Answer :
To determine how the graph of the function [tex]\( g(x) = \frac{1}{2}e^x - 4 \)[/tex] compares to the graph of the function [tex]\( f(x) = e^x - 4 \)[/tex], let's analyze how the transformations affect the graph.
1. Identify the Functions and Their Basic Forms:
- The function [tex]\( f(x) \)[/tex] is given as [tex]\( f(x) = e^x - 4 \)[/tex].
- The function [tex]\( g(x) \)[/tex] is given as [tex]\( g(x) = \frac{1}{2}e^x - 4 \)[/tex].
2. Examine the Transformation:
- In [tex]\( f(x) = e^x - 4 \)[/tex], the term [tex]\( e^x \)[/tex] is the exponential function.
- In [tex]\( g(x) = \frac{1}{2}e^x - 4 \)[/tex], the term [tex]\( \frac{1}{2}e^x \)[/tex] represents a multiplication of [tex]\( e^x \)[/tex] by a factor of [tex]\( \frac{1}{2} \)[/tex].
3. Interpret the Coefficient:
- The coefficient [tex]\( 1 \)[/tex] in [tex]\( f(x) \)[/tex] indicates the normal exponential growth rate of [tex]\( e^x \)[/tex].
- The coefficient [tex]\( \frac{1}{2} \)[/tex] in [tex]\( g(x) \)[/tex] signifies that the exponential growth rate of [tex]\( e^x \)[/tex] is halved.
4. Vertical Compression:
- Multiplying [tex]\( e^x \)[/tex] by [tex]\( \frac{1}{2} \)[/tex] results in a vertical compression of the graph. This means that the values of [tex]\( g(x) \)[/tex] will be half of the values of [tex]\( f(x) \)[/tex] for any [tex]\( x \)[/tex].
- This compression happens because the transformation [tex]\( e^x \to \frac{1}{2}e^x \)[/tex] reduces the y-values of the function, effectively compressing the graph vertically.
From these observations, we can conclude that:
- The graph of function [tex]\( g \)[/tex] is not a horizontal shift to the left or right, so options A and D are incorrect.
- The graph of function [tex]\( g \)[/tex] is a vertical compression (not a vertical stretch) of the graph of function [tex]\( f \)[/tex], so option B is incorrect.
Therefore, the correct answer is:
C. The graph of function [tex]\( g \)[/tex] is a vertical compression of the graph of function [tex]\( f \)[/tex].
1. Identify the Functions and Their Basic Forms:
- The function [tex]\( f(x) \)[/tex] is given as [tex]\( f(x) = e^x - 4 \)[/tex].
- The function [tex]\( g(x) \)[/tex] is given as [tex]\( g(x) = \frac{1}{2}e^x - 4 \)[/tex].
2. Examine the Transformation:
- In [tex]\( f(x) = e^x - 4 \)[/tex], the term [tex]\( e^x \)[/tex] is the exponential function.
- In [tex]\( g(x) = \frac{1}{2}e^x - 4 \)[/tex], the term [tex]\( \frac{1}{2}e^x \)[/tex] represents a multiplication of [tex]\( e^x \)[/tex] by a factor of [tex]\( \frac{1}{2} \)[/tex].
3. Interpret the Coefficient:
- The coefficient [tex]\( 1 \)[/tex] in [tex]\( f(x) \)[/tex] indicates the normal exponential growth rate of [tex]\( e^x \)[/tex].
- The coefficient [tex]\( \frac{1}{2} \)[/tex] in [tex]\( g(x) \)[/tex] signifies that the exponential growth rate of [tex]\( e^x \)[/tex] is halved.
4. Vertical Compression:
- Multiplying [tex]\( e^x \)[/tex] by [tex]\( \frac{1}{2} \)[/tex] results in a vertical compression of the graph. This means that the values of [tex]\( g(x) \)[/tex] will be half of the values of [tex]\( f(x) \)[/tex] for any [tex]\( x \)[/tex].
- This compression happens because the transformation [tex]\( e^x \to \frac{1}{2}e^x \)[/tex] reduces the y-values of the function, effectively compressing the graph vertically.
From these observations, we can conclude that:
- The graph of function [tex]\( g \)[/tex] is not a horizontal shift to the left or right, so options A and D are incorrect.
- The graph of function [tex]\( g \)[/tex] is a vertical compression (not a vertical stretch) of the graph of function [tex]\( f \)[/tex], so option B is incorrect.
Therefore, the correct answer is:
C. The graph of function [tex]\( g \)[/tex] is a vertical compression of the graph of function [tex]\( f \)[/tex].
