Modelling with the right graph

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What this lesson covers

  • A steady rate suggests a straight line.
  • Rising then falling suggests a parabola.
  • Levelling off suggests an exponential with an asymptote.
Worked example

Matching story to shape

  • Ask how the quantity changes.
SituationModel
Steady hourly payLinear
Path of a thrown ballParabola
Hot drink cooling to room temperatureExponential with an asymptote
Wait time as staff numbers riseHyperbola
Worked example

A ball's path

  • A ball reaches its highest point of 5 m after travelling 10 m, and it starts on the ground.
  1. h = a(x − 10)² + 5vertex (10, 5)
  2. 0 = 100a + 5starts at (0, 0)
  3. a = −0.05
  4. h = −0.05(x − 10)² + 5
  5. It lands at x = 20 m, by symmetry
Worked example

A cooling drink

  • T = 20 + 60(0.5)^(t/10), with t in minutes.
  1. t = 0: T = 80 °C
  2. t = 10: T = 50 °C
  3. t = 20: T = 35 °C
  4. It levels off at 20 °C, the asymptote

Common mistakes

  • Using a model outside the situation.
    A height below zero after landing is not part of the model, so state the domain.
  • Giving the numbers with no units.
    Finish in the context of the question.

Your turn

  • 1. An arch is 8 m wide and 6 m high. With the origin at the middle of its base, find its equation.
  • 2. How high is the arch 2 m from the middle?
Show answer

1. h = −0.375x² + 6 2. h = −1.5 + 6 = 4.5 m

Summary

What to take away

  • Let the behaviour choose the family.
  • State where the model stops making sense.

More in AS 91257

Families of graphsNotesTransforming graphsNotesFinding the equation from a graphNotesModelling with the right graphNotes

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