Closest point on a graph Problem set A One question at a time

Rational functions have asymptotes.

1. (a) Write down the equations of the vertical and horizontal asymptotes of the graph of \(f(x)=\dfrac{9x}{x-6}\), where \(x\neq6\).

   • Denominator zero: \(x-6=0\), so the vertical asymptote is \(x=6\)
   • For large x the function tends to the ratio of the x coefficients, so the horizontal asymptote is \(y=9\)

2. (b) The graph of \(g(x)=\dfrac{8x}{x-q}\) has asymptotes that intersect at the point \(Q(-2,\,8)\). Find the value of q.

   • Vertical asymptote \(x=q\); horizontal asymptote \(y=8\)
   • They meet at \((q,\,8)\)
   • Compare with \((-2,\,8)\): q = -2

The point \(P(x,\,y)\) lies on the graph of \(f(x)=\dfrac{4x}{x-3}\). The asymptotes of the graph meet at \(Q(3,\,4)\). Show that \(PQ=\sqrt{(x-3)^2+\left(\dfrac{12}{x-3}\right)^2}\).

• \(PQ=\sqrt{(x-3)^2+(y-4)^2}\)
• \(y-4=\dfrac{4x}{x-3}-4=\dfrac{4x-4(x-3)}{x-3}=\dfrac{12}{x-3}\)
• Substitute into the distance formula to give the result

The graph of \(f(x)=\dfrac{4x}{x-3}\) has asymptotes that meet at \(Q(3,\,4)\). A point \(P(x,\,y)\) on the graph is a distance \(PQ=\sqrt{(x-3)^2+\left(\dfrac{12}{x-3}\right)^2}\) from Q.

1. (a) Use your calculator to find the x-coordinates of the two points on the graph closest to Q.

   • Graph \(Y1=\sqrt{(x-3)^2+(12/(x-3))^2}\)
   • From the calculator: minimum at x = −0.464 and at x = 6.46

2. (b) Find the coordinates of these points and the shortest distance PQ.

   • \(y=\dfrac{4x}{x-3}\): x = −0.464 gives y = 0.536; x = 6.46 gives y = 7.46
   • The minimum value of PQ from the calculator is 4.90

A drone flies along a path modelled by \(y=\sqrt{x}\), where x and y are in kilometres. A radio mast stands at the point M(9, 0).

1. (a) Write an expression for the distance \(d\) between the point \((x,\,\sqrt{x})\) on the curve and the mast M(9, 0).

   • Distance between \((x,\sqrt{x})\) and \((9,0)\)
   • \(d=\sqrt{(x-9)^2+(\sqrt{x})^2}=\sqrt{(x-9)^2+x}\)

2. (b) Find the shortest distance from the mast M(9, 0) to the curve.

   • Graph \(Y1=\sqrt{(x-9)^2+x}\)
   • From the calculator: minimum at x = 8.50
   • Shortest distance = 2.96 km

A bird colony on an island has \(P(t)=\dfrac{600t}{t+4}\) birds, where t is the number of years since monitoring began, \(t\ge0\).

1. (a) Write down the equations of the vertical and horizontal asymptotes of the graph of P.

   • Denominator zero: \(t+4=0\), so \(t=-4\)
   • For large t, \(P\to600\), so \(P=600\)

2. (b) What does the horizontal asymptote tell you about the colony?

   • As t grows, P approaches 600 from below
   • The model predicts a long-term colony size of 600

The point \(P(x,\,y)\) lies on the curve \(y=\dfrac{18}{x}\), where \(x>0\). O is the origin.

1. (a) Show that \(OP=\sqrt{x^2+\dfrac{324}{x^2}}\).

   • \(OP=\sqrt{x^2+y^2}\) and \(y=\dfrac{18}{x}\)
   • \(y^2=\dfrac{324}{x^2}\), so \(OP=\sqrt{x^2+\dfrac{324}{x^2}}\)

2. (b) Find the coordinates of the point on the curve closest to O, and the distance OP.

   • Graph \(Y1=\sqrt{x^2+324/x^2}\) for \(x>0\)
   • From the calculator: minimum at x = 4.24, OP = 6.00
   • \(y=18/4.24=4.24\)

The point \(P(x,\,y)\) lies on the curve \(y=x^2\). The point A is \((0,\,5)\). Find the coordinates of the points on the curve that are closest to A.

• A point on the curve is \((x,\,x^2)\); the distance to \((0,\,5)\) is \(d=\sqrt{x^2+(x^2-5)^2}\)
• Graph \(Y1=\sqrt{x^2+(x^2-5)^2}\)
• From the calculator: minimums at x = −2.12 and x = 2.12
• \(y=x^2=4.50\): the points are (2.12, 4.50) and (−2.12, 4.50)