Closest point on a graph Problem set B Worked solutions

Rational functions have asymptotes.

1. (a) Write down the equations of the vertical and horizontal asymptotes of the graph of \(f(x)=\dfrac{4x}{x+3}\), where \(x\neq-3\).

   • Denominator zero: \(x+3=0\), so the vertical asymptote is \(x=-3\)
   • For large x the function tends to the ratio of the x coefficients, so the horizontal asymptote is \(y=4\)

2. (b) The graph of \(g(x)=\dfrac{6x}{x-q}\) has asymptotes that intersect at the point \(Q(5,\,6)\). Find the value of q.

   • Vertical asymptote \(x=q\); horizontal asymptote \(y=6\)
   • They meet at \((q,\,6)\)
   • Compare with \((5,\,6)\): q = 5

The point \(P(x,\,y)\) lies on the graph of \(f(x)=\dfrac{3x}{x+2}\). The asymptotes of the graph meet at \(Q(-2,\,3)\). Show that \(PQ=\sqrt{(x+2)^2+\left(\dfrac{-6}{x+2}\right)^2}\).

• \(PQ=\sqrt{(x+2)^2+(y-3)^2}\)
• \(y-3=\dfrac{3x}{x+2}-3=\dfrac{3x-3(x+2)}{x+2}=\dfrac{-6}{x+2}\)
• Substitute into the distance formula to give the result

The graph of \(f(x)=\dfrac{3x}{x+2}\) has asymptotes that meet at \(Q(-2,\,3)\). A point \(P(x,\,y)\) on the graph is a distance \(PQ=\sqrt{(x+2)^2+\left(\dfrac{-6}{x+2}\right)^2}\) from Q.

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

   • Graph \(Y1=\sqrt{(x+2)^2+(-6/(x+2))^2}\)
   • From the calculator: minimum at x = −4.45 and at x = 0.449

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

   • \(y=\dfrac{3x}{x+2}\): x = −4.45 gives y = 5.45; x = 0.449 gives y = 0.551
   • The minimum value of PQ from the calculator is 3.46

A fence runs along the curve \(y=\sqrt{x}\), where x and y are in metres. A sensor is fixed at the point S(11, 0).

1. (a) Write an expression for the distance \(d\) between the point \((x,\,\sqrt{x})\) on the curve and the sensor S(11, 0).

   • Distance between \((x,\sqrt{x})\) and \((11,0)\)
   • \(d=\sqrt{(x-11)^2+(\sqrt{x})^2}=\sqrt{(x-11)^2+x}\)

2. (b) Find the shortest distance from the sensor S(11, 0) to the curve.

   • Graph \(Y1=\sqrt{(x-11)^2+x}\)
   • From the calculator: minimum at x = 10.5
   • Shortest distance = 3.28 m

The concentration of a drug in the blood is modelled by \(C(t)=\dfrac{12t}{t+3}\) mg per litre, where t is the number of hours after the dose, \(t\ge0\).

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

   • Denominator zero: \(t+3=0\), so \(t=-3\)
   • For large t, \(C\to12\), so \(C=12\)

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

   • As t grows, C approaches 12 from below
   • The model predicts a long-term concentration of 12 mg per litre

The point \(P(x,\,y)\) lies on the curve \(y=\dfrac{32}{x}\), where \(x>0\). O is the origin.

1. (a) Show that \(OP=\sqrt{x^2+\dfrac{1024}{x^2}}\).

   • \(OP=\sqrt{x^2+y^2}\) and \(y=\dfrac{32}{x}\)
   • \(y^2=\dfrac{1024}{x^2}\), so \(OP=\sqrt{x^2+\dfrac{1024}{x^2}}\)

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

   • Graph \(Y1=\sqrt{x^2+1024/x^2}\) for \(x>0\)
   • From the calculator: minimum at x = 5.66, OP = 8.00
   • \(y=32/5.66=5.66\)

The point \(P(x,\,y)\) lies on the curve \(y=x^2\). The point A is \((0,\,3)\). 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 \((0,\,3)\) is \(d=\sqrt{x^2+(x^2-3)^2}\)
• Graph \(Y1=\sqrt{x^2+(x^2-3)^2}\)
• From the calculator: minimums at x = −1.58 and x = 1.58
• \(y=x^2=2.50\): the points are (1.58, 2.50) and (−1.58, 2.50)