Transformations of graphs (translation, reflections, stretches): 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 $p$.

The following diagram shows the graph of $y = \arctan\left(\frac{x-3}{2}\right) -\frac{\pi}{4}$ for $x \in \mathbb{R}$, with asymptotes at $y = -\frac{3\pi}{4}$ and $y = \frac{\pi}{4}$.

Describe a sequence of transformations that transforms the graph of $y = \arctan x$ to the graph of $y = \arctan\left(\frac{x-3}{2}\right) -\frac{\pi}{4}$ for $x \in \mathbb{R}$.

Show that $\arctan p + \arctan q = \arctan\left(\frac{p+q}{1-pq}\right)$ where $p, q > 0$ and $pq < 1$.

Using mathematical induction and the result from part (b), prove that

$\sum_{r=1}^{n} \arctan\left(\frac{1}{r^2+r+1}\right) = \arctan\left(\frac{n}{n+2}\right)$ for $n \in \mathbb{Z}^+$.

Consider the function $f(x) = e^{2x} - \ln(x)$ and the function $g(x)$ which is obtained through horizontally translating $f(x)$ 1 unit to the right also vertically translating it by 2 units downwards.

$a$ Find $g(x)$.

$b$ The function $g(x)$ has a minimum at $x = a$. Find $a$.

The following diagram shows the graph of $y = f(x)$, where $f(x) = \frac{Cx + D}{x+2}$, for $x \in \mathbb{R}$, $x \neq -2$ and $C, D \in \mathbb{Z}$.

The graph has a y-intercept at $(0, -3)$ and an x-intercept at $(3, 0)$.

(a) Find the value of $C$ and the value of $D$.

(b) Describe a sequence of transformations that maps the graph of $g(x) = \frac{1}{x}$ onto the graph of $y = f(x)$.

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 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.

A company models the efficiency of a new production line by the function $f(x) = \frac{2x+3}{x+2}$, where $f(x)$ represents the output rate (in units per hour) after $x$ hours of operation. For mathematical analysis, we consider the function over its natural domain $x \in \mathbb{R}$, $x \ne -2$.

For the graph of f,

write down the equation of the vertical asymptote;

find the equation of the horizontal asymptote.

Find $f^{-1}(x)$.

Using an algebraic approach, show that the graph of $f^{-1}$ is obtained by a reflection of the graph of f in the y-axis followed by a reflection in the x-axis.

The graphs of f and $f^{-1}$ intersect at $x = p$ and $x = q$, where $p < q$.

Find the value of p and the value of q.

Hence, find the area enclosed by the graph of f and the graph of $f^{-1}$.

Consider the function $f(x) = e^x - 3x - 6$.

On the following axes, sketch the graph of $f$ for $-3 \le x \le 3$.

The function $g$ is defined by $g(x) = e^{2x} - 6x - 10$.

The graph of $g$ is obtained from the graph of $f$ (from part a) by a horizontal stretch with scale factor $k$, followed by a vertical translation of $c$ units.

Find the value of $k$ and the value of $c$.

Describe the sequence of transformations that maps the graph of $y = g(x)$ onto the graph of $y = g(x+1) - 4$.

Describe the transformation that maps the graph of $y = g(x)$ onto the graph of $y = -g(x)$.

Describe the sequence of transformations that maps the graph of $y = g(x)$ onto the graph of $y = g(2x) + 3$.

Describe the sequence of transformations that maps the graph of $y = g(x)$ onto the graph of $y = 3g(-x)$.

Describe the sequence of transformations that maps the graph of $y = g(x)$ onto the graph of $y = g(\frac{x}{2} - 1)$.

A scientist is studying the growth of a certain bacterial culture. The population, in thousands, at time $t$ hours is modeled by the function $P(t) = e^t - 4t - 5$.

On the following axes, sketch the graph of $P(t)$ for $-2 \leq t \leq 4$.

Another bacterial culture, observed under different conditions, has its population modeled by the function $Q(t) = e^{2t} - 8t - 12$.

The graph of $Q(t)$ is obtained from the graph of $P(t)$ by a horizontal stretch with scale factor $k$, followed by a vertical translation of $c$ units.

Find the value of $k$ and the value of $c$.

In a simulation of a rotating antenna, its orientation is modeled by a complex number $w$ on the Argand diagram. Initially, the antenna's orientation is given by $w = \cos \left( \frac{\pi}{10} \right) + i \sin \left( \frac{\pi}{10} \right)$. The antenna's controller performs a series of rotations, and after $n$ such operations, its orientation is required to be pointing directly upwards along the imaginary axis, represented by the complex number $i$.

Find the smallest positive integer value of $n$ for which the antenna's orientation $w^n$ is equal to $i$.

Hence or otherwise, describe a single geometric transformation on the Argand diagram that maps $w$ to $w^{12}$.

A scientist is modeling the growth of a certain bacterial colony. The population size, $P(t)$, at time $t$ hours, is given by the function $f(t) = 2^t - 2t - 5$ for $-3 \leq t \leq 4$.

(a) On the following axes, sketch the graph of $f$ for $-3 \leq t \leq 4$.

Another bacterial colony, under different conditions, has its population size modeled by $g(t) = 4^t - 4t - 8$. It is observed that the growth curve of this second colony can be obtained from the graph of $f$ by a horizontal stretch with scale factor $k$, followed by a vertical translation of $c$ units.

(b) Find the value of $k$ and the value of $c$.

The graph of the function $g(x) = \frac{3}{x-1} + 2$ is translated by the vector $\begin{pmatrix} -3 \\ 4 \end{pmatrix}$.

(a) Describe this translation in words.

(b) Find the equation of the translated graph, giving your answer in the form $y = \frac{ax+b}{cx+d}$.

(c) State the equations of the vertical and horizontal asymptotes of the translated graph.

A function $f$ is defined by $f(x) = -2(x + 3)^2 + 8, x \in \mathbb{R}$.

(a) Write $f(x)$ in the form $ax^2 + bx + c$, where $a$, $b$ and $c$ are constants.

(b) Find the coordinates of the vertex of the graph of $f$.

(c) Find the equation of the axis of symmetry of the graph of $f$.

(d) State the range of $f$.

The graph of $f$ is translated by the vector $\begin{pmatrix} -1 \\ 5 \end{pmatrix}$ to form a curve representing a new function $g(x)$.

(e) Find $g(x)$ in the form $px^2 + qx + r$, where $p$, $q$ and $r$ are constants.

A function $f$ is defined by $f(x) = -3x^2 + 6x + 5$ for $x \in \mathbb{R}$.

(a) Express $f(x)$ in the form $f(x) = a(x-h)^2+k$.

(b) The graph of $y=f(x)$ is obtained from the graph of $y=x^2$ by a sequence of transformations. Describe fully this sequence of transformations.

The graph of a function $y = g(x)$ has a local maximum at P$(2, 8)$, a local minimum at Q$(-4, -6)$, a y-intercept at R$(0, 5)$ and an x-intercept at S$(5, 0)$.

(a) The graph of $g(x)$ is transformed to the graph of $y = g(x-3) - 2$. Write down the coordinates of the images of the points P, Q, R, and S.

(b) The graph of $g(x)$ is transformed to the graph of $y = -g(\frac{x}{2})$. Write down the coordinates of the images of the points P, Q, R, and S.

A function is defined by $f(x) = -2x^2 + 12x - 11$.

(a) Express $f(x)$ in the form $f(x) = a(x-h)^2 + k$, where $a, h, k$ are integers.

(b) Write down the coordinates of the vertex of the graph of $y=f(x)$.

(c) The graph of a function $g$ is obtained from the graph of $f$ by a translation of $\begin{pmatrix} -4 \\ 1 \end{pmatrix}$. Find the coordinates of the vertex of the graph of $y=g(x)$.

(a) A scientist is studying the population sizes of two bacterial cultures, A and B, in a controlled environment. The population sizes, in thousands, are modelled by the functions $f(x) = \ln(x+1)$ and $g(x) = \ln(5-x)$ respectively, where $x$ is the time in hours. Sketch the graphs of $f(x)$ and $g(x)$ on the same set of axes for $x \in \mathbb{R}$. Clearly indicate any asymptotes and axis intercepts.

(b) Write down the equation of the vertical asymptote of the graph of $g(x)$.

(c) Describe a single transformation that maps the graph of $f(x)$ onto the graph of $g(x)$.

(d) Use your sketch to find the time, $x$, when the populations of culture A and culture B are equal.