Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
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].
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
