https://file.aiquickdraw.com/mathbot/file/17270631769299xj6e6em.jpg

Let's break down the problem visible in the image:

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**Given:**

Let \( P = (x, y) \) be a point on the graph of \( y = \sqrt{x} \).  
We are asked to solve several parts based on this graph.

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**Part (a):**  
**Express the distance \( d \) from point \( P \) to point \( (4,0) \) as a function of \( x \).**

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Here, \( P = (x, \sqrt{x}) \) and \( (4,0) \) is the fixed point, so we substitute into the distance formula:

\[
d(x) = \sqrt{(4 - x)^2 + (0 - \sqrt{x})^2}
\]

Simplifying:

\[
d(x) = \sqrt{(4 - x)^2 + (\sqrt{x})^2} = \sqrt{(4 - x)^2 + x}
\]

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**Part (b):**  
You are asked to choose the correct graph for \( d(x) = \sqrt{(4 - x)^2 + x} \).  
The graph represents the distance function.

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If you'd like, I can further explain the graph behavior for this equation or provide more details on finding the smallest distance.

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Would you like a breakdown of how to graph this function? Feel free to ask for further clarifications.

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Here are 5 additional related questions to deepen your understanding:

1. How would you compute the derivative of \( d(x) \) to find the critical points?
2. What does minimizing the distance function \( d(x) \) tell us about the geometry of the problem?
3. What are the possible physical interpretations of the graph?
4. How do transformations affect the shape of the distance function graph?
5. What happens to the distance function as \( x \to 0 \) or \( x \to 4 \)?

**Tip:** When dealing with distance problems, always use the distance formula and try simplifying expressions as much as possible to make graphing or solving easier!

Let P = (x, y) be a point on the graph of y = √x. (a) Express the distance d from P to the point (4, 0) as a function of x. (b) Use a graphing utility to graph d = d(x). (c) For what value(s) of x is d smallest? (d) What is the smallest distance?

Learn how to express the distance between a point on the graph of y = √x and the point (4, 0) as a function of x. This problem covers key algebraic concepts, including the distance formula, radical functions, and graphing. Suitable for high school students.

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