Let's solve this step by step.

a) Write an equation for $g(x)$.

Given the function $f(x) = x$, let's apply the transformations:

1. Vertical compression by $\frac{1}{2}$: This transformation scales the function vertically by $\frac{1}{2}$. So, the new function becomes: $f_1(x) = \frac{1}{2}x$

2. Shift left by 1: Shifting the function left by 1 unit means replacing $x$ with $x + 1$. The function becomes: $f_2(x) = \frac{1}{2}(x + 1)$

3. Shift up by 3: Shifting the function up by 3 units means adding 3 to the function. The final function is: $g(x) = \frac{1}{2}(x + 1) + 3$

Simplifying this, we get: $g(x) = \frac{1}{2}x + \frac{1}{2} + 3$ $g(x) = \frac{1}{2}x + \frac{7}{2}$

b) The slope of this line is:

The slope of the line is the coefficient of $x$ in the equation for $g(x)$. From the equation $g(x) = \frac{1}{2}x + \frac{7}{2}$, the slope $m$ is: $\text{slope} = \frac{1}{2}$

c) The vertical intercept of this line is:

The vertical intercept (also known as the y-intercept) is the constant term in the equation for $g(x)$. In this case, the y-intercept is: $\text{vertical intercept} = \frac{7}{2} = 3.5$

Summary:

• a) The equation for $g(x)$ is $g(x) = \frac{1}{2}x + \frac{7}{2}$.
• b) The slope of the line is $\frac{1}{2}$.
• c) The vertical intercept is $\frac{7}{2}$ or 3.5.

Would you like further details or have any other questions?

Here are some related questions:

1. How would the equation change if the function was reflected over the x-axis?
2. What would the equation be if the function was shifted right by 2 units instead of left by 1 unit?
3. How would the slope change if the function had a vertical stretch by a factor of 3?
4. What is the equation for $g(x)$ if the function was shifted down by 4 units instead of up by 3 units?
5. How would the y-intercept change if the slope were doubled?

Tip: Always apply transformations in the correct order: stretches/compressions first, then shifts.