Transformations of graphs (translation, reflections, stretches): notes and practice questions

A landscape architect is designing a decorative water channel. The cross-section of the channel is modelled by the function $f(x) = x^3 - 4x$, for $-2 \leq x \leq 2$. The shaded region, $R$, represents the cross-sectional area of the channel, bounded by the graph of $y = f(x)$ and the $x$-axis.

and below for x in [0,2].]

Write down an integral that represents the area of $R$.

Find the area of $R$.

The architect considers a modified design, where the cross-section is given by $g(x) = 0.5f(x+1)$.

On the following set of axes, the graph of $y = f(x)$ has been drawn. On the same set of axes, sketch the graph of $y = g(x)$.

The region $R$ (the original cross-section) is rotated through $2\pi$ radians about the $x$-axis to form a three-dimensional decorative element. Find the volume of this element.

The graph of $y = f(x)$ represents the concentration of a certain chemical in a solution (in mol/L) at time $x$ (in hours). The graph passes through the points $(1, 10)$ and $(3, 4)$, and has a horizontal asymptote at $y = 2$.

Let $g(x) = 3f(x + 1) - 5$ represent the concentration in a different experiment.

Find $g(0)$.

On a new set of axes, sketch the graph of $y = g(x)$, clearly indicating its horizontal asymptote and the $y$-intercept. You do not need to show the graph of $f(x)$.

(a) A designer is creating a prototype for a decorative vase. The cross-section of the vase can be modelled by the function $f(x) = x\sqrt{4-x^2}$, for $-2 \le x \le 2$.

Sketch the graph of $y = f(x)$ on the following pair of axes.

(b) The region enclosed by the graph of $y = f(x)$ and the x-axis is rotated $360^\circ$ about the x-axis to form the body of the vase.

(i) Write down an integral that represents the volume of this vase.

(ii) Calculate the value of this integral.

(c) The designer decides to create a new, larger version of the vase, $y = g(x)$, by applying the following transformations to the original cross-section $y = f(x)$:

• A horizontal stretch by a scale factor of 3, parallel to the x-axis.

• A vertical stretch by a scale factor of 0.75, parallel to the y-axis.

Find the volume of this new vase.

A civil engineer is designing a new road. A preliminary section of the road is modelled by a straight line with equation $y = kx + d$.

Find the vectors $\mathbf{a}$ and $\mathbf{b}$ such that the equation of the line can be expressed in vector form $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$ in terms of $k$ and/or $d$.

Before construction, a ground transformation is applied to the design. This transformation is described by the matrix $T = \begin{pmatrix} 4 & 2 \\ 6 & 3 \end{pmatrix}$.

Calculate the value of $\det T$.

The preliminary road section $y = kx + d$ (where $k \neq -2$) undergoes the transformation described by matrix $T$.

Show that the equation of the resulting transformed path does not depend on $k$ or $d$.

A landscape architect is designing a large decorative fountain for a new public park. The outer structure of the fountain consists of a cylindrical base topped by a conical section. The inner part of the fountain, which holds the water, is a hollow space created by rotating a specific curve around the vertical axis.

The shape of the inner hollow is based on a transformation of the graph $y = -x^3$. The curve that defines the profile of the inner hollow is given by $y = 40 - 0.064x^3$. This transformation involves a vertical translation of $a$ units and a stretch parallel to the x-axis with a scale factor of $b$.

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

(a.ii) Find the value of $b$.

The cylindrical base of the fountain has a radius of $15$ m and a height of $5$ m. The conical section on top has the same base radius of $15$ m and a height of $8$ m. The inner hollow, described by the curve $y = 40 - 0.064x^3$, extends from the base of the fountain ($y=0$) up to the total height of the outer structure.

Find the volume of the solid material that makes up the fountain (i.e., the volume of the outer structure minus the volume of the inner hollow).

A civil engineer is designing a section of a new roller coaster track. She collects data points from a preliminary design sketch to model a specific curved section.

Here are the coordinates of five points on the track's profile:

The engineer thinks a cubic curve will be a good model for this section of the track.

Find the equation of the cubic regression curve for this data.

For another section of the roller coaster, the engineer first creates a small-scale model. The equation for the profile of this model section is given by $y = 0.02x^3 - 0.3x^2 + 1.5x + 5$.

The full-size roller coaster track will be an enlargement of this model, with a scale factor of 5, centered at the origin (0,0).

Determine the equation of the cubic curve that models the full-size roller coaster track.

The "SkyGazer" is a new observation wheel in a city park. The wheel has a diameter of $70$ m. To begin the ride, a passenger enters a capsule at the lowest point on the wheel, which is $3$ m above the ground. A ride consists of multiple revolutions, and the wheel makes $2$ revolutions per minute.

The height of a capsule above the ground, $h$, measured in metres, during a ride on the SkyGazer can be modelled by the function $h(t) = -a \cos (bt) + d$, where $t$ is the time, in seconds, since a passenger began their ride.

(a) Calculate the value of

(i) $a$;

(a)(ii) $b$;

(a)(iii) $d$.

(b) A ride on the SkyGazer lasts for $10$ minutes in total.

Calculate the number of revolutions of the wheel per ride.

(c) For exactly one ride on the SkyGazer, suggest

(i) an appropriate domain for $h(t)$;

(c)(ii) an appropriate range for $h(t)$.

(d) A $20$ metre-tall building stands on the horizontal ground next to the SkyGazer.

By considering the graph of $h(t)$, determine the length of time during one revolution of the wheel for which the capsule is higher than the building.

(e) There is a plan to relocate the SkyGazer onto a taller platform which will increase the maximum height of the wheel to $75$ m. This will change the value of one parameter, $a$, $b$ or $d$, found in part (a).

(i) Identify which parameter will change.

(e)(ii) Find the new value of the parameter identified in part (e)(i).

A drone is launched vertically upwards from a platform. Its height, $h$ metres above the ground, $t$ seconds after launch, is given by the formula $h = c + ut - 5t^2$, where $c$ is the initial height of the platform and $u$ is the initial upward velocity.

The drone is launched from a platform $2.5$ m above the ground with an initial upward velocity of $40$ ms$^{-1}$.

Calculate the time taken for the drone to return to the height of the launch platform.

Find the time it takes for the drone to reach its maximum height.

Calculate the maximum height of the drone.

Another drone operator, Liam, launches an identical drone with the same initial upward velocity, but from a platform $3.0$ m above the ground.

By considering the difference in the two height-time graphs for the drones, write down the answers to parts (a), (b), and (c) for Liam's drone.

(a) (i) Consider the function $f(x) = x^3$. Find $f'(x)$.

(ii) The first section of the stone wall's profile is given by $y = f(x)$ for $0 \le x \le 0.5$. A straight glass panel is to be installed tangent to this section of the wall at the point where $x=0.5$. Find the equation of this tangent line.

The full profile of the stone wall, $F(x)$, is defined by:

$F(x) = \begin{cases} x^3 & 0 \le x \le 0.5 \\ 0.75x - 0.25 & 0.5 < x \le 1.0 \end{cases}$

A smaller, decorative stone insert is designed using a transformation of $F(x)$. The graph of $G(x)$ is obtained from the graph of $F(x)$ by:

• a stretch scale factor of $\frac{1}{2}$ in the $x$ direction,

• followed by a stretch scale factor of $\frac{1}{2}$ in the $y$ direction,

• followed by a translation of $0.5$ units to the right.

Point P lies on the graph of $F(x)$ and has coordinates $(1.0, 0.5)$. Point Q is the image of P under the given transformations and has coordinates $(q_x, q_y)$.

Find the value of $q_x$ and the value of $q_y$.

The piecewise function $G(x)$ is given by

$G(x) = \begin{cases} k(x) & c \le x \le d \\ mx + n & d < x \le q_x \end{cases}$

(c) Find

(i) an expression for $k(x)$.

(ii) the value of $d$.

(iii) the value of $n$.

(d) (i) Calculate the total area of the profile of the stone wall, enclosed by $y = F(x)$, the $x$-axis, and the line $x = 1.0$.

The decorative insert $G(x)$ is placed within the main wall profile $F(x)$. The region of the main wall profile that is not covered by the insert is to be painted a contrasting colour. This region is bounded by $y=F(x)$, the $x$-axis, and the lines $x=0$ and $x=1$, excluding the area under $G(x)$ from $x=0.5$ to $x=1.0$. Find the area of this region.

Consider the function $f(x) = |x|$.

(a) The graph of the function is translated vertically to define a new function $g(x) = f(x) + k$. Find the value of $k$ such that the graph of $g(x)$ passes through the point $(2, 7)$.

(b) The graph of the function is translated horizontally to define a new function $h(x) = f(x - l)$. Find the values of $l$ such that the graph of $h(x)$ passes through the point $(5, 3)$.

(c) The graph of the function is translated to define a new function $m(x) = f(x - r) + s$. Find the values of $r$ and $s$ such that the minimum point of $m(x)$ is $(-1, 6)$.

(d) The graph of the function is transformed to define a new function $n(x) = af(x - b) + c$. Find the values of $a, b$ and $c$ such that the maximum point of $n(x)$ is $(4, 10)$ and the graph passes through the point $(6, 6)$.

A design studio is manufacturing a decorative wall sconce. The cross-sectional profile of the sconce is defined by two mathematical curves. The sconce is formed by rotating the region between these curves through $\pi$ radians about the $x$-axis, creating a flat rear surface that mounts flush against a wall. All linear dimensions are in centimetres.

The curve of the outer profile is given by $f(x) = 6 \cos\left(\frac{\pi}{12}x\right)$, for $0 \le x \le 6$.

The curve of the inner profile, $g(x)$, is formed by translating the graph of $f$ by $0.6$ units to the left and $1.2$ units down.

(a) Write down an expression for $g(x)$.

(b) The curve $g(x)$ intersects the $x$-axis at $(a, 0)$, where $0 \le x \le a$ defines the inner boundary of the sconce.

Find the value of $a$. Give your answer to three significant figures.

(c.i) Write down an expression for the volume of the solid formed.

(c.ii) Hence find the volume of material used in the sconce. Give your answer to three significant figures.

The depth of water, $D$ metres, at the entrance to a harbour can be modelled by the function $D(t) = A + B\cos(Ct)$, where $t$ is the time in hours after midnight and $C$ is a constant in radians per hour. The graph below shows the depth of water over a 24-hour period.

(a) On the same axes, sketch the graph of the function $D_2(t) = (A-1) + \frac{B}{2}\cos(Ct)$.

(b) Determine the values of the constants $A$, $B$, and $C$.

A design studio is developing a wall-mounted decorative bracket. The bracket is designed to mount flush against a flat vertical wall. The cross-sectional profile of the bracket in the $x y$-plane is defined by two mathematical curves, where all linear dimensions are measured in centimetres.

The curve of the outer profile is given by $f(x) = 8 \cos\left(\frac{\pi}{16}x\right)$, for $0 \le x \le 8$.

The curve of the inner profile, $g(x)$, is formed by translating the graph of $f$ by $0.8$ units to the left and $1.2$ units downwards.

(a) Write down an expression for $g(x)$.

The inner profile curve $g(x)$ intersects the $x$-axis at the point $(a, 0)$. The relevant portion of the inner profile is restricted to $0 \le x \le a$.

(b) Find the value of $a$. Give your answer to three significant figures.

The bracket is modelled by the solid formed when the region between the profiles is rotated through $\pi$ radians about the $x$-axis. This region is bounded by $f(x)$ from $x = 0$ to $x = 8$, the line $x = 0$, the $x$-axis, and $g(x)$ from $x = 0$ to $x = a$.

(c.i) Write down an expression for the volume of the solid formed.

(c.ii) Hence find the volume of material used in the bracket. Give your answer to three significant figures.

(a) The population $P$ of a rare species of orchid in a botanical garden, $t$ years after its introduction, is modelled by the logistic function $P(t) = \frac{120}{1+e^{-0.5t+10}}$.

Determine the time $t$ (in years) at which the rate of population growth of the orchid species is at its maximum.

(b) Hence, find the coordinates of the point of inflexion for a modified model $P(t-3)$, which represents the population if the initial conditions were set 3 years earlier.

(a) A Ferris wheel has a radius of $20$ metres, and its centre is $22$ metres above the ground. A passenger boards the wheel at its highest point at $t = 0$ minutes. The height, $H$ metres, of the passenger above the ground at time $t$ minutes is modelled by the function $H(t) = 22 + 20 \cos\left(\frac{\pi}{5}t\right)$.

Write down the amplitude of the Ferris wheel's motion.

(b) Calculate the minimum height above the ground a passenger reaches. Determine the first positive time, in minutes, this height occurs.

(c) Find the period of the Ferris wheel's rotation.

A company monitors the daily temperature fluctuations in a specialized industrial oven. The temperature $T$, in degrees Celsius, at time $t$ hours after midnight, is modelled by the function $T(t) = A - B \cos(Ct)$, where $A$, $B$, $C$ are positive constants. The graph below shows the temperature fluctuations over a 24-hour period.

On the same axes, sketch the graph of $T_{\text{new}}(t) = A + \frac{B}{2} \cos(Ct)$.

Determine the values of the constants $A$, $B$ and $C$.

The temperature, $T$ in degrees Celsius, in a controlled environment can be modelled by a sinusoidal function of time, $h$ hours after monitoring began. At $h=100$ hours, the temperature reaches its maximum of $15^{\circ}C$. At $h=250$ hours, the temperature drops to its minimum of $5^{\circ}C$. The monitoring takes place over $600$ hours.

(a) Find the equation of the function in the form $T(h) = A \cos(B(h-C) ) + K$, for $h \in [0, 600]$.

During the monitoring period, the temperature $T(h) = \beta$ is recorded exactly five times. Find the value of $\beta$.

A company is designing a decorative stand for a new line of luxury smart speakers. The profile of the stand's base is defined by two mathematical curves. The stand itself is formed by rotating the region between these curves through $\pi$ radians about the $x$-axis. All dimensions are in centimetres.

The curve of the outer profile is given by $f(x) = 5 \cos\left(\frac{\pi}{10}x\right)$, for $0 \le x \le 5$.

The curve of the inner profile, $g(x)$, is formed by translating the graph of $f$ by $0.4$ units to the left and $0.6$ units down.

(a) Write down an expression for $g(x)$.

The inner profile curve $g(x)$ intersects the $x$-axis at $(a, 0)$. The relevant portion of $g(x)$ for the stand is restricted to $0 \le x \le a$.

(b) Find the value of $a$. Give your answer to three significant figures.

The decorative stand is modelled by the solid formed when the region $R$ is rotated through $\pi$ radians about the $x$-axis. The region $R$ is defined by the area under $f(x)$ from $x=0$ to $x=5$, excluding the area under $g(x)$ from $x=0$ to $x=a$.

(c.i) Write down an expression for the volume of the solid formed.

(c.ii) Hence find the volume of material used in the stand. Give your answer to three significant figures.