If g(x) is the f(x)=x after a vertical compression by 14, a shift right by 1, and a shift up by 2,

a) write an equation for g(x):  g(x) =

b) The slope of this line is:

c) The vertical intercept of this line is:

c) The vertical intercept of this line is

Let's solve the problem step by step.

### a) Equation for \( g(x) \):

1. **Start with the function \( f(x) = x \).**
   
2. **Vertical Compression by 14:**
   - A vertical compression by a factor of 14 multiplies the function by \( \frac{1}{14} \).
   - The function becomes \( \frac{1}{14}x \).

3. **Shift Right by 1:**
   - Shifting the function to the right by 1 involves replacing \( x \) with \( (x - 1) \).
   - The function becomes \( \frac{1}{14}(x - 1) \).

4. **Shift Up by 2:**
   - Shifting the function up by 2 involves adding 2 to the function.
   - The final function becomes \( g(x) = \frac{1}{14}(x - 1) + 2 \).

So, the equation for \( g(x) \) is:
\[ g(x) = \frac{1}{14}(x - 1) + 2 \]

### b) Slope of the Line:
- The slope of a linear function \( g(x) = mx + b \) is the coefficient of \( x \).
- The slope of \( g(x) \) here is \( \frac{1}{14} \).

### c) Vertical Intercept of the Line:
- The vertical intercept (also known as the y-intercept) occurs when \( x = 0 \).
- Substitute \( x = 0 \) into \( g(x) = \frac{1}{14}(x - 1) + 2 \):

\[ g(0) = \frac{1}{14}(0 - 1) + 2 = \frac{-1}{14} + 2 = \frac{-1 + 28}{14} = \frac{27}{14} \]

Thus, the vertical intercept is \( \frac{27}{14} \).

### Final Answers:
- **a)** \( g(x) = \frac{1}{14}(x - 1) + 2 \)
- **b)** The slope of this line is \( \frac{1}{14} \).
- **c)** The vertical intercept of this line is \( \frac{27}{14} \).

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### Related Questions:
1. What happens to the slope if there is a horizontal stretch instead of a compression?
2. How would the equation change if the function was shifted left instead of right?
3. What if the function \( f(x) \) was originally \( f(x) = 2x + 3 \)?
4. How would a vertical stretch affect the vertical intercept?
5. What is the geometric significance of a vertical intercept?

**Tip:** Always apply transformations in the order: stretches/compressions, reflections, translations (shifts) for consistency.

Explore how to derive the function g(x) from f(x) = x after a vertical compression by 14, a shift right by 1, and a shift up by 2. Learn about function transformations and their application in this high school-level math problem.

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