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To graph the reflection of the function [tex]\( f(x) = -3 (2)^x \)[/tex] across the x-axis, we can follow several steps carefully.
### Step 1: Determine the Reflected Function
When reflecting a function across the x-axis, each value [tex]\( y \)[/tex] of the function [tex]\( f(x) \)[/tex] will change its sign. Thus, given [tex]\( f(x) = -3 (2)^x \)[/tex], the reflected function [tex]\( g(x) \)[/tex] can be calculated as follows:
[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: Calculate the Initial Value of the Reflected Function
Now, let's find the initial value of the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 3 (2)^0 = 3 \][/tex]
So, the initial value [tex]\( g(0) \)[/tex] is [tex]\( 3 \)[/tex].
### Step 3: Generate Some Points for the Reflected Function
To graph the function, it's helpful to calculate several points. We can use [tex]\( x \)[/tex] values ranging from [tex]\( -2 \)[/tex] to [tex]\( 2 \)[/tex] for demonstration:
For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = 3 (2)^{-2} = 3 \cdot \frac{1}{4} = 0.75 \][/tex]
For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = 3 (2)^{-1} = 3 \cdot \frac{1}{2} = 1.5 \][/tex]
For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 3 (2)^0 = 3 \][/tex]
For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 3 (2)^1 = 3 \cdot 2 = 6 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 3 (2)^2 = 3 \cdot 4 = 12 \][/tex]
Now we can summarize these calculations in a table:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y = g(x)$ \\ \hline -2 & 0.75 \\ \hline -1 & 1.5 \\ \hline 0 & 3 \\ \hline 1 & 6 \\ \hline 2 & 12 \\ \hline \end{tabular} \][/tex]
### Summary of Points
We have generated the following points for the function [tex]\( g(x) = 3 (2)^x \)[/tex]:
- At [tex]\( x = -2 \)[/tex], [tex]\( y = 0.75 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( y = 1.5 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( y = 6 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( y = 12 \)[/tex]
These points can now be plotted on the coordinate plane to graph the reflected function [tex]\( g(x) = 3 (2)^x \)[/tex].
### Step 1: Determine the Reflected Function
When reflecting a function across the x-axis, each value [tex]\( y \)[/tex] of the function [tex]\( f(x) \)[/tex] will change its sign. Thus, given [tex]\( f(x) = -3 (2)^x \)[/tex], the reflected function [tex]\( g(x) \)[/tex] can be calculated as follows:
[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: Calculate the Initial Value of the Reflected Function
Now, let's find the initial value of the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 3 (2)^0 = 3 \][/tex]
So, the initial value [tex]\( g(0) \)[/tex] is [tex]\( 3 \)[/tex].
### Step 3: Generate Some Points for the Reflected Function
To graph the function, it's helpful to calculate several points. We can use [tex]\( x \)[/tex] values ranging from [tex]\( -2 \)[/tex] to [tex]\( 2 \)[/tex] for demonstration:
For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = 3 (2)^{-2} = 3 \cdot \frac{1}{4} = 0.75 \][/tex]
For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = 3 (2)^{-1} = 3 \cdot \frac{1}{2} = 1.5 \][/tex]
For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 3 (2)^0 = 3 \][/tex]
For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 3 (2)^1 = 3 \cdot 2 = 6 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 3 (2)^2 = 3 \cdot 4 = 12 \][/tex]
Now we can summarize these calculations in a table:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y = g(x)$ \\ \hline -2 & 0.75 \\ \hline -1 & 1.5 \\ \hline 0 & 3 \\ \hline 1 & 6 \\ \hline 2 & 12 \\ \hline \end{tabular} \][/tex]
### Summary of Points
We have generated the following points for the function [tex]\( g(x) = 3 (2)^x \)[/tex]:
- At [tex]\( x = -2 \)[/tex], [tex]\( y = 0.75 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( y = 1.5 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( y = 6 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( y = 12 \)[/tex]
These points can now be plotted on the coordinate plane to graph the reflected function [tex]\( g(x) = 3 (2)^x \)[/tex].
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