To graph the reflection of the function [tex]\( f(x) = -3(2)^x \)[/tex] across the [tex]\( x \)[/tex]-axis, we need to follow these steps:
### Step 1: Determine the Reflected Function
Reflection across the [tex]\( x \)[/tex]-axis changes the sign of the function [tex]\( f(x) \)[/tex]. Thus, if [tex]\( f(x) = -3(2)^x \)[/tex], the reflected function will be:
[tex]\[
g(x) = -f(x) = -(-3(2)^x) = 3(2)^x
\][/tex]
So the reflected function is:
[tex]\[
g(x) = 3(2)^x
\][/tex]
### Step 2: Create a Table of Values for [tex]\( g(x) \)[/tex]
We will select a range of [tex]\( x \)[/tex] values and calculate the corresponding [tex]\( y \)[/tex] values for [tex]\( g(x) = 3(2)^x \)[/tex]. Let's use the [tex]\( x \)[/tex] values [tex]\(-2, -1, 0, 1, 2\)[/tex].
1. [tex]\( x = -2 \)[/tex]
[tex]\[
g(-2) = 3 (2)^{-2} = 3 \left(\frac{1}{4}\right) = \frac{3}{4} = 0.75
\][/tex]
2. [tex]\( x = -1 \)[/tex]
[tex]\[
g(-1) = 3 (2)^{-1} = 3 \left(\frac{1}{2}\right) = \frac{3}{2} = 1.5
\][/tex]
3. [tex]\( x = 0 \)[/tex]
[tex]\[
g(0) = 3 (2)^0 = 3 (1) = 3
\][/tex]
4. [tex]\( x = 1 \)[/tex]
[tex]\[
g(1) = 3 (2)^1 = 3 (2) = 6
\][/tex]
5. [tex]\( x = 2 \)[/tex]
[tex]\[
g(2) = 3 (2)^2 = 3 (4) = 12
\][/tex]
### Step 3: Complete the Table
Let's arrange these values in a table format:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-2 & 0.75 \\
-1 & 1.5 \\
0 & 3 \\
1 & 6 \\
2 & 12 \\
\hline
\end{tabular}
\][/tex]
### Step 4: Check the Reflection
By comparing the table of values for the original function [tex]\( f(x) \)[/tex] with the table of values for the reflected function [tex]\( g(x) \)[/tex], we can verify that all the [tex]\( y \)[/tex]-values of [tex]\( g(x) \)[/tex] represent the negative of the [tex]\( y \)[/tex]-values of [tex]\( f(x) \)[/tex].
Here is the reflection of [tex]\( f(x) = -3(2)^x \)[/tex]:
[tex]\[
g(x) = 3(2)^x
\][/tex]
with the table completed as shown.
