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The homework problem provided involves analyzing a function to express the distance from a point \( P \) on the graph of \( y = x^2 - 4 \) to the origin and solving related questions.

Let's go through it step by step:

### Given:
- The curve: \( y = x^2 - 4 \)
- The distance formula from a point \( (x, y) \) to the origin is \( d(x) = \sqrt{x^2 + y^2} \).
- Substituting \( y = x^2 - 4 \) into the distance formula gives:
  \[
  d(x) = \sqrt{x^2 + (x^2 - 4)^2}
  \]
  Simplifying this, you'll get:
  \[
  d(x) = \sqrt{x^2 + (x^4 - 8x^2 + 16)}
  \]
  \[
  d(x) = \sqrt{x^4 - 7x^2 + 16}
  \]
  This is the expression for the distance \( d(x) \) from point \( P(x, y) \) to the origin.

Now, the questions:

### (a) **Express the distance d from P to the origin as a function of x**:
You've already done this part. The function for distance is:
\[
d(x) = \sqrt{x^4 - 7x^2 + 16}
\]

### (b) **What is \( d \) if \( x = 0 \)?**
Plug \( x = 0 \) into the function:
\[
d(0) = \sqrt{0^4 - 7(0)^2 + 16} = \sqrt{16} = 4
\]
So, \( d(0) = 4 \).

### (c) **What is \( d \) if \( x = 1 \)?**
Substitute \( x = 1 \) into the function:
\[
d(1) = \sqrt{1^4 - 7(1)^2 + 16} = \sqrt{1 - 7 + 16} = \sqrt{10} \approx 3.16
\]
So, \( d(1) \approx 3.16 \).

### (d) **For what positive value of \( x \) is \( d(x) \) smallest?**
To find the minimum distance, you would need to graph the function \( d(x) = \sqrt{x^4 - 7x^2 + 16} \) and observe the value of \( x \) where \( d(x) \) is smallest. Based on the function's shape, this could be done using a graphing utility. The answer will be some positive value of \( x \), which you can find using the graph or calculus (by finding the derivative and solving for critical points).

Would you like me to assist in graphing \( d(x) \) or finding the derivative for more precision?

---

### 5 Related Questions:
1. How would you determine the derivative of \( d(x) \) to find critical points?
2. What happens to the distance function as \( x \to \infty \)?
3. How would the distance formula change if the origin were at a different point, like \( (1, 1) \)?
4. What is the geometric significance of the points where the distance is minimized?
5. How can you verify that the distance is minimized at a critical point?

**Tip**: Graphing functions and checking critical points by calculus can both help confirm whether a function has a minimum or maximum value at a given point.

Let P = (x, y) be a point on the graph of y = x^2 - 4. (a) Express the distance d from P to the origin as a function of x. (b) What is d if x = 0? (c) What is d if x = 1? (d) For what positive value of x is d(x) smallest?

Learn how to express the distance from a point on the graph of y = x^2 - 4 to the origin as a function of x, and find the minimum distance using graphing techniques. This problem explores key mathematical concepts like the distance formula, function graphing, and function minimization. Suitable for students in Grades 9-12.

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