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To solve this problem, we need to find the distance from the point \( P = (x, y) \), which lies on the graph of \( y = x^2 - 4 \), to the origin (0, 0). This distance is expressed as a function of \( x \).

### Step-by-Step Solution:

1. **Distance formula**: 
   The distance \( d \) from a point \( (x_1, y_1) \) to another point \( (x_2, y_2) \) is given by the formula:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   Here, the point \( P = (x, y) = (x, x^2 - 4) \) is on the graph, and we are finding the distance from \( P \) to the origin \( (0, 0) \).

2. **Substitute the coordinates of \( P \) and the origin into the formula**:
   \[
   d(x) = \sqrt{(x - 0)^2 + ((x^2 - 4) - 0)^2}
   \]
   Simplify:
   \[
   d(x) = \sqrt{x^2 + (x^2 - 4)^2}
   \]

3. **Expand the squared term**:
   \[
   d(x) = \sqrt{x^2 + (x^4 - 8x^2 + 16)}
   \]

4. **Combine like terms**:
   \[
   d(x) = \sqrt{x^4 - 7x^2 + 16}
   \]

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

Would you like further details or explanations on any of the steps?

Here are some related questions for further exploration:
1. How does the distance change as \( x \) increases or decreases?
2. What are the minimum and maximum values of \( d(x) \) for real values of \( x \)?
3. Can the distance ever be zero? If yes, for which value(s) of \( x \)?
4. How can you find the derivative of \( d(x) \) to understand the rate of change of the distance?
5. How would the distance function change if the graph equation was different, say \( y = x^3 - 4 \)?

**Tip**: Always expand and simplify any squared terms carefully when using the distance formula to avoid mistakes!

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 P on the graph y = x^2 - 4 to the origin using the distance formula. This problem covers key algebraic and distance concepts, perfect for Grades 10-12.

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