IB Mathematics Applications & Interpretation SL 2.5—modelling Functions Notes

Why A Small Toolbox Of Shapes Models Almost Anything

1. Real data rarely comes with its equation attached; you are handed a scatter of points or a described situation and asked what rule fits.
2. The trick is that most situations follow one of a handful of recognisable shapes, and each shape is a family of functions with a standard form.
3. Modelling is a two-step habit: first choose the family whose shape matches the situation, then find the parameters that pin down the exact member of that family.
4. Your GDC does the fitting once you have chosen the family, so the thinking you are assessed on is the choice of shape and the meaning of each parameter in context.

Linear Models Describe A Constant Rate Of Change

1. A linear model is $f(x) = mx + c$, and it fits any quantity that changes by the same amount for each unit increase in the input.
2. The gradient $m$ is the constant rate of change and $c$ is the starting value when $x = 0$, exactly as in SL 2.1.
3. A piecewise linear model joins two or more straight-line pieces, each with its own gradient, to describe a rate that changes at set points.
4. Piecewise linear models fit stepped situations such as a phone tariff that charges one rate up to a cap and a higher rate beyond it.

Quadratic Models Have A Single Turning Point

1. A quadratic model is $f(x) = ax^2 + bx + c$, and its graph is a parabola with exactly one turning point.
2. The sign of $a$ sets the direction: $a > 0$ opens upward with a minimum, and $a < 0$ opens downward with a maximum.
3. The turning point is the vertex, and it sits on the axis of symmetry, the vertical line given in the formula booklet: $$ x = -\frac{b}{2a} $$
4. Substituting that $x$ back into the model gives the $y$-value of the vertex, which is the model's maximum or minimum.
5. The zeros, where $f(x) = 0$, are the $x$-intercepts, and the $y$-intercept is simply $c$.
6. Quadratics fit projectile heights, the area of a rectangle with a fixed perimeter, and profit that rises then falls.

Cubic Models Bend Twice And Grow Without Limit

1. A cubic model is $f(x) = ax^3 + bx^2 + cx + d$, and its graph can turn twice, giving one local maximum and one local minimum.
2. For large $|x|$ a cubic grows much faster than a quadratic, so one end shoots up and the other shoots down when $a > 0$.
3. Cubics fit quantities built from a length cubed, such as the volume of a box as a side length changes, and data that rises, dips, then rises again.
4. You locate the turning points and any intercepts of a cubic model on the GDC, just as for any other curve in SL 2.4.

Exponential Models Grow Or Decay By A Constant Factor

1. An exponential model multiplies by the same factor for each unit step, rather than adding the same amount as a linear model does.
2. The formula booklet gives two equivalent forms for the SL exponential model: $$ f(x) = k a^{x} + c \qquad f(x) = k e^{rx} + c $$
3. The base $a$ (or the rate $r$) controls the speed: $a > 1$ or $r > 0$ gives growth, and $0 < a < 1$ or $r < 0$ gives decay.
4. The constant $c$ is a horizontal asymptote $y = c$, the level the curve approaches but never reaches as the process settles.
5. The coefficient $k$ scales the curve and, together with $c$, fixes the starting value $f(0) = k + c$.
6. Exponential models fit populations, compound growth, radioactive or drug decay, and a hot drink cooling towards room temperature.

Direct And Inverse Variation Scale By A Power Of x

1. A variation model has the form $y = a x^{n}$, where $n$ is an integer and $a$ is a constant.
2. Direct variation uses a positive power, so $y$ grows as $x$ grows; the simplest case $y = ax$ is a straight line through the origin.
3. Inverse variation uses a negative power, most commonly $y = \dfrac{a}{x} = a x^{-1}$, so $y$ shrinks as $x$ grows.
4. The graph of $y = \dfrac{a}{x}$ has the axes as asymptotes, because $y$ blows up as $x \to 0$ and levels towards $0$ as $x$ grows large.
5. Inverse variation fits situations where a fixed total is shared out, such as the time for a journey against speed, or pressure against volume at fixed temperature.

Sinusoidal Models Describe Anything That Repeats

1. A sinusoidal model is $f(x) = a\sin(bx) + d$ or $f(x) = a\cos(bx) + d$, and its graph is a wave that repeats forever.
2. At SL the input is measured in degrees, so all of the numbers below use degrees.
3. The amplitude is $|a|$, the distance from the middle of the wave up to a peak or down to a trough.
4. The principal axis is the horizontal line $y = d$, the middle level the wave oscillates about.
5. The period is the horizontal length of one full cycle, and for these models it is: $$ \text{period} = \frac{360^\circ}{b} $$
6. So $b$ controls how many cycles fit in $360^\circ$: a larger $b$ squeezes the wave into a shorter period.
7. Sinusoidal models fit tides, daily temperature, hours of daylight through the year, and the height of a point on a Ferris wheel.

Choosing The Right Model From The Shape

1. Before fitting anything, match the described behaviour to a family, because each family has a signature shape.
2. The table below lines up each model with its standard form, its tell-tale shape, and a typical real-world use.

What The Formula Booklet Gives You, And What It Does Not

1. The functions section of the AI formula booklet prints the standard forms for you: the axis of symmetry $x = -\dfrac{b}{2a}$ of a quadratic, both exponential forms $ka^x + c$ and $ke^{rx} + c$, and the sinusoidal forms $a\sin(bx)+d$ and $a\cos(bx)+d$.
2. Because the forms are given, the assessed skill is not recalling them but choosing the right one and interpreting its parameters in context.
3. The meaning of each parameter, the gradient as a rate, $c$ as an asymptote, $|a|$ as amplitude, is not written in the booklet and must be understood, not looked up.