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Sagot :
To compare the graphs of the functions [tex]\( f(x) = e^x - 4 \)[/tex] and [tex]\( g(x) = \frac{1}{2} e^x - 4 \)[/tex], we need to analyze the differences in their functional forms.
1. Start by examining the general form of each function:
- The function [tex]\( f(x) \)[/tex] has the form [tex]\( f(x) = e^x - 4 \)[/tex], which is essentially an exponential function [tex]\( e^x \)[/tex] that has been shifted downward by 4 units.
- The function [tex]\( g(x) \)[/tex] has the form [tex]\( g(x) = \frac{1}{2} e^x - 4 \)[/tex], which is an exponential function [tex]\( e^x \)[/tex] that has been scaled by a factor of [tex]\(\frac{1}{2}\)[/tex] and then shifted downward by 4 units.
2. Determine the effect of the scaling factor [tex]\( \frac{1}{2} \)[/tex] in [tex]\( g(x): \ - A scaling factor of \(\frac{1}{2}\)[/tex] affects the vertical stretch or compression of the graph. Specifically, multiplying the exponential part [tex]\( e^x \)[/tex] by [tex]\(\frac{1}{2}\)[/tex] compresses the graph vertically. Each y-value on the graph of [tex]\( f(x) \)[/tex] will be halved in [tex]\( g(x) \)[/tex].
3. Compare the two graphs:
- For any given [tex]\( x \)[/tex]:
- In [tex]\( f(x) \)[/tex], the y-value is [tex]\( e^x - 4 \)[/tex].
- In [tex]\( g(x) \)[/tex], the y-value is [tex]\( \frac{1}{2} e^x - 4 \)[/tex].
- The term [tex]\( \frac{1}{2} e^x \)[/tex] indicates that the output of [tex]\( g(x) \)[/tex] is a vertical compression (a "squeeze" towards the x-axis) of the output of [tex]\( f(x) \)[/tex] before the -4 shift is applied. This means the graph of [tex]\( g(x) \)[/tex] is compressed vertically by a factor of [tex]\(\frac{1}{2}\)[/tex] compared to the graph of [tex]\( f(x) \)[/tex].
4. Check the given options based on this interpretation:
- Option A: A horizontal shift to the right is not correct because there is no horizontal translation involved between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
- Option B: A vertical compression correctly describes the transformation from [tex]\( f(x) \)[/tex] to [tex]\( g(x) \)[/tex].
- Option C: A horizontal shift to the left is not correct for the same reason as option A.
- Option D: A vertical stretch is incorrect because the transformation is a compression, not a stretch.
Therefore, the correct answer is:
B. The graph of function [tex]\( g \)[/tex] is a vertical compression of the graph of function [tex]\( f \)[/tex].
1. Start by examining the general form of each function:
- The function [tex]\( f(x) \)[/tex] has the form [tex]\( f(x) = e^x - 4 \)[/tex], which is essentially an exponential function [tex]\( e^x \)[/tex] that has been shifted downward by 4 units.
- The function [tex]\( g(x) \)[/tex] has the form [tex]\( g(x) = \frac{1}{2} e^x - 4 \)[/tex], which is an exponential function [tex]\( e^x \)[/tex] that has been scaled by a factor of [tex]\(\frac{1}{2}\)[/tex] and then shifted downward by 4 units.
2. Determine the effect of the scaling factor [tex]\( \frac{1}{2} \)[/tex] in [tex]\( g(x): \ - A scaling factor of \(\frac{1}{2}\)[/tex] affects the vertical stretch or compression of the graph. Specifically, multiplying the exponential part [tex]\( e^x \)[/tex] by [tex]\(\frac{1}{2}\)[/tex] compresses the graph vertically. Each y-value on the graph of [tex]\( f(x) \)[/tex] will be halved in [tex]\( g(x) \)[/tex].
3. Compare the two graphs:
- For any given [tex]\( x \)[/tex]:
- In [tex]\( f(x) \)[/tex], the y-value is [tex]\( e^x - 4 \)[/tex].
- In [tex]\( g(x) \)[/tex], the y-value is [tex]\( \frac{1}{2} e^x - 4 \)[/tex].
- The term [tex]\( \frac{1}{2} e^x \)[/tex] indicates that the output of [tex]\( g(x) \)[/tex] is a vertical compression (a "squeeze" towards the x-axis) of the output of [tex]\( f(x) \)[/tex] before the -4 shift is applied. This means the graph of [tex]\( g(x) \)[/tex] is compressed vertically by a factor of [tex]\(\frac{1}{2}\)[/tex] compared to the graph of [tex]\( f(x) \)[/tex].
4. Check the given options based on this interpretation:
- Option A: A horizontal shift to the right is not correct because there is no horizontal translation involved between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
- Option B: A vertical compression correctly describes the transformation from [tex]\( f(x) \)[/tex] to [tex]\( g(x) \)[/tex].
- Option C: A horizontal shift to the left is not correct for the same reason as option A.
- Option D: A vertical stretch is incorrect because the transformation is a compression, not a stretch.
Therefore, the correct answer is:
B. The graph of function [tex]\( g \)[/tex] is a vertical compression of the graph of function [tex]\( f \)[/tex].
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