SOLUTION: Let P = (x,y) be a point on the graph of y = 1/x. A. Express the distance d from P to the origin as a function of x. B. Use a graphing utility to graph d = d(x).

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Let P = (x,y) be a point on the graph of y = 1/x. A. Express the distance d from P to the origin as a function of x. B. Use a graphing utility to graph d = d(x).      Log On


   

Question 1208130: Let P = (x,y) be a point on the graph of y = 1/x.
A. Express the distance d from P to the origin as a function of x.
B. Use a graphing utility to graph d = d(x).
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Part A

(x1,y1) = origin = (0,0)
(x2,y2) = point P = (x,y) = (x,1/x) since y = 1/x

Distance formula from (x1,y1) to (x2,y2)
d+=+sqrt%28+%28x2-x1%29%5E2+%2B+%28y2-y1%29%5E2+%29+

d+=+sqrt%28+%28x-0%29%5E2+%2B+%28y-0%29%5E2+%29+

d+=+sqrt%28+x%5E2+%2B+y%5E2+%29+

d%28x%29+=+sqrt%28+x%5E2+%2B+%281%2Fx%29%5E2+%29+
Optionally you can simplify this, but I'll let the student take over from here.
Note that the distance formula is based on the Pythagorean Theorem.

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Part B

This is what the graph of d(x) looks like
graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-100%2Csqrt%28+x%5E2+%2B+%281%2Fx%29%5E2+%29%29
There are many graphing tools to pick from.
My favorite is GeoGebra, but Desmos is really good as well.
If you are familiar with something like a TI83, then it might be best to stick to that.

Things to pay attention to:
  • x = 0 isn't allowed since it causes a division by zero error. This explains the vertical asymptote at x = 0.
  • The curve is symmetric about the y axis. The function d(x) is even. We can prove this algebraically by showing that d(-x) = d(x) for all x in the domain. I'll leave the proof for the reader.
  • The smallest distance possible is d = sqrt(2) = 1.41421356 approximately, which occurs when either x = 1 or x = -1. Use calculus to find this, or use the graphing calculator's minimum locator feature.