Periodic functions (amplitude, period, shift, transformations): notes and practice questions

Consider the functions $f(x) = -3\cos x + 5$ and $g(x) = -3\cos\left(x+\frac{\pi}{2}\right) + 5 - k$, where $x \in \mathbb{R}$ and $k > 0$.

The graph of $g$ is obtained by two transformations of the graph of $f$.

Describe these two transformations.

The $y$-intercept of the graph of $g$ is at $(0, p)$.

Given that the maximum value of $g(x)$ is less than or equal to 1, find the largest possible value of $p$.

Consider the functions $f(x) = -2\cos(x) + 5$ and $g(x) = -2\cos(x - \frac{\pi}{4}) + 5 - k$, where $x \in \mathbb{R}$ and $k > 0$.

The graph of $g$ is obtained by two transformations of the graph of $f$.

(a) Describe these two transformations.

The $y$-intercept of the graph of $g$ is at $(0, s)$.

(b) Given that the maximum value of $g(x)$ is 1, find the value of $s$.

The depth of water, $D(t)$ metres, in a harbour on a particular day is modelled by the function $D(t) = a \cos(bt) + c$, for $0 \le t \le 12$, where $t$ is the number of hours after midnight.

The graph of $y=D(t)$ is shown below.

The graph has a maximum point at $(0, 12)$ and a minimum point at $(6, 4)$.

(a) State:

i) the range of $D$.

ii) the period of $D$.

(b) Hence, find the values of $a$, $b$ and $c$.

(c) A ship requires a water depth of at least 10 metres to be able to enter the harbour. Find the time intervals, during the first 12 hours, when the ship can enter.

A rotating arm on an industrial machine has a sensor attached to its end. The arm is 8 metres long. The lowest point reached by the sensor is 2 metres above the ground. The arm completes one full rotation in 15 seconds.

At time $t=0$ seconds, the sensor is at its lowest point.

The height, $h$ metres, of the sensor above the ground after $t$ seconds is modelled by the function $h(t) = a\cos (bt) + c$, where $a, b, c \in \mathbb{R}$.

Find the values of $a$, $b$, and $c$.

(b) Calculate the height of the sensor after 5 seconds.

(c) Find the first time, in seconds, when the height of the sensor is 7 metres.

The graph shows the height of two floating platforms, Platform A and Platform B, in a wave pool over a 24-second interval. The height, $h$ metres, is measured from the bottom of the pool. Time $t$ is in seconds.

The height of Platform A is modelled by the function $h_A(t) = p \cos(qt) + r$.

The height of Platform B is modelled by the function $h_B(t) = a \sin(bt) + d$.

Find the values of $p, q, r, a, b$ and $d$.

Solve the equation $h_A(t) = h_B(t)$ for $0 \le t \le 24$.

The daily temperature, in degrees Celsius, in a coastal city is modelled by the function $T(t) = a \sin(b (t - c) ) + d$, where $t$ is the number of hours after midnight, and $a, b, c$ and $d$ are constants, where $a > 0$, $b > 0$ and $c > 0$.

The following graph shows the temperature for a 24-hour period, starting at midnight.

The first peak temperature occurs at 14:00 (2 PM) and the next peak temperature occurs 24 hours later. Throughout the day, the temperature fluctuates between 14.5 °C and 28.5 °C.

All temperatures are given correct to one decimal place.

Show that $b = \frac{\pi}{12}$.

(b) Find the value of $a$.

(c) Find the value of $d$.

(d) Find the smallest possible value of $c$.

(e) Find the temperature at 06:00 (6 AM).

(f) Determine the number of hours, over a 24-hour period, for which the temperature is higher than 25 °C.

This question asks you to investigate the motion of a buoy bobbing up and down in the water.

A buoy bobs up and down in the water.

A fixed origin $\text{O}$ is the equilibrium position of the buoy (the water level).

The buoy's displacement, $y$ metres, from $\text{O}$ at time $t$ seconds is given by

$y = 6\sin\left(2t + \frac{\pi}{6}\right), \text{ for } 0 \le t \le \pi.$

Determine

the amplitude of the buoy's motion;

the buoy's initial displacement from $\text{O}$;

the value of $t$ when the buoy first passes through $\text{O}$.

Now consider the general case of a buoy bobbing up and down.

The buoy's acceleration is always directed towards a fixed origin $\text{O}$ at its equilibrium position.

The buoy's acceleration, $a$, at a displacement, $y$, from $\text{O}$ satisfies the differential equation

$a = -\omega^2 y, \text{ where } \omega > 0.$

The buoy's displacement, $y$, from $\text{O}$ at time $t$ is given by

$y = H\sin(\omega t + c), \text{ where } t \ge 0,\ H, \omega > 0 \text{ and } -\pi \le c \le \pi.$

By finding expressions for $\frac{\mathrm{d}y}{\mathrm{d}t}$ and $\frac{\mathrm{d}^2y}{\mathrm{d}t^2}$, verify that $y = H\sin(\omega t + c)$ satisfies the differential equation $a = -\omega^2 y$.

Use the chain rule to show that $a = v\frac{\mathrm{d}v}{\mathrm{d}y}$, where $v$ is velocity.

By solving the differential equation, $v\frac{\mathrm{d}v}{\mathrm{d}y} = -\omega^2 y$, show that $v^2 = \omega^2(H^2 - y^2)$.

Hence, or otherwise, find the buoy's maximum speed.

The continuous random variable $Y$ denotes the buoy's displacement, $y$, from $\text{O}$ at time $t$.

The probability density function $f$ of $Y$ is defined by

$f(y) = \begin{cases} \frac{1}{\pi\sqrt{H^2 - y^2}}, & -H < y < H \\ 0, & \text{otherwise.} \end{cases}$

Show that $\mathrm{P}\left(0 \le Y \le \frac{H\sqrt{3}}{2}\right) = \frac{1}{3}$.

For $-H < y < H$, the function $f(y)$ can be expressed in the form $\frac{m}{|v(y)|}$, where $m > 0$ and $v(y)$ is the buoy's velocity at a displacement, $y$, from $\text{O}$.

Find the value of $m$.

Determine $\mathrm{E}(Y)$, justifying your answer.

Interpret the result found in part (f)(i) in the context of the buoy's motion.

Consider the function $g(x) = p\sin(qx)$, where $p, q$ are positive integers. The following diagram shows part of the graph of $g$.

(a) Write down the value of $p$.

(b) (i) Find the period of $g$.

(b) (ii) Hence, find the value of $q$.

(c) Find the value of $g\left(\frac{\pi}{12}\right)$.

(a) A new Ferris wheel is being constructed. From a point A on the ground, 15 metres from the base of the support tower, the angle of elevation to the centre C of the Ferris wheel is 33.7 degrees. Find the height of point C above the ground.

(b) An engineer walks 5 metres closer to the support tower to point B. Find the angle of elevation of point C from point B, giving your answer in radians.

(c) The lowest point a passenger cabin reaches is 1.5 metres above the ground, allowing for easy boarding. Calculate the radius of the Ferris wheel.

(d) The height $h$, in metres, of a passenger cabin above the ground can be modelled by the function $h(t) = q - p \cos(kt)$, where $t$ is the time in seconds after the cabin starts moving from its lowest point. The Ferris wheel completes one full rotation in 40 seconds.

Find the values of $p$, $q$, and $k$.

(e) An observer watches the Ferris wheel for 5 minutes. How many times does a specific cabin pass its highest point during this time?

The vertical displacement, $h$ metres, of a weight oscillating on a spring at time $t$ seconds, is modelled by the function $h(t) = a \cos(bt)$, where $b > 0$. The graph of $y = h(t)$ for $0 \le t \le 5$ is shown below.

The graph has a maximum point at $(0, 5)$ and a minimum point at $(2, -5)$.

(a) Find the period of the function.

(b) Write down the amplitude of the function.

(c) Write down the value of $a$.

(d) Find the value of $b$.

The depth of water, $D$ metres, in a harbour is modelled by the function $D(t) = a \cos(bt) + k$, where $t$ is the number of hours after midnight.

The graph below shows the depth of water over a 24-hour period.

(a) Find the value of $k$, the mean water level.

(b) Find the value of $a$, the amplitude.

(c) Find the value of $b$.

(a) A scientist is modelling the population dynamics of a certain species of microorganism in a controlled environment. The growth rate is modelled by the function $f(x) = \sin x$, where $x$ is time in hours. Simultaneously, a decay factor due to a limited nutrient supply is modelled by $g(x) = e^{-x}$. The scientist is interested in finding the time $x$ when the growth rate equals the decay factor within the first $\pi$ hours of the experiment.

Use a sketch to help you solve $\sin x = e^{-x}$ for $0 \le x \le \pi$.

(b) An engineer is designing a new component where two critical dimensions, represented by functions $D_1(x) = 2 \ln x$ and $D_2(x) = x^2 - 3$, must be equal for optimal performance. The variable $x$ represents a design parameter, and its valid range is $0.5 \le x \le 2.5$.

Solve $2 \ln x = x^2 - 3$ for $0.5 \le x \le 2.5$.

The height, $h$ metres, of a passenger on a Ferris wheel $t$ minutes after the ride starts is given by $h(t) = a \cos(bt) + d$. The ride starts at $t=0$ when the passenger is at the lowest point. The lowest point is 1 metre above the ground. The diameter of the wheel is 20 metres. The wheel completes one revolution in 16 minutes.

(a) Find the maximum height reached by the passenger and hence find the value of $d$.

(b) Find the value of $a$.

(c) Show that $b = \frac{\pi}{8}$.

(d) The function $h(t)$ is transformed to a new function $g(t)$. The graph of $h$ is first translated by the vector $\begin{pmatrix} 4 \\ -1 \end{pmatrix}$ and then reflected in the $t$-axis. Find the equation of $g(t)$.

The height, $h$ metres, of a passenger on a Ferris wheel at time $t$ seconds, is modelled by the function $h(t) = a \cos(b(t-c) ) + d$, for $t \ge 0$. The graph of $y=h(t)$ for the first two revolutions is shown below.

(a) Write down the maximum and minimum height of the passenger.

(b) Find the time it takes for the Ferris wheel to complete one full revolution.

(c) Find the value of $d$ and the value of $a$.

(d) Show that $b = \frac{\pi}{30}$.

(e) Find a possible value for $c$.

(f) During the first two revolutions shown on the graph, find the times when the passenger is at a height of 7 metres.

The height, $h$ metres, of a passenger on a Ferris wheel at time $t$ minutes after the ride starts is modelled by the function $h(t) = a \sin(bt) + c$, for $0 \le t \le 8$.

The graph of $h$ for the first revolution is shown below.

The graph shows a maximum point at $(1, 90)$ and a minimum point at $(3, 10)$.

(a) (i) Write down the range of $h$.

(a) (ii) Find the period of $h$.

(b) Hence, find the values of $a, b$ and $c$.

(c) Find the times during the first 8 minutes when the passenger is at a height of 70 metres.

The intensity of light, $I$, emitted by a pulsating star is modelled by the function $I(t) = 2 + \cos(\pi \sin t)$, where $t$ is time in seconds.

(a) Sketch the graph of $y = I(t)$ for $0 \le t \le 2\pi$. Clearly label any maximum and minimum points and the $y$-intercept.

(b) (i) Find the coordinates of all minimum and maximum points of $I(t)$ in the given domain $0 \le t \le 2\pi$.

(ii) State the range of $I(t)$.

(iii) State the $y$-intercept of the graph of $y = I(t)$.

(c) The total light energy emitted by the star during a time interval is given by the area under the curve of $I(t)$.

(i) Write down an expression for the total light energy emitted by the star during the first $2\pi$ seconds.

(ii) Hence, determine the total light energy emitted by the star during the first $2\pi$ seconds.

(d) Show that the light intensity $I(t)$ is always positive for all $t \in \mathbb{R}$.