Let P (x,y) be a point on a graph of y=x^2-8

Question

A) express the distance d From p to the origin as a function of x

B) what is d if x=o

c) what is d if x=1

d) what is the equation of d=d(x)

e) for what values of x is do the smallest

Answer 1

Since y=x² -8, the coordinate of P can be expressed as (x, x² - 8)

a) The distance to the origin can be expressed as d(x) =sqrt(x² + (x²-8)²)

b) if x =0, d = 8

c) If x = 1, d = sqrt(1+49) = sqrt(50)

d) if d = sqrt(x² + (x²-8)²), then d² = x⁴ - 15 x² + 64 ==> x⁴ - 15 x² + 64 - d² = 0

e) d = sqrt( x⁴ - 15 x² + 64) = sqrt( (x² - 15/2)² - 225/4 + 64 ), when x² = 15/2, x = sqrt(30)/2 or -sqrt(30)/2, d is the smallest at sqrt(31)/2.