Connecting graphs, equations and contexts

What "connecting representations" means

• The standard names "connecting different representations of relations" as one of its methods.

• The four representations of the same relationship are:

  • a table of values
  • a graph
  • an equation
  • a description in words or a real context

• Merit and Excellence are earned by moving between them, not by working within one. A question that gives you a context and asks for a graph is testing exactly this.

Choosing the right family for a context

• Learn these shapes by sight. Most of the marks in this topic come from matching the shape of the data to the right family before any algebra begins — a model from the wrong family cannot be rescued by good arithmetic.
• Reading the situation tells you the family before any algebra:

Situation	Family
A constant rate of change	Linear
A quantity changing by a fixed percentage each period	Exponential
Something thrown or dropped; area against a length	Quadratic
Volume against a length	Cubic
Two quantities with a constant product (fixed area, fixed distance)	Hyperbolic
Something that grows quickly then flattens off	Logarithmic

• The give-away for exponential is "per cent" or "doubles" or "halves". The give-away for hyperbolic is "the more of one, the less of the other, with the product fixed".

Building a model from data

• Identify the family from the wording or the shape of a plot.

• Write the general form with unknown constants.

• Substitute the given data points to make equations.

• Solve for the constants.

• State the model, and state its practical domain.

• Worked through — a population is 500 at the start and grows 8% a year:

  • Exponential, so $P = a\,b^t$
  • $a = 500$ (the starting value) and $b = 1.08$ (the multiplier)
  • $P = 500(1.08)^t$, valid for $t \geq 0$

Interpreting the features in context

• Every graphical feature means something in the situation, and saying so is what earns Merit:

Feature	Typical meaning
$y$-intercept	the starting value, at time zero
$x$-intercept	when the quantity reaches zero
Gradient	the rate of change, with units from the axes
Vertex (maximum)	the greatest value, and when it occurs
Vertex (minimum)	the least value
Horizontal asymptote	a limiting value the quantity approaches
Vertical asymptote	a value the input can never take

• Give the units. "The gradient is $-4$" is incomplete; "the volume falls by 4 litres per minute" is the answer.

Stating the practical domain

• The mathematical domain is what the equation allows; the practical domain is what the situation allows.
• A model $h = 30t - 5t^2$ for a ball works only while the ball is airborne, so $0 \leq t \leq 6$.
• Beyond the practical domain the mathematics keeps producing numbers, but they describe nothing. Saying where a model stops applying is a genuine Excellence-level observation.

Evaluating a model

• At Excellence you may be asked whether a model is reasonable:

  • Does it fit the data at the points you were given?
  • Does it behave sensibly between and beyond them?
  • What does it predict at extreme values, and is that plausible?
  • What assumptions does it make?

• An exponential population model, for instance, predicts unbounded growth — which cannot continue indefinitely with finite food and space. Naming that limitation is worth more than another calculation.

Reading a graph accurately

• Check the scales on both axes before reading anything. They are often different, and often do not start at zero.
• Read the axis labels and units.
• Interpolating — reading between plotted points — is usually reasonable; extrapolating beyond the data is much less reliable.