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To solve this problem, we are looking for the equation of the line tangent to the graph of \( f \) at \( x = 2 \), using the graph of \( f' \), which represents the derivative of \( f \).

Here’s the process step-by-step:

### 1. Identify the slope of the tangent line at \( x = 2 \)
The slope of the tangent line to the graph of \( f \) at any point is given by the value of \( f'(x) \). We are asked to find the slope at \( x = 2 \).

From the graph of \( f' \), at \( x = 2 \), the value of \( f'(2) = 5 \) (from the height of the graph at \( x = 2 \)).

Thus, the slope \( m \) of the tangent line at \( x = 2 \) is:
\[
m = f'(2) = 5
\]

### 2. Use the point-slope form of the tangent line
We are given the point \( (2, 6) \) on the graph of \( f \). The equation of a line can be written in point-slope form:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the point on the line. Substituting \( m = 5 \) and the point \( (2, 6) \), we get:
\[
y - 6 = 5(x - 2)
\]
Simplifying:
\[
y - 6 = 5x - 10
\]
\[
y = 5x - 4
\]

Thus, the equation of the tangent line is \( y = 5x - 4 \).

### 3. Conclusion
The correct answer is **B** \( y = 5x - 4 \).

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Would you like a deeper explanation, or do you have any follow-up questions? 

Here are 5 related questions to further explore the topic:
1. How is the derivative related to the slope of the tangent line?
2. Why do we use the point-slope form for finding the tangent line?
3. What is the significance of \( f' \) being positive, negative, or zero at a point?
4. How do you find the equation of a tangent line if the derivative is given as a function, not a graph?
5. Can the tangent line intersect the graph of the function at multiple points?

**Tip:** Always double-check the values from the graph when calculating the slope and ensure you are reading the derivative correctly for precise results.

The graph of f', the derivative of a function f, is shown above. The points (2, 6) and (4, 18) are on the graph of f. Which of the following is an equation of the line tangent to the graph of f at x = 2?

Learn how to find the equation of the tangent line to a function at x = 2 using the derivative. This problem involves understanding the derivative as the slope of the tangent line and using the point-slope form to find the equation. Suitable for students in Grades 11-12.

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