To find the transformed function and determine which graph matches it, we start with the original function \( f(x) = e^x \).
Step 1: Translate 3 units to the right
Translating the function 3 units to the right involves replacing \( x \) with \( x - 3 \): \[ f(x) = e^{x - 3} \]
Step 2: Vertically compress by a factor of \( \frac{1}{4} \)
Next, we compress the function vertically by a factor of \( \frac{1}{4} \): \[ f(x) = \frac{1}{4} e^{x - 3} \]
Step 3: Translate 6 units down
Finally, we translate the function downwards by 6 units: \[ f(x) = \frac{1}{4} e^{x - 3} - 6 \]
Analyze the transformed function:
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Horizontal Asymptote: The horizontal asymptote of the function occurs where \( y \) approaches a constant value as \( x \) approaches infinity. For the current function, this asymptote is: \[ y = -6 \]
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Behavior of the function: The exponential function \( e^{x - 3} \) translates to the right, and the vertical compression makes it grow slower. The \( -6 \) shifts everything down.
Now, check the graphs:
- All the graphs mentioned have a horizontal asymptote at \( y = -6 \).
- Let's look for the behavior of the function. A vertically compressed exponential function shifted to the right will not have positive y-values near \( x = 3 \).
To check the points through which the transformed function passes:
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If we evaluate at \( x = 3 \): \[ f(3) = \frac{1}{4} e^{3 - 3} - 6 = \frac{1}{4} - 6 = -\frac{23}{4} \approx -5.75 \]
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If we evaluate at \( x = 4 \): \[ f(4) = \frac{1}{4} e^{4 - 3} - 6 = \frac{1}{4} e^1 - 6 \approx 0.67 - 6 = -5.33 \]
From the numerical approximations:
- The y-values were approximately \( -5.75 \) for \( x = 3 \) and \( -5.33 \) for \( x = 4 \).
Conclusion:
The second graph matches the behavior of our transformed function, as it passes through points near \( (-3, -5.75) \) and \( (-2, -5.32) \), and has the correct horizontal asymptote.
Thus, the correct graph is:
- A graph of an exponential function that begins in quadrant 3, has a horizontal asymptote at \( y = -6 \), and passes through the points \( \left(-3, -5.75\right) \) and \( \left(-2, -5.32\right) \).
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