Question Type 3: Formulating sinusoidal models through descriptions Exercises

A function has amplitude $3$, period $\pi$, and principal axis $y=2$. Find a model of the form $f(x)=2+3\cos(bx+\phi)$ such that $f\left(\frac{\pi}{2}\right)=-1$ is a minimum value.

The tide has an amplitude of $2.5\text{ m}$, a period of $12\text{ hours}$, and a midline of $3\text{ m}$. At $t = 4$, the tide is at its minimum height and is rising.

Find a model for $H(t)$.

Amplitude 2, period 4 and principal axis -1. Find a model of the form $f(x)=-1+2\sin\left(b(x-c)\right)$ so that $f(1)=-1$.

Given an amplitude of 4, period 3 and principal axis 3, find a sinusoidal model of the form $f(x)=3+4\sin(bx+\phi)$ such that $f(0)=3$.

Express $\sin(5x+9\pi)$ in the form $\cos\bigl(5(x-c)\bigr)$.

Given amplitude 5, period $\frac{2\pi}{3}$ and principal axis 1, find a model of the form $f(x)=1+5\cos(bx+\phi)$ satisfying $f(0)=6$.

Write the expression $\sin \left( 3x - \frac{\pi}{4} \right)$ in the form $\cos(3(x - c))$. Hence, state the value of $c$.

A daily temperature ($^\circ\text{C}$) is modelled by a sinusoid with amplitude $10$, period $24\text{ h}$, and midline $15\,^\circ\text{C}$. The maximum temperature occurs at 3 pm ($t = 15\text{ h}$). Write a model $T(t)$.

Express $\cos\left(2x - \frac{\pi}{3}\right)$ in the form $\sin\left(2(x - c)\right)$.

A trigonometric function $f(x)$ has amplitude 3, period 4, and midline $y = 1$. At $x = 2$, the function attains its maximum value and is decreasing for values of $x$ immediately greater than 2.

Find an expression for $f(x)$.

A function $f$ has amplitude $6$, period $\pi$, and midline $-2$. The function is defined by $f(x) = -2 + 6\sin(bx + \phi)$ where $b > 0$ and $-\pi < \phi \le \pi$.

Given that $f\left(\frac{\pi}{4}\right) = -2 + \frac{6}{\sqrt{2}}$ and that the function is increasing at $x = \frac{\pi}{4}$, find the expression for $f(x)$.

Express $\cos\left(4x + \frac{\pi}{6}\right)$ in the form $\sin\left(4(x + c)\right)$.