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, including their graphs and transformations, and to apply them to growth-and-decay and periodic phenomena. These function families are the modelling backbone of the course and reappear with calculus in Year 12.

Exponential functions

An exponential function $y = a^x$ (with $a > 0$, $a \neq 1$) is always positive, has horizontal asymptote $y = 0$, and passes through $(0, 1)$. When $a > 1$ it increases (growth); when $0 < a < 1$ it decreases (decay). Writing the base as $e$ gives the natural exponential $y = y_0 e^{kt}$, whose constant $k$ directly measures the proportional rate of change.

Exponential growth uses $y = y_0 e^{kt}$ with $k > 0$, and the doubling time is $t = \dfrac{\ln 2}{k}$. Exponential decay uses $y = y_0 e^{-kt}$ with $k > 0$, and the half-life is $t = \dfrac{\ln 2}{k}$. The doubling-time and half-life formulas both come from solving $2 = e^{kt}$ and $\tfrac{1}{2} = e^{-kt}$ respectively.

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.

• $\sin(x)$: wave, amplitude 1, period $2\pi$.
• $\cos(x)$: shifted sin, $y$-intercept 1.
• $\tan(x)$: 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 $D$.

The CAST diagram and exact values

Knowing the signs of the three ratios in each quadrant (the CAST rule: all positive in the first, sine in the second, tangent in the third, cosine in the fourth) lets you place every solution without a calculator. Combined with the exact values at $\tfrac{\pi}{6}$, $\tfrac{\pi}{4}$ and $\tfrac{\pi}{3}$ (for example $\sin\tfrac{\pi}{6} = \tfrac{1}{2}$, $\cos\tfrac{\pi}{4} = \tfrac{1}{\sqrt 2}$, $\tan\tfrac{\pi}{3} = \sqrt 3$), this is the foundation of calculator-free trigonometry in Paper 1. The reference angle (the acute angle to the $x$-axis) gives the size of the solution, and the quadrant gives its sign and position.

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

Exponentials model anything changing by a constant factor per period: compound interest, population growth and decay, and carbon dating (where the known half-life pins down the decay constant). Trigonometric functions model anything that repeats: tides, sound waves, daylight hours and oscillating systems. In a model $y = A\sin(B(x - C)) + D$, the amplitude $A$ is half the peak-to-trough range, $D$ is the midline (average value), the period $\tfrac{2\pi}{B}$ is the time for one cycle, and $C$ is the horizontal shift, so reading these off a worded context is the core modelling skill.