Answer :
To solve the problem, we need to understand and apply two transformations to the linear parent function [tex]\( f(x) = x \)[/tex]:
1. Vertically stretching the function by a factor of 14.
2. Flipping the function over the [tex]\( x \)[/tex]-axis.
Let's proceed step-by-step:
### Step 1: Vertically Stretch by a Factor of 14
When we vertically stretch a function [tex]\( f(x) \)[/tex] by a factor of 14, each [tex]\( y \)[/tex]-value of the function gets multiplied by 14. Therefore, the transformed function [tex]\( f'(x) \)[/tex] will be:
[tex]\[ f'(x) = 14 \cdot f(x) = 14 \cdot x \][/tex]
Now, our function has been transformed into:
[tex]\[ f'(x) = 14x \][/tex]
### Step 2: Flip Over the [tex]\( x \)[/tex]-Axis
When we flip a function over the [tex]\( x \)[/tex]-axis, we multiply the entire function by -1. Therefore, applying this transformation to [tex]\( f'(x) = 14x \)[/tex], we get the new function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = -1 \cdot 14x = -14x \][/tex]
So, the final equation of the new function after applying both transformations is:
[tex]\[ g(x) = -14x \][/tex]
### Conclusion
Among the given choices:
- A. [tex]\( g(x) = 14x - 1 \)[/tex]
- B. [tex]\( g(x) = -\frac{1}{14}x \)[/tex]
- C. [tex]\( g(x) = -14x \)[/tex]
- D. [tex]\( g(x) = \frac{14}{x} \)[/tex]
The correct answer is:
C. [tex]\( g(x) = -14x \)[/tex]
1. Vertically stretching the function by a factor of 14.
2. Flipping the function over the [tex]\( x \)[/tex]-axis.
Let's proceed step-by-step:
### Step 1: Vertically Stretch by a Factor of 14
When we vertically stretch a function [tex]\( f(x) \)[/tex] by a factor of 14, each [tex]\( y \)[/tex]-value of the function gets multiplied by 14. Therefore, the transformed function [tex]\( f'(x) \)[/tex] will be:
[tex]\[ f'(x) = 14 \cdot f(x) = 14 \cdot x \][/tex]
Now, our function has been transformed into:
[tex]\[ f'(x) = 14x \][/tex]
### Step 2: Flip Over the [tex]\( x \)[/tex]-Axis
When we flip a function over the [tex]\( x \)[/tex]-axis, we multiply the entire function by -1. Therefore, applying this transformation to [tex]\( f'(x) = 14x \)[/tex], we get the new function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = -1 \cdot 14x = -14x \][/tex]
So, the final equation of the new function after applying both transformations is:
[tex]\[ g(x) = -14x \][/tex]
### Conclusion
Among the given choices:
- A. [tex]\( g(x) = 14x - 1 \)[/tex]
- B. [tex]\( g(x) = -\frac{1}{14}x \)[/tex]
- C. [tex]\( g(x) = -14x \)[/tex]
- D. [tex]\( g(x) = \frac{14}{x} \)[/tex]
The correct answer is:
C. [tex]\( g(x) = -14x \)[/tex]
