91Ó°ÊÓ

To understand the distance formula, consider two points in a coordinate plane: Point A \((x_1, y_1)\) and Point B \((x_2, y_2)\). The distance \(d\) between these points is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula is derived from the Pythagorean theorem applied to the horizontal and vertical distances between the points. It calculates the straight-line distance.
In our exercise, we use the distance formula to find the distance between a point \(P = (x, y)\) on the curve \((y=x^2-8)\) and the fixed point \( (0, -1)\).
By substituting \((x, x^2 - 8)\) for \(P\) and \((0, -1)\) for the other point, our formula becomes:
\[ d = \sqrt{x^2 + ((x^2 - 8) + 1)^2} \]
Simplifying this, we get:
\[ d = \sqrt{x^4 - 13x^2 + 49} \]
This expression helps in finding specific distances when \(x\) is 0 or -1, and also aids in graphing the distance function.

Graphing functions is a central concept in algebra and trigonometry. It involves plotting the values of a function to visualize its behavior. In this exercise, we graph the distance function:
\[ d(x) = \sqrt{x^4 - 13x^2 + 49} \]
To graph a function, follow these steps:

• Identify the domain of the function—values for which the function is defined.
• Calculate specific points by substituting various \(x\) values into the function.
• Plot these points on a coordinate system.
• Connect the points smoothly following the function's trend.

Using a graphing utility simplifies this process, as it can automatically produce the graph when you input the function.
In our case, using such a utility, we can see how \d(x)\ behaves for different \x\ values and identify where the function reaches its minimum value, which helps answer part (e) of the exercise.

Minimizing a function means finding the input value (or values) where the function reaches its lowest point. This is often crucial in problems involving optimization.
For the function \d(x) = \sqrt{x^4 - 13x^2 + 49}\, we want to determine the \x\ value(s) that yield the smallest distance.
Here's the approach to minimize the function:

• Graph the function to visually identify the minima.
• Use calculus techniques, such as finding the derivative \(d'(x)\), and setting it to 0 to find critical points: \[ d'(x) = \frac{d}{dx}\left( \sqrt{x^4 - 13x^2 + 49} \right) = 0 \] .
• Analyze these critical points to determine which gives the minimum value.

In our exercise, using these methods, we determine that \d(x)\ has its smallest value at specific \x\ values, providing a clear solution to part (e). Understanding minima gives insights into the behavior of functions and their practical applications in various fields.