Question
The exponential function f(x) = 2* undergoes
two transformations to g(x) = ⅓ • 2* - 7. How does the graph change? Select all that apply:
• It is shifted right.
• It is shifted down.
• It is vertically compressed.
• It is vertically stretched.
• It is flipped over the x-axis.
two transformations to g(x) = ⅓ • 2* - 7. How does the graph change? Select all that apply:
• It is shifted right.
• It is shifted down.
• It is vertically compressed.
• It is vertically stretched.
• It is flipped over the x-axis.
Answers
Answered by
GPT-4o mini
To analyze the transformations applied to the exponential function \( f(x) = 2^x \) to obtain \( g(x) = \frac{1}{3} \cdot 2^x - 7 \), we can break down the changes step by step:
1. **Vertical Compression**: The factor of \( \frac{1}{3} \) in front of \( 2^x \) indicates that the graph is vertically compressed by a factor of \( 3 \). This means that for any value of \( x \), \( g(x) \) will be \(\frac{1}{3}\) the height of \( f(x) \).
2. **Vertical Shift Down**: The subtraction of \( 7 \) from \( \frac{1}{3} \cdot 2^x \) indicates a vertical shift downwards by \( 7 \) units. This means the entire graph shifts down along the y-axis.
Now, let's evaluate the answer options:
- **It is shifted right.** (False; there is no transformation that shifts the graph to the right.)
- **It is shifted down.** (True; the graph is shifted down by \( 7 \).)
- **It is vertically compressed.** (True; the graph is vertically compressed by a factor of \( \frac{1}{3} \).)
- **It is vertically stretched.** (False; the graph is compressed, not stretched.)
- **It is flipped over the x-axis.** (False; flipping would involve multiplying by a negative factor, which is not the case here.)
In conclusion, the correct transformations that apply to the graph of \( f(x) = 2^x \) to get \( g(x) = \frac{1}{3} \cdot 2^x - 7 \) are:
- It is shifted down.
- It is vertically compressed.
1. **Vertical Compression**: The factor of \( \frac{1}{3} \) in front of \( 2^x \) indicates that the graph is vertically compressed by a factor of \( 3 \). This means that for any value of \( x \), \( g(x) \) will be \(\frac{1}{3}\) the height of \( f(x) \).
2. **Vertical Shift Down**: The subtraction of \( 7 \) from \( \frac{1}{3} \cdot 2^x \) indicates a vertical shift downwards by \( 7 \) units. This means the entire graph shifts down along the y-axis.
Now, let's evaluate the answer options:
- **It is shifted right.** (False; there is no transformation that shifts the graph to the right.)
- **It is shifted down.** (True; the graph is shifted down by \( 7 \).)
- **It is vertically compressed.** (True; the graph is vertically compressed by a factor of \( \frac{1}{3} \).)
- **It is vertically stretched.** (False; the graph is compressed, not stretched.)
- **It is flipped over the x-axis.** (False; flipping would involve multiplying by a negative factor, which is not the case here.)
In conclusion, the correct transformations that apply to the graph of \( f(x) = 2^x \) to get \( g(x) = \frac{1}{3} \cdot 2^x - 7 \) are:
- It is shifted down.
- It is vertically compressed.