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The problem presented here is from a mathematics exam (CE 2019 TV1) and focuses on analyzing the behavior of a function \( f_p(x) \) for \( p \geq 1 \), with the function defined as:

\[
f_p(x) = p + \sqrt{x - p}
\]

The figure shows graphs for various values of \( p \) as well as the line \( k \) given by the equation:

\[
y = x + \frac{1}{4}
\]

### The main tasks:
1. **Interpret the graph**: The function \( f_p(x) \) shifts for different values of \( p \), and the line \( k \) intersects or is tangent to these graphs. The question states that line \( k \) touches the graph of \( f_p \) for all \( p \geq 1 \).
   
2. **The key question to address (4th question)**:
   "Prove that for \( p \geq 1 \), the endpoint of the graph of \( f_p(x) \) lies on the graph of \( f_{p-1}(x) \)."

### Explanation of what needs to be done:
The endpoint of \( f_p(x) \) occurs when the argument of the square root becomes zero, that is when:

\[
x = p
\]

At this point, \( f_p(p) = p \), so the endpoint of the graph of \( f_p(x) \) is \( (p, p) \). The task is to show that this point \( (p, p) \) also lies on the graph of the function \( f_{p-1}(x) \).

Substituting \( x = p \) into \( f_{p-1}(x) \):

\[
f_{p-1}(p) = (p - 1) + \sqrt{p - (p - 1)} = (p - 1) + \sqrt{1} = p
\]

Thus, the point \( (p, p) \) indeed lies on the graph of \( f_{p-1}(x) \), proving the required statement.

---

### Would you like more details on this explanation, or do you have any questions about specific steps?

Here are 5 related questions to expand on this:

1. How does the function \( f_p(x) = p + \sqrt{x - p} \) change when \( p \) increases?
2. What is the significance of the square root term in \( f_p(x) \)?
3. How do you find the tangent line at a specific point on a function graph?
4. Why is the condition \( p \geq 1 \) important in this context?
5. How can you verify graphically that line \( k \) touches the graph of \( f_p(x) \)?

**Tip**: When proving properties of functions involving parameters, always substitute key values and check boundary conditions to simplify the problem.

Voor p ≥ 1 is de functie f_p gegeven door: f_p(x) = p + √(x - p). In figuur 1 is voor enkele waarden van p de grafiek van f_p weergegeven en ook lijn k met vergelijking y = x + 1/4. Bewijs dat inderdaad voor p ≥ 1 geldt: het randpunt van de grafiek van f_p ligt op de grafiek van f_{p-1}.

Explore how to prove that for p ≥ 1, the endpoint of the function f_p(x) = p + √(x - p) lies on the graph of f_{p-1}(x). This problem involves analyzing the behavior of function graphs and their tangency with the line y = x + 1/4. Suitable for high school students.

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