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AI HL · 12.5

Functions

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12.5.8

Closest point on a graph

12.5.8

Closest point on a graph

Target: I can write down the vertical and horizontal asymptotes of \(f(x)=\dfrac{ax}{x-q}\) and use them to find a missing parameter. (+3 more)

By the end of this section

Key points

Asymptotes of a rational function

  • For \(f(x)=\dfrac{ax}{x-q}\), the vertical asymptote is \(x=q\) (the denominator is 0) and the horizontal asymptote is \(y=a\).
  • The asymptotes meet at \((q,\,a)\). Use this to find q or a.
  • Write an asymptote as an equation, such as \(x=2\), not just the number.

A distance function

  • The distance between \((x_1,y_1)\) and \((x_2,y_2)\) is \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\).
  • A point on the curve \(y=f(x)\) is \((x,\,f(x))\). Put f(x) in place of y so the distance depends on x only.
  • For the curve \(y=\dfrac{ax}{x-q}\) and \(Q(q,a)\): \(y-a=\dfrac{aq}{x-q}\), so \(PQ=\sqrt{(x-q)^2+\left(\dfrac{aq}{x-q}\right)^2}\).
  • In a "show that" question, write each line. The result you are given is the check.

Minimising with the calculator

  • Graph the distance as Y1 and find the minimum. The minimum value is the shortest distance. The x-value is where it happens.
  • Read the question: it may ask for the distance, the x-coordinate, or the coordinates of the point. For coordinates, put x into f to find y.
  • A graph can have more than one minimum. Check each low point, and compare the distances.
  • Give answers to 3 significant figures.

On the calculator

Graphing a distance function

  • Enter the distance as Y1, using the square root key and brackets round each part.
  • Choose a window that shows the low points of the graph.
  • A distance is never negative, so the graph stays above the x-axis.

Screenshot and video to come

Finding a minimum

  • Use the graph's minimum tool. Set a lower bound left of the low point and an upper bound right of it.
  • Read both the x-value and the minimum value of the distance.
  • If the graph has two low points, find each one.

Screenshot and video to come

Explore: the closest point on a curve

Drag P along the curve. The orange line joins P to the fixed point, Q(2, 5) or A(7, 0). The small graph plots its length against x. Predict where the shortest line is, then check.

x
P

Drag P, or select it (Tab) and use the arrow keys.

Drag P along the curve.

  1. Move P towards the fixed point and then away from it. Where on the small graph is the shortest distance?
  2. On the small graph, what happens to the distance as x gets close to 2 (first curve)? Why?
  3. The first curve has two low points on the small graph. Are they equal? What does that say about the curve and Q?
  4. Is the closest point always the one with x nearest to the x-coordinate of the fixed point? Test it on the second curve.

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
The asymptotes of a rational function

Write down the equations of the vertical and horizontal asymptotes of the graph of \(f(x)=\dfrac{5x}{x-2}\), where \(x\neq2\).

−4−22468−5510xy
Working
  1. Vertical asymptote

    \(x-2=0\), so \(x=2\)

    The function is undefined where the denominator is 0. The curve climbs without limit beside that line.

  2. Horizontal asymptote

    \(f(x)=\dfrac{5}{1-2/x}\to 5\) as \(x\to\pm\infty\), so \(y=5\)

    Divide top and bottom by x. For large x, \(2/x\) is almost 0.

  3. State both

    Vertical asymptote \(x=2\); horizontal asymptote \(y=5\)

    Write each as an equation, not just a number.

Video walkthrough coming
Finding a parameter from the asymptotes

The graph of \(f(x)=\dfrac{7x}{x-q}\) has vertical and horizontal asymptotes that intersect at the point \(Q(-4,\,7)\). Find the value of q.

Working
  1. Asymptotes in terms of q

    Vertical: \(x=q\). Horizontal: \(y=7\)

    The denominator is 0 when \(x=q\).

  2. Intersection

    The asymptotes meet at \((q,\,7)\)

    A vertical line \(x=q\) and a horizontal line \(y=7\) meet at \((q, 7)\).

  3. Match Q

    \((q,\,7)=(-4,\,7)\), so \(q=-4\)

    Compare the x-coordinates. The function is \(\dfrac{7x}{x+4}\).

Video walkthrough coming
The distance from a point on the curve to Q

The point \(P(x,\,y)\) lies on the graph of \(f(x)=\dfrac{5x}{x-2}\). The asymptotes of the graph meet at \(Q(2,\,5)\). Show that \(PQ=\sqrt{(x-2)^2+\left(\dfrac{10}{x-2}\right)^2}\).

Working
  1. Distance formula

    \(PQ=\sqrt{(x-2)^2+(y-5)^2}\)

    P is \((x, y)\) and Q is \((2, 5)\).

  2. Write y - 5 in terms of x

    \(y-5=\dfrac{5x}{x-2}-5=\dfrac{5x-5(x-2)}{x-2}=\dfrac{10}{x-2}\)

    P is on the curve, so \(y=\dfrac{5x}{x-2}\). Put both terms over \(x-2\).

  3. Substitute

    \(PQ=\sqrt{(x-2)^2+\left(\dfrac{10}{x-2}\right)^2}\)

    This is the result to show. It depends on x only, so it can be minimised.

Video walkthrough coming
Closest points on the curve

The graph of \(f(x)=\dfrac{5x}{x-2}\) has asymptotes that meet at \(Q(2,\,5)\). The distance from a point \(P(x,\,y)\) on the graph to Q is \(PQ=\sqrt{(x-2)^2+\left(\dfrac{10}{x-2}\right)^2}\). Find the coordinates of the points on the graph that are closest to Q.

−4−22468−224681012xyQ
Working
  1. Enter the distance

    Y1 = \(\sqrt{(x-2)^2+(10/(x-2))^2}\)

    Graph PQ as a function of x. Choose a window that shows both low points.

  2. Minimum, right-hand branch

    From the calculator: minimum at x = 5.16, PQ = 4.47

    Set the bounds either side of the low point. The graph has two minimums.

  3. Minimum, left-hand branch

    From the calculator: minimum at x = −1.16, PQ = 4.47

    The two distances are equal: the curve is symmetrical about Q.

  4. −4−22468−224681012xyQ
    Find y

    \(y=\dfrac{5x}{x-2}\): points (5.16, 8.16) and (−1.16, 1.84)

    The question asks for coordinates, so put each x into f.

Video walkthrough coming
Shortest distance to a curve

What is the shortest distance from the point \(A(7,\,0)\) to the curve \(y=\sqrt{x}\)?

246810−2−112345xyA
Working
  1. A point on the curve

    \(P(x,\,\sqrt{x})\)

    Any point on the curve has this form.

  2. Distance function

    \(AP=\sqrt{(x-7)^2+(\sqrt{x})^2}=\sqrt{(x-7)^2+x}\)

    Use the distance between two points. \((\sqrt{x})^2=x\).

  3. Minimum

    From the calculator: minimum at x = 6.50, AP = 2.60

    Graph \(AP\) against x and find the minimum. Round to 3 significant figures.

  4. State

    Shortest distance = 2.60 units

    The distance, not the x-coordinate, is what is asked for.

Video walkthrough coming