 \)[/tex].
3. When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 0.1^0
\][/tex]
We know any number raised to the power of 0 is 1.
Therefore, [tex]\( y = 1 \)[/tex].
4. When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 0.1^1
\][/tex]
We know that any number raised to the power of 1 is the number itself.
Therefore, [tex]\( y = 0.1 \)[/tex].
Now the table is completed as follows:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-2 & 100 \\
\hline
-1 & 10 \\
\hline
0 & 1 \\
\hline
1 & 0.1 \\
\hline
\end{tabular}
\][/tex]
Next, we need to graph the function [tex]\( y = 0.1^x \)[/tex].
To plot the function, we plot the points [tex]\((-2, 100)\)[/tex], [tex]\((-1, 10)\)[/tex], [tex]\((0, 1)\)[/tex], and [tex]\((1, 0.1)\)[/tex].
Here’s a step-by-step guide to sketching the graph manually:
1. Draw the coordinate axes:
Label the [tex]\( x \)[/tex]-axis and the [tex]\( y \)[/tex]-axis.
2. Plot each point:
- For [tex]\( (-2, 100) \)[/tex], place a point far above the origin, as [tex]\( y = 100 \)[/tex] is a large value.
- For [tex]\( (-1, 10) \)[/tex], place a point above the origin at [tex]\( x = -1 \)[/tex] and [tex]\( y = 10 \)[/tex].
- For [tex]\( (0, 1) \)[/tex], place a point at the origin’s immediate right, as [tex]\( y = 1 \)[/tex].
- For [tex]\( (1, 0.1) \)[/tex], place a point slightly above the [tex]\( x \)[/tex]-axis at [tex]\( x = 1 \)[/tex].
3. Draw a smooth curve:
Connect these points with a smooth and continuous curve, ensuring that the curve decreases rapidly as [tex]\( x \)[/tex] increases and increases rapidly as [tex]\( x \)[/tex] decreases.
4. Label the graph:
Add a title “Graph of [tex]\( y = 0.1^x \)[/tex]” and label the respective axes.
The resulting graph will show an exponential decay function. The function has a steep decline on the positive side of the [tex]\( x \)[/tex]-axis and surges upward very rapidly on the negative side of the [tex]\( x \)[/tex]-axis.
