Asked by 54t4n45
Analyze the effects on the graph of f(x)=ex
when it is replaced by af(x)
and a is greater than one. Which of the following statements best describes the effects on the graph for af(x)
?(1 point)
Responses
The graph of af(x)
has the same horizontal asymptote as f(x)
.
The graph of a f left parenthesis x right parenthesis has the same horizontal asymptote as f left parenthesis x right parenthesis .
The graph of af(x)
has a y-intercept at (0,0)
.
The graph of a f left parenthesis x right parenthesis has a y -intercept at left parenthesis 0 comma 0 right parenthesis .
The graph of af(x)
has been reflected with respect to the x-axis.
The graph of a f left parenthesis x right parenthesis has been reflected with respect to the x -axis.
The graph of af(x)
has been vertically compressed by a factor of a.
The graph of a f left parenthesis x right parenthesis has been vertically compressed by a factor of a .
when it is replaced by af(x)
and a is greater than one. Which of the following statements best describes the effects on the graph for af(x)
?(1 point)
Responses
The graph of af(x)
has the same horizontal asymptote as f(x)
.
The graph of a f left parenthesis x right parenthesis has the same horizontal asymptote as f left parenthesis x right parenthesis .
The graph of af(x)
has a y-intercept at (0,0)
.
The graph of a f left parenthesis x right parenthesis has a y -intercept at left parenthesis 0 comma 0 right parenthesis .
The graph of af(x)
has been reflected with respect to the x-axis.
The graph of a f left parenthesis x right parenthesis has been reflected with respect to the x -axis.
The graph of af(x)
has been vertically compressed by a factor of a.
The graph of a f left parenthesis x right parenthesis has been vertically compressed by a factor of a .
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To understand the effects of replacing \( f(x) = e^x \) with \( af(x) \) where \( a > 1 \), let's analyze what this does to the graph of \( f(x) \).
1. **Vertical Stretch**: When the function \( f(x) \) is multiplied by a constant factor \( a \) (where \( a > 1 \)), the graph of the function is vertically stretched. This means that for every \( x \), the value of \( af(x) = ae^x \) will be greater than the corresponding \( f(x) = e^x \).
2. **Horizontal Asymptote**: The horizontal asymptote of the original function \( f(x) = e^x \) is \( y = 0 \) as \( x \to -\infty \). The asymptote remains the same for \( af(x) = ae^x \) because as \( x \to -\infty \), \( ae^x \) also approaches 0. Thus, \( af(x) \) has the same horizontal asymptote of \( y = 0 \).
3. **Y-Intercept**: The y-intercept of the original function \( f(x) = e^x \) is at \( (0, 1) \) because \( e^0 = 1 \). For \( af(x) \), the y-intercept becomes \( a \cdot e^0 = a \cdot 1 = a \), which will not be at (0,0) unless \( a = 0 \), which is not the case here since \( a > 1 \).
4. **Reflection**: There is no reflection with respect to the x-axis since we are multiplying by a positive factor \( a \).
5. **Vertical Compression**: The function \( af(x) = ae^x \) is not vertically compressed; instead, it is vertically stretched by a factor \( a \).
With this analysis in mind, the correct statement that best describes the effects on the graph of \( af(x) \) is:
**The graph of \( af(x) \) has the same horizontal asymptote as \( f(x) \).**
This is reflective of the fact that both functions approach the same horizontal asymptote, \( y = 0 \), as \( x \to -\infty \).
1. **Vertical Stretch**: When the function \( f(x) \) is multiplied by a constant factor \( a \) (where \( a > 1 \)), the graph of the function is vertically stretched. This means that for every \( x \), the value of \( af(x) = ae^x \) will be greater than the corresponding \( f(x) = e^x \).
2. **Horizontal Asymptote**: The horizontal asymptote of the original function \( f(x) = e^x \) is \( y = 0 \) as \( x \to -\infty \). The asymptote remains the same for \( af(x) = ae^x \) because as \( x \to -\infty \), \( ae^x \) also approaches 0. Thus, \( af(x) \) has the same horizontal asymptote of \( y = 0 \).
3. **Y-Intercept**: The y-intercept of the original function \( f(x) = e^x \) is at \( (0, 1) \) because \( e^0 = 1 \). For \( af(x) \), the y-intercept becomes \( a \cdot e^0 = a \cdot 1 = a \), which will not be at (0,0) unless \( a = 0 \), which is not the case here since \( a > 1 \).
4. **Reflection**: There is no reflection with respect to the x-axis since we are multiplying by a positive factor \( a \).
5. **Vertical Compression**: The function \( af(x) = ae^x \) is not vertically compressed; instead, it is vertically stretched by a factor \( a \).
With this analysis in mind, the correct statement that best describes the effects on the graph of \( af(x) \) is:
**The graph of \( af(x) \) has the same horizontal asymptote as \( f(x) \).**
This is reflective of the fact that both functions approach the same horizontal asymptote, \( y = 0 \), as \( x \to -\infty \).
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