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The question asks to express the distance \( d \) from the point \( P(x, y) \) on the graph of \( y = x^2 - 4 \) to the origin as a function of \( x \).

### Step-by-Step Solution:

1. **Point on the Graph:**
   The point \( P \) is given as \( (x, y) \), and the graph is described by \( y = x^2 - 4 \). Therefore, the coordinates of the point are \( (x, x^2 - 4) \).

2. **Distance Formula:**
   The distance \( d(x) \) from a point \( (x_1, y_1) \) to the origin \( (0, 0) \) is given by the formula:
   \[
   d(x) = \sqrt{x_1^2 + y_1^2}
   \]
   For the point \( (x, y) = (x, x^2 - 4) \), we substitute \( x_1 = x \) and \( y_1 = x^2 - 4 \).

3. **Substitute into the Distance Formula:**
   \[
   d(x) = \sqrt{x^2 + (x^2 - 4)^2}
   \]
   
4. **Simplify:**
   Expand \( (x^2 - 4)^2 \):
   \[
   (x^2 - 4)^2 = x^4 - 8x^2 + 16
   \]
   Now substitute back into the equation:
   \[
   d(x) = \sqrt{x^2 + x^4 - 8x^2 + 16}
   \]
   Combine like terms:
   \[
   d(x) = \sqrt{x^4 - 7x^2 + 16}
   \]

Thus, the distance function \( d(x) \) is:
\[
d(x) = \sqrt{x^4 - 7x^2 + 16}
\]

Would you like further clarification or details on any step?

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Here are 5 related questions to further explore this topic:

1. How would the distance function change if the graph were \( y = x^2 + 4 \)?
2. What are the points on the curve where the distance to the origin is minimized?
3. Can you calculate the distance for a specific value of \( x \), such as \( x = 2 \)?
4. How would the formula change if the point were to measure distance to a different point, say \( (1, 1) \)?
5. How does the shape of the curve \( y = x^2 - 4 \) affect the distance function?

### Tip:
When simplifying distance-related functions, always watch out for algebraic errors in squaring and combining terms, as they can easily complicate your calculations.

Let P = (x, y) be a point on the graph of y = x^2 - 4. Express the distance d from P to the origin as a function of x.

Learn how to express the distance from a point on the curve y = x^2 - 4 to the origin as a function of x. This problem uses the distance formula and quadratic functions, suitable for high school algebra students.

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