Answer :
To sketch the graph of [tex]\( g(x) = -\frac{1}{2}(3^x) + 3 \)[/tex] using transformations, let's break down the transformations applied to the base function [tex]\( f(x) = 3^x \)[/tex].
### Step-by-Step Transformations
1. Start with the base function [tex]\( f(x) = 3^x \)[/tex].
2. Vertical Scaling and Reflection:
- The function is scaled vertically by [tex]\( -\frac{1}{2} \)[/tex]. This means every value of the function [tex]\( 3^x \)[/tex] is multiplied by [tex]\( -\frac{1}{2} \)[/tex].
- This will both compress the graph vertically and reflect it across the x-axis. The new function can be written as [tex]\( g_1(x) = -\frac{1}{2}(3^x) \)[/tex].
3. Vertical Translation:
- Finally, add 3 to the result of the previous step, which shifts the entire graph upwards by 3 units. This gives us the final function [tex]\( g(x) = -\frac{1}{2}(3^x) + 3 \)[/tex].
### Calculating Key Points
Let's find the values of [tex]\( g(x) \)[/tex] for some key points:
- When [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = -\frac{1}{2}(3^{-2}) + 3 = -\frac{1}{2} \cdot \frac{1}{9} + 3 = -\frac{1}{18} + 3 \approx 2.944 \][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = -\frac{1}{2}(3^{-1}) + 3 = -\frac{1}{2} \cdot \frac{1}{3} + 3 = -\frac{1}{6} + 3 \approx 2.833 \][/tex]
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -\frac{1}{2}(3^0) + 3 = -\frac{1}{2} \cdot 1 + 3 = -\frac{1}{2} + 3 = 2.5 \][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -\frac{1}{2}(3^1) + 3 = -\frac{1}{2} \cdot 3 + 3 = -1.5 + 3 = 1.5 \][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = -\frac{1}{2}(3^2) + 3 = -\frac{1}{2} \cdot 9 + 3 = -4.5 + 3 = -1.5 \][/tex]
### Analyzing the Transformation Effect
- Asymptote: The horizontal asymptote of the original function [tex]\( 3^x \)[/tex] is [tex]\( y = 0 \)[/tex]. After the vertical transformation and translation, the new horizontal asymptote for [tex]\( g(x) \)[/tex] becomes [tex]\( y = 3 \)[/tex].
- Behavior: As [tex]\( x \to \infty \)[/tex], [tex]\( 3^x \)[/tex] grows very large. Scaling it by [tex]\(-\frac{1}{2}\)[/tex] will make it very negative, and shifting it by 3 will result in the function approaching [tex]\( y = 3 - \infty \)[/tex], remaining negative. Thus, [tex]\( g(x) \)[/tex] will tend towards negative infinity as [tex]\( x \)[/tex] increases.
### Summary: Plotting the Points
Using the key points and transformation understanding:
- [tex]\( (-2, 2.944) \)[/tex]
- [tex]\( (-1, 2.833) \)[/tex]
- [tex]\( (0, 2.5) \)[/tex]
- [tex]\( (1, 1.5) \)[/tex]
- [tex]\( (2, -1.5) \)[/tex]
Next, draw the transformed curve by connecting these points and ensuring the curve reflects the vertical scaling, reflection, and upward shift.
### Sketch the Graph
The graph will:
- Pass through the calculated points.
- Be downward-sloping due to the reflection.
- Approach the horizontal asymptote [tex]\( y = 3 \)[/tex] as [tex]\( x \rightarrow -\infty \)[/tex].
- Plummet towards negative infinity as [tex]\( x \rightarrow \infty \)[/tex].
### Conclusion
The transformations serve as a systematic way of sketching [tex]\( g(x) = -\frac{1}{2}(3^x) + 3 \)[/tex], ensuring accuracy and understanding of exponential function behavior.
### Step-by-Step Transformations
1. Start with the base function [tex]\( f(x) = 3^x \)[/tex].
2. Vertical Scaling and Reflection:
- The function is scaled vertically by [tex]\( -\frac{1}{2} \)[/tex]. This means every value of the function [tex]\( 3^x \)[/tex] is multiplied by [tex]\( -\frac{1}{2} \)[/tex].
- This will both compress the graph vertically and reflect it across the x-axis. The new function can be written as [tex]\( g_1(x) = -\frac{1}{2}(3^x) \)[/tex].
3. Vertical Translation:
- Finally, add 3 to the result of the previous step, which shifts the entire graph upwards by 3 units. This gives us the final function [tex]\( g(x) = -\frac{1}{2}(3^x) + 3 \)[/tex].
### Calculating Key Points
Let's find the values of [tex]\( g(x) \)[/tex] for some key points:
- When [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = -\frac{1}{2}(3^{-2}) + 3 = -\frac{1}{2} \cdot \frac{1}{9} + 3 = -\frac{1}{18} + 3 \approx 2.944 \][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = -\frac{1}{2}(3^{-1}) + 3 = -\frac{1}{2} \cdot \frac{1}{3} + 3 = -\frac{1}{6} + 3 \approx 2.833 \][/tex]
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -\frac{1}{2}(3^0) + 3 = -\frac{1}{2} \cdot 1 + 3 = -\frac{1}{2} + 3 = 2.5 \][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -\frac{1}{2}(3^1) + 3 = -\frac{1}{2} \cdot 3 + 3 = -1.5 + 3 = 1.5 \][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = -\frac{1}{2}(3^2) + 3 = -\frac{1}{2} \cdot 9 + 3 = -4.5 + 3 = -1.5 \][/tex]
### Analyzing the Transformation Effect
- Asymptote: The horizontal asymptote of the original function [tex]\( 3^x \)[/tex] is [tex]\( y = 0 \)[/tex]. After the vertical transformation and translation, the new horizontal asymptote for [tex]\( g(x) \)[/tex] becomes [tex]\( y = 3 \)[/tex].
- Behavior: As [tex]\( x \to \infty \)[/tex], [tex]\( 3^x \)[/tex] grows very large. Scaling it by [tex]\(-\frac{1}{2}\)[/tex] will make it very negative, and shifting it by 3 will result in the function approaching [tex]\( y = 3 - \infty \)[/tex], remaining negative. Thus, [tex]\( g(x) \)[/tex] will tend towards negative infinity as [tex]\( x \)[/tex] increases.
### Summary: Plotting the Points
Using the key points and transformation understanding:
- [tex]\( (-2, 2.944) \)[/tex]
- [tex]\( (-1, 2.833) \)[/tex]
- [tex]\( (0, 2.5) \)[/tex]
- [tex]\( (1, 1.5) \)[/tex]
- [tex]\( (2, -1.5) \)[/tex]
Next, draw the transformed curve by connecting these points and ensuring the curve reflects the vertical scaling, reflection, and upward shift.
### Sketch the Graph
The graph will:
- Pass through the calculated points.
- Be downward-sloping due to the reflection.
- Approach the horizontal asymptote [tex]\( y = 3 \)[/tex] as [tex]\( x \rightarrow -\infty \)[/tex].
- Plummet towards negative infinity as [tex]\( x \rightarrow \infty \)[/tex].
### Conclusion
The transformations serve as a systematic way of sketching [tex]\( g(x) = -\frac{1}{2}(3^x) + 3 \)[/tex], ensuring accuracy and understanding of exponential function behavior.
