🏗️Piecewise Functions & Modelling

Cartographer Iris draws the shoreline of Graph Lake in two strokes, because the shore does two different things. Up to the boathouse it runs straight; past the boathouse it curves away. One line would be a lie about half the lake.

A piecewise function is written the same way: one function, several rules, each with the stretch of the number line it applies to.

f(x) = 2x + 1 if x < 3 f(x) = x² - 4 if x >= 3

To evaluate it, do the two jobs in this order — and never the other way round.

• ◆Choose the piece. Test the input against each condition until one holds.
• ◆Then calculate. Substitute into that rule alone.

So f(1) uses the first line, because 1 < 3, giving 3. And f(5) uses the second, because 5 >= 3, giving 21.

The interesting input is always the boundary. Exactly one condition may claim it, and you can see which by the inequality signs: <= and >= include the endpoint, < and > exclude it. Here x = 3 satisfies x >= 3, so f(3) = 3² - 4 = 5 — the first rule stops just short of 3 and never gets a say.

On a graph this shows as two kinds of dot at each boundary: a filled dot where the piece includes its endpoint, and a hollow dot where it stops just short. A well-drawn piecewise graph never has two filled dots stacked above the same x, because a function is allowed only one output per input.

A step function is a piecewise function whose pieces are all constants — the graph is a flight of stairs. Step pricing is everywhere: a locker at $4 for each hour or part of an hour, postage by 100-gram band, a car park that rounds up. "Or part of an hour" means the count is rounded up, never to the nearest, so 2 hours 5 minutes is charged as 3 hours.

A straight line has one slope everywhere. A curve does not — it steepens and flattens as it goes. The average rate of change measures the curve the honest way: draw the straight line between the two ends of an interval and take *its* slope.

average rate of change on [a, b] = (f(b) - f(a)) / (b - a)

That is exactly rise over run, with the rise measured in whatever the output measures and the run in whatever the input measures. Which is why the answer always carries a meaning: metres per second, dollars per kilogram, degrees per hour.

Three sources give the same calculation.

• ◆From a rule. Work out f(a) and f(b), then subtract and divide.
• ◆From a table. Only the two rows at the ends of the interval matter. The rows in between tell you the function is curving, but they play no part.
• ◆From two readings in a story. "At 2 seconds it had gone 5 metres; at 6 seconds it had gone 23 metres" is a table with two rows.

Two things go wrong, and both are worth naming before you meet them. The first is stopping after the subtraction: f(b) - f(a) is a *change*, not a *rate*, and it only becomes a rate once it is divided. The second is dividing by b instead of b - a. The denominator is the width of the interval, so on [2, 6] you divide by 4, never by 6.

The sign carries information too. A negative average rate of change means the quantity fell across that interval, and a rate of zero means it ended where it started — which does not mean it stayed there in between.

Most of Grade 12 modelling is one recipe, applied to a drawing.

1. Name the variable. Pick the one length, price or time everything else depends on, and call it x. 2. Write every other quantity in terms of x, using the constraint the question gives you. 3. Combine them the way the quantity you want is defined — multiply for an area or a volume, multiply price by quantity for revenue. 4. State the domain that makes sense, which is rarely all of the real numbers.

Three models come up again and again.

A rectangle with a fixed perimeter. If the perimeter is 40 cm, then width + length = 20, because a perimeter counts each side twice. With width x the length is 20 - x, so

A(x) = x(20 - x)

A garden fenced on three sides. With a wall along the fourth side and 50 metres of fencing, the two sides running out from the wall use 2x metres, leaving 50 - 2x for the side parallel to the wall:

A(x) = x(50 - 2x)

An open box cut from a sheet. Cut a square of side x from each corner of a 20 by 12 sheet and fold the flaps up. The flaps have height x, so x is the depth. Each dimension loses a corner square at both ends, which is 2x off each:

V(x) = x(20 - 2x)(12 - 2x)

Revenue is the same recipe with money. If a price of p dollars sells 400 - 30p lanterns, then revenue is price times quantity: R(p) = p(400 - 30p).

The domain is part of the model, not an afterthought. A width cannot be negative and cannot use up the whole half-perimeter, so the rectangle above only makes sense for 0 < x < 20. In the box, 12 - 2x must stay positive too, so 0 < x < 6. An answer outside the sensible domain is a signal that something was set up wrongly, not a discovery.

Once a model exists there are only four things anyone ever asks of it.

• ◆A value. Substitute. N(3) for N(t) = 500(1.08)^t means raising 1.08 to the power 3 first, then multiplying by 500.
• ◆A largest or smallest value. For a quadratic, that is the vertex, which sits exactly halfway between the two zeros. A(x) = x(20 - x) is zero at x = 0 and x = 20, so the vertex is at x = 10 — and the largest area is A(10) = 100. Notice that "which width" and "what area" are different questions with different answers.
• ◆A return to zero. Factor and set each factor to zero. Then read the roots in context: for a thrown stone the useful root is the positive one.
• ◆The meaning of a parameter. In N(t) = 500(1.08)^t the 500 is the starting population, because N(0) = 500. The 1.08 is what one year *does* — it multiplies. Multiplying by 1 would change nothing, so the growth is the 0.08 above 1, that is 8 % a year.

Choosing a family is the same question asked backwards: what stays constant as the input steps forward?

• ◆constant first differences — an equal amount added each step — is linear
• ◆constant second differences is quadratic
• ◆a constant ratio between neighbouring values is exponential
• ◆values that repeat on a fixed cycle are sinusoidal
• ◆a constant product xy, so the output is a fixed number divided by the input, is rational
• ◆two rules in two regions is piecewise

Given a table, take the first differences. If they are equal, stop — it is linear. If not, try the second differences and the ratios. If neither settles, multiply the two columns together and see whether that is constant.

Variation language is a compact way of saying which letters sit above the line and which sit below it. The constant k, called the constant of variation, always multiplies.

• ◆y varies directly as x: y = kx
• ◆y varies inversely as x: y = k/x
• ◆z varies jointly as x and y: z = kxy
• ◆y varies directly as x and inversely as z (combined): y = kx/z
• ◆y varies inversely as the square of x: y = k/x²

Every variation problem has the same two stages. Find k from the pair of values you are given, then use that k with the new values. For direct variation with y = 12 when x = 3, substituting gives 12 = 3k, so k = 4, and the model is y = 4x.

Inverse variation is where the sign of your instincts has to flip. Because y = k/x rearranges to xy = k, the product of the two quantities is the same for every pair — so when x doubles, y halves. Finding k means multiplying the given pair, not dividing it.

An inverse square is fiercer still. In y = k/x², doubling x divides y by four, and trebling it divides y by nine. That is why light and sound fade so quickly with distance.