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 \( f(x) \).
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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 \( f(x) = e^x \).
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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 \( y = 0 \).
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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 \( a > 1 \).
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Reflection: There is no reflection with respect to the x-axis since we are multiplying by a positive factor \( a \).
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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 \( x \to -\infty \).