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We are given that \( G(x) = 4f(x) - g(x) \), and we need to find \( G'(2) \), which is the derivative of \( G(x) \) evaluated at \( x = 2 \). 

Using basic differentiation rules, we know:

\[
G'(x) = 4f'(x) - g'(x)
\]

Thus, to find \( G'(2) \), we need to compute \( 4f'(2) - g'(2) \).

### Steps:
1. From the graph, we will estimate the slopes (derivatives) \( f'(x) \) and \( g'(x) \) at \( x = 2 \).
2. We will multiply \( f'(2) \) by 4, then subtract \( g'(2) \) from it to get \( G'(2) \).

Looking at the graph:

- **For \( f'(2) \):** 
  The graph of \( f(x) \) appears to be a straight line at \( x = 2 \), and the slope is the change in \( y \) over the change in \( x \). By estimation from the graph, we see that for \( f(x) \), the slope appears to be negative. A rough estimate gives \( f'(2) \approx -1 \).

- **For \( g'(2) \):**
  Similarly, for \( g(x) \), the slope at \( x = 2 \) also looks negative but steeper than \( f(x) \). A rough estimate gives \( g'(2) \approx -2 \).

### Now substitute these estimates into the equation for \( G'(2) \):

\[
G'(2) = 4(-1) - (-2) = -4 + 2 = -2
\]

So, \( G'(2) \approx -2 \).

Would you like more details on how to estimate slopes from graphs? 

Here are some related questions:
1. How is the derivative of a function defined geometrically?
2. How do you differentiate a function with multiple terms like \( G(x) = 4f(x) - g(x) \)?
3. What is the significance of the slope of a function at a particular point?
4. How do you compute the derivative when dealing with piecewise or non-linear graphs?
5. Can we use numerical methods to estimate derivatives from graphical data?

**Tip:** When estimating slopes from graphs, focus on identifying two clear points on the line and use the formula for the slope: \( \text{slope} = \frac{\Delta y}{\Delta x} \).

Let G = 4f - g, where the graphs of f and g are shown. Find G'(2).

Learn how to find the derivative of G(x) = 4f(x) - g(x) at x = 2 by estimating the slopes from the graph. This problem involves differentiation, slope estimation, and basic calculus concepts. Suitable for Grades 11-12.

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