2.5 Modelling with functionsIB Maths: Applications and Interpretation HL: Revision notes

A linear model $f(x)=mx+c$ has a constant rate of change: the gradient $m$ is the change in $f$ per unit of $x$, and $c$ is the value when $x=0$ (the $y$-intercept). It links to the equation of a straight line (SL 2.1) and to an arithmetic sequence, whose terms increase by a constant common difference. A piecewise linear model uses a different line on different parts of the domain, for example mobile phone charges, the depth of a swimming pool or the horizontal distance of an object to a wall. Example: a taxi charges 12 AED for the first 2 km, then 2.50 AED per km. $F(x)=12$ for $0\leq x\leq2$ and $F(x)=12+2.5(x-2)$ for $x>2$. For a 10 km journey, $F(10)=12+2.5(8)=32$ AED.

A quadratic model is $f(x)=ax^2+bx+c$, $a\neq0$. Its graph is a parabola with:

• the $y$-intercept $(0,c)$
• the axis of symmetry $x=-\frac{b}{2a}$
• the vertex on that axis, a maximum if $a<0$ and a minimum if $a>0$
• zeros (roots) where $f(x)=0$, which you can find with your GDC. Example: profit $P(x)=-2x^2+24x-54$ thousand AED from selling $x$ hundred items. The GDC gives zeros $x=3$ and $x=9$, the axis is $x=-\frac{24}{2(-2)}=6$ and the maximum profit is $P(6)=18$ thousand AED. The model breaks even at 300 and 900 items.

An exponential model has the form $f(x)=ka^x+c$ (growth if $a>1$, decay if $0<a<1$; $f(x)=ka^{-x}+c$ with $a>1$ is also decay) or $f(x)=ke^{rx}+c$ ($r>0$ growth, $r<0$ decay). The graph has a horizontal asymptote $y=c$: it gets closer but never reaches it. Links: compound interest (SL 1.4), geometric sequences (SL 1.3) and amortization (SL 1.7). Example: 2000 AED invested at 4% compound interest gives $A(t)=2000(1.04)^t$, so $A(10)=2960.49$ AED. A cooling drink $T=20+60e^{-0.1t}$ has asymptote $T=20$, the room temperature.

A power model $f(x)=ax^n$ with $n\in\mathbb{Z}$ describes direct variation when $n>0$ (for example $y=ax$ or $y=ax^2$, area against radius) and inverse variation when $n<0$ (for example $y=\frac{a}{x}$). When $n<0$ the $y$-axis is a vertical asymptote, $x=0$. Example: $n$ workers take $T=\frac{60}{n}$ hours: 4 workers take 15 hours, and doubling the workers halves the time. A cubic model is $f(x)=ax^3+bx^2+cx+d$: it has up to 3 zeros and up to 2 turning points, and $d$ is the $y$-intercept. Example: cutting squares of side $x$ cm from the corners of a 20 cm by 15 cm sheet to make an open box gives $V=x(20-2x)(15-2x)=4x^3-70x^2+300x$. The GDC shows a maximum $V=379$ cm$^3$ at $x=2.83$, with domain $0<x<7.5$.

A sinusoidal model $f(x)=a\sin(bx)+d$ or $f(x)=a\cos(bx)+d$ repeats regularly. At SL you only need to find or predict:

• the amplitude $|a|$ (half the distance between the maximum and minimum)
• the period $\frac{360^\circ}{b}$
• the principal axis $y=d$, midway between the maximum $d+a$ and the minimum $d-a$. You do not need to convert between $\sin x$ and $\cos x$. Example: $T(t)=6\sin(15t^\circ)+24$ models temperature $t$ hours after 06:00. The amplitude is 6, the period is $\frac{360}{15}=24$ hours, the axis is $T=24$, the maximum is 30 °C and the minimum is 18 °C.

Choose the model from the shape and the context:

• constant rate of change: linear
• one turning point, symmetric: quadratic
• multiplying by a constant factor, or levelling off towards an asymptote: exponential
• decreasing curve with a vertical asymptote at $x=0$: inverse variation
• regular repeating pattern: sinusoidal. Always state the domain that makes sense, and answer in context with units, giving 3 significant figures (money to 2 d.p.).