Question
Wrics 3 Quarter CC \#5 \( f(x)=2^{\mathrm{x}} \) after a vertical stretch by a factor of 3 and a reflection across the \( x \)-axis. Graph both functions. Writion \( g(x) \) that represents the exponential function
Ask by Reeves Lowe. in the United States
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( g(x) = -3 \cdot 2^x \).
Solution
It seems that the system does not currently support plotting functions. However, I can describe how to graph both functions.
### Steps to Graph the Functions
1. **Graph \( f(x) = 2^x \)**:
- This is an exponential growth function.
- It passes through the point (0, 1) because \( 2^0 = 1 \).
- As \( x \) increases, \( f(x) \) increases rapidly.
- As \( x \) decreases, \( f(x) \) approaches 0 but never touches the x-axis (asymptote at \( y = 0 \)).
2. **Graph \( g(x) = -3 \cdot 2^x \)**:
- This function is a reflection of \( f(x) \) across the x-axis and vertically stretched by a factor of 3.
- It passes through the point (0, -3) because \( g(0) = -3 \cdot 2^0 = -3 \).
- As \( x \) increases, \( g(x) \) decreases rapidly (since it is negative).
- As \( x \) decreases, \( g(x) \) approaches 0 from below (asymptote at \( y = 0 \)).
### Summary of the Graphs
- **\( f(x) = 2^x \)**: Exponential growth, passing through (0, 1), approaching 0 as \( x \) decreases.
- **\( g(x) = -3 \cdot 2^x \)**: Exponential decay (reflected), passing through (0, -3), approaching 0 as \( x \) decreases.
You can use graphing software or a graphing calculator to visualize these functions based on the descriptions provided. If you have any further questions or need additional assistance, feel free to ask!
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Simplify this solution Extra Insights
To understand how the transformations affect the function \( f(x) = 2^x \), first, we apply the vertical stretch by a factor of 3, resulting in \( 3f(x) = 3 \cdot 2^x \). Then, reflecting this function across the \( x \)-axis gives us \( g(x) = -3 \cdot 2^x \). To visualize this, plot \( f(x) = 2^x \) which steadily rises, and then graph \( g(x) = -3 \cdot 2^x \). You'll notice that \( g(x) \) starts at 0 and dives downward, creating an interesting contrast between the original function and its transformed version. Draw some key points, like the intercepts and asymptotes, to make your graph clearer!
