2.9 Further modelling functionsIB Maths: Applications and Interpretation HL: Revision notes

Exponential models such as $f(x)=ka^{x}+c$ or $ke^{rx}+c$ extend to decay with a half-life: the time for a quantity to halve. If the half-life is $T$ and the initial amount is $m_0$: $m=m_0\left(\tfrac12\right)^{t/T}.$ Example: $m=80\left(\frac12\right)^{t/6}$ has half-life 6 hours; after 18 hours, $80\left(\frac12\right)^3=10$ mg. To find the time for a given mass, solve $80\left(\frac12\right)^{t/6}=5$ by GDC or algebra: $\left(\frac12\right)^{t/6}=\frac1{16}$, so $t=24$. The half-life is constant whatever the starting amount.

A logarithmic model is $f(x)=a+b\ln x$, with $x>0$. Because $\ln1=0$, the constant $a$ is the value at $x=1$. For $b>0$ the function increases without bound, but ever more slowly, and the $y$-axis ($x=0$) is a vertical asymptote. To find $a$ and $b$, substitute two known points. Example: $H=a+b\ln x$ with $H=2$ at $x=1$ and $H=5$ at $x=e^2$ gives $a=2$ and $5=2+2b$, so $b=1.5$. Then $H=7$ when $2+1.5\ln x=7$, $x=e^{10/3}=28.0$ (GDC).

A sinusoidal model is $f(x)=a\sin(b(x-c))+d$. Radians are assumed unless a degree sign is shown, for example $f(x)=\sin x^\circ$.

• Amplitude: $|a|$
• Period: $\frac{2\pi}{b}$ (radians)
• Phase shift: $c$, a horizontal translation to the right by $c$
• Principal axis: $y=d$, so maximum $d+|a|$ and minimum $d-|a|$ Example: $d(t)=2.5\sin\left(\frac{\pi}{6}(t-1)\right)+6$ has amplitude 2.5, period 12, phase shift 1 and range $3.5\le d\le8.5$. To find when $d\ge7$ in one cycle, solve $d(t)=7$ with the GDC in radian mode: $t=1.79$ and $t=6.21$.

A logistic model is $f(x)=\frac{L}{1+Ce^{-kx}}$ with $L,C,k>0$. It describes growth with a limit, such as a population on an island, bacteria in a dish or the height of a seedling.

• $y=L$ is the horizontal asymptote, the carrying capacity.
• $f(0)=\frac{L}{1+C}$ is the starting value.
• The graph is S-shaped: growth is slow at first, fastest in the middle, then slows towards $L$. Example: $N(t)=\frac{480}{1+11e^{-0.9t}}$ has $N(0)=40$ and carrying capacity 480. $N=300$ when $1+11e^{-0.9t}=1.6$, so $t=3.23$.

A piecewise model uses different rules on different intervals. To avoid a jump, make both rules give the same value at the join. Example: $f(x)=1+x$ for $0\le x<2$ and $f(x)=ax^2+x$ for $x\ge2$. The first rule gives $3$ at $x=2$, so $4a+2=3$ and $a=\frac14$. Always check which piece applies before you substitute or solve: a value of $t$ found from the wrong piece must be rejected.

Match the model to the context: sinusoidal for repeating cycles (tides, temperature, daylight), logistic for limited growth, logarithmic for growth that keeps slowing, exponential for constant percentage change and half-life. In the exam you may also be given an unfamiliar model; treat it like the others by substituting values, using your GDC to solve, and interpreting each parameter in context. State units, give answers to 3 significant figures, and comment on limits such as extrapolation beyond the data.