Answered

Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

Select the correct answer.

Which statement correctly compares the graph of function [tex]$g$[/tex] with the graph of function [tex]$f$[/tex]?

[tex]$
\begin{array}{l}
f(x)= e^x - 4 \\
g(x)=\frac{1}{2} e^x - 4
\end{array}
$[/tex]

A. The graph of function [tex]$g$[/tex] is a horizontal shift of the graph of function [tex]$f$[/tex] to the right.
B. The graph of function [tex]$g$[/tex] is a vertical compression of the graph of function [tex]$f$[/tex].
C. The graph of function [tex]$g$[/tex] is a horizontal shift of the graph of function [tex]$f$[/tex] to the left.
D. The graph of function [tex]$g$[/tex] is a vertical stretch of the graph of function [tex]$f$[/tex].

Sagot :

GinnyAnswer
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].
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.