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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Would you like more details on any part of this solution?

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.