Question

Applications and Ext
²8.
L. Let P = (x, y) be a point on the graph of y =
(a) Express the distance d from P to the origin as a func-
tion of x.

1. The distance d from p to the origin as a function of x is;  d = √(x⁴ + 17x² + 64)

### How to use Pythagoras Theorem?

This question uses the Pythagorean Theorem to determine the length of a hypotenuse of a right triangle.
The base of the triangle goes x distance along the x-axis, then y distance up to the point in question.
In order to find the distance (the hypotenuse of that triangle) as a function of x, we need two points.
Point P = (x , x² + 8)
Point 2 = (0, 0)
The formula for distance is;
d = √[(xP – x²)² + (yP – y²)²]
Plugging in the relevant Values gives;
d = √[(x – 0)² + (x² + 8 – 0)²]
d = √[x² + (x⁴ + 16x² + 64)]
d = √(x⁴ + 17x² + 64)
Thus, the distance d from p to the origin as a function of x is;  d = √(x⁴ + 17x² + 64)
Complete Question is;
Let P = (x, y) be a point on the graph of y = x² + 8. Express the distance d from p to the origin as a function of x.