IB Mathematics Applications & Interpretation 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 $f(x)=2+3\cos (bx+\varphi )$ 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 $$f(x)=3+4\sin (bx+\varphi )$$ such that $$f(0)=3$$.

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

Given amplitude 5, period $\frac{2\pi }{3}$ and principal axis 1, find a model of the form $f(x)=1+5\cos (bx+\varphi )$ 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 $$f(x)=-2+6\sin (bx+\varphi )$$ where $b>0$ and $-\pi <\varphi \le \pi$.

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 $\sin \left(4(x+c)\right)$.