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AI SL · 12.5

Functions

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12.5.5

Review: the approach model and graph transformations

12.5.5

Review: the approach model and graph transformations

Target: I can explain why the constant a in \(D=a+bk^{-t}\) is the total distance, using the graph. (+3 more)

By the end of this section

Key points

The approach model

  • \(D=a+bk^{-t}\) with \(k>1\): as t grows, \(k^{-t}\to0\), so D tends to a.
  • a is the horizontal asymptote and the total distance. D gets close to a but never reaches it.
  • Put \(t=0\) to find b: \(D(0)=a+b\). If the start is 0, then \(b=-a\).
  • Remaining distance = \(a-D\).

Finding k and t

  • Use one more value, such as D(1), to get an equation in k. Remember \(k^{-1}=1/k\).
  • Check that k > 1. If you get k < 1, the power has gone wrong.
  • To find a time, write D = value and solve for t on the calculator. Round only at the end, and to the nearest hour when asked.

Reviewing the unit

  • Composite \(f(g(x))\): work from the inside out. Inverse: swap x and y, and the graph reflects in \(y=x\).
  • Transformations of a graph: \(f(x)+c\) up, \(f(x+c)\) left, \(pf(x)\) vertical stretch, \(f(qx)\) horizontal stretch by \(1/q\), \(-f(x)\) and \(f(-x)\) reflections.
  • For the closest point on a curve, write the distance as a function and find its minimum with the calculator.

On the calculator

Solving the model for t

  • Write the equation D = value.
  • Solve for t on the calculator, with t as the variable.
  • Round at the end, to the nearest hour if the question says so.

Screenshot and video to come

Checking with a graph

  • Graph the model and the line D = value.
  • The t-value where they cross is the time.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Why a is the total distance

A sailing boat heads for a buoy. Its distance travelled, D km, after t hours is modelled by \(D=12-12\times1.5^{-t}\) for \(t\ge0\). The graph is shown. The model has the form \(D=a+bk^{-t}\) with \(k>1\). Explain why a is the total distance the boat can travel, and state its value.

2468101224681012t (hours)D (km)D
Working
  1. Large t

    \(t\to\infty\): \(1.5^{-t}=\dfrac{1}{1.5^{t}}\to0\)

    Since k > 1, the power k^t grows without limit, so k^(−t) gets as close to 0 as you like.

  2. The limit

    \(D\to12+(-12)\times0=12\)

    The curve levels off at the line D = 12 (the dashed line). It gets near 12 but never passes it.

  3. Meaning

    a = 12 km

    D cannot go above 12, and it comes as close to 12 as you want. So a is the total distance.

Video walkthrough coming
Finding b

A drone flies towards a rock. Its distance travelled, D metres, after t hours is modelled by \(D=8+bk^{-t}\), where \(k>1\). At the start, \(D=0\). Find b.

Working
  1. Put t = 0

    \(D(0)=8+bk^{0}=8+b\)

    Any number to the power 0 is 1, so k disappears.

  2. Use D(0) = 0

    \(8+b=0\)

    The drone has travelled nothing at the start.

  3. Solve

    b = −8

    b is negative. That is why D rises towards 8 instead of falling.

Video walkthrough coming
Finding k

The model is now \(D=8-8k^{-t}\) with \(k>1\). After 1 hour the drone has travelled 3 m. Find k.

Working
  1. Put t = 1

    \(8-8k^{-1}=3\)

    D(1) = 3.

  2. Isolate the power

    \(8k^{-1}=5\), so \(\dfrac{8}{k}=5\)

    Subtract 3 from 8, then use k^(−1) = 1/k.

  3. Solve

    k = 8 ÷ 5 = 1.6

    k is greater than 1, as the model needs.

Video walkthrough coming
Remaining distance

Using \(D=8-8\times1.6^{-t}\) (D in metres, t in hours), find how much further the drone has to travel after 4 hours to reach the total distance.

Working
  1. Distance travelled

    \(D(4)=8-8\times1.6^{-4}=6.779\)

    Evaluate on the calculator.

  2. Remaining

    8 − 6.779 = 1.22 m

    Remaining = a − D. Here it is the term \(8\times1.6^{-4}\) itself.

Video walkthrough coming
Time to reach a distance

Using \(D=8-8\times1.6^{-t}\), find the time, to the nearest hour, when the drone has travelled 7.9 m.

Working
  1. Write the equation

    \(8-8\times1.6^{-t}=7.9\)

    Set D equal to 7.9.

  2. Solve for t

    Solve for t: t = 9.32

    Solve on the calculator. A graph of the two sides crossing gives the same value.

  3. Round

    t = 9 hours

    Round to the nearest hour, as asked.

Video walkthrough coming