Exponential, logarithmic and trigonometric functions (including their graphs and transformations), and applications to growth and decay and periodic phenomena

What this dot point is asking

QCAA wants Year 11 students to extend their understanding of exponential, logarithmic and trigonometric functions with applications.

Exponential functions

$y = a^x$ for $a > 0, a \neq 1$. Always positive, horizontal asymptote $y = 0$, $y$-intercept $(0, 1)$.

Exponential growth: $y = y_0 e^{kt}$ with $k > 0$. Doubling time $t = \ln 2 / k$.

Exponential decay: $y = y_0 e^{-kt}$ with $k > 0$. Half-life $t = \ln 2 / k$.

Logarithmic functions

$y = \log_a(x)$ inverse of $y = a^x$. Defined for $x > 0$, vertical asymptote $x = 0$, $x$-intercept $(1, 0)$.

Natural log: $\ln x = \log_e x$.

Trigonometric functions

Unit circle. Point at angle $\theta$: $(\cos\theta, \sin\theta)$.

Exact values at $0, \pi/6, \pi/4, \pi/3, \pi/2, \pi, 3\pi/2, 2\pi$.

Graphs.

• IMATH_21 : wave, amplitude 1, period $2\pi$.
• IMATH_23 : shifted sin, $y$-intercept 1.
• IMATH_25 : period $\pi$, asymptotes at $\pi/2 + \pi k$.

Identities. $\sin^2\theta + \cos^2\theta = 1$. Even/odd: $\sin(-\theta) = -\sin\theta$, $\cos(-\theta) = \cos\theta$.

Transformations. $y = A\sin(B(x - C)) + D$: amplitude $|A|$, period $2\pi/|B|$, phase shift $C$, vertical shift $D$.

Solving trig equations

Find principal solutions, then use symmetry/periodicity for all in range.

For $\sin x = k$: principal $x_1 = \arcsin(k)$; second $x_2 = \pi - x_1$.

For $\cos x = k$: principal $x_1 = \arccos(k)$; second $x_2 = -x_1$ or $2\pi - x_1$.

For $\tan x = k$: principal $x_1 = \arctan(k)$; add multiples of $\pi$.

Applications

• Compound interest (exponential).
• Population growth/decay (exponential).
• Carbon dating (exponential decay).
• Tides, sound, oscillating systems (trigonometric).

Common errors

Calculator in degrees. VCE/QCE Methods uses radians.

Missing solutions. $\sin x = 1/2$ has two solutions per period.

Reciprocal vs inverse. $\sin^{-1}$ is inverse function (arcsin), not $1/\sin$.

In one sentence

Unit 2 extends function families to exponential (growth and decay), logarithmic (inverse of exponential), and trigonometric (unit circle, exact values, graphs of $\sin$, $\cos$ and $\tan$ with their transformations); solving trig equations uses principal solutions plus symmetry / periodicity.