Let $f(x)=e^x$. The graph of $f$ is translated by the vector

to form the graph of $g$. The graph of $g$ passes through $(0,3)$ and $(\ln 4,9)$.

Let

A new function is defined by

A function $f$ has domain $[-2,5]$ and range $[-1,7]$. A function $g$ is defined by

It is also known that $f(1)=6$.

Let $f(x)=x^2$. A transformed function has the form

where $p>0$ and $q>0$. The graph of $g$ has vertex $(3,-2)$, is obtained using a horizontal stretch with scale factor $2$, and passes through $(5,6)$.

The graph of $y=f(x)$ is transformed to the graph of

Let $f(x)=\dfrac1x$. A transformed function is defined by

where $a>0$ and $b>0$. The graph of $g$ has vertical asymptote $x=2$, horizontal asymptote $y=-1$, and is obtained using a horizontal stretch with scale factor $2$. It also passes through $(3,1)$.

Let $T$ denote a translation $3$ units to the right and let $S$ denote a horizontal stretch with scale factor $2$. These transformations are applied to the graph of $y=f(x)$ in different orders.

Let

and define

Let $f(x)=|x|$ and define

The vertical displacement of a test platform is modelled by

where $t$ is measured in seconds. A second sensor produces the transformed signal

The graph of $f$ has a maximum point at $(1,e^{-1})$.

The depth of water above a reference level at a harbour is modelled by

where $t$ is measured in hours and $0\leq t\leq 2+4\pi$.

A calibration curve is obtained from the graph of $f(x)=\ln x$ using

The temperature of a container is modelled by

A second container begins cooling after a delay of $5$ minutes. It cools $1.4$ times as slowly, its temperature difference from the original baseline is multiplied by $0.8$, and its new baseline is $22^\circ\text{C}$.

A retailer models weekly demand by $Q=f(P)$, where $P$ is the price in dollars and $Q$ is the number of units demanded. Two points on the original demand curve are $(18,260)$ and $(30,140)$. A new market has demand

Let $f(x)=\sqrt{x}$ and define

A numerical model is defined by

Values of $x$ used in the model are close to $1\,000\,000$.

The intensity profile of a light pulse is modelled by $f(t)=e^{-t^2}$. A modified pulse is modelled by

A graph-design program transforms $y=f(x)$ into
$g(x)=p f(q(x-a))+b$
where $p>0$ and $q>0$. The points $U(-2,5)$ and $V(4,-1)$ on the original graph are mapped to $U'(1,12)$ and $V'(4,0)$ respectively.

A recorded seismic signal is represented by $A=f(t)$, where $t$ is measured in seconds and $A$ in millimetres. A processed signal is defined by
$G(t)=-1.5f(2(t-0.8))+0.2$
The original signal has a local maximum at $(1.6,0.9)$ and crosses the time axis at $t=0.4$ and $t=2.8$.

A graphics filter uses the parent curve
$f(x)=\frac{2}{x+1}-3$
to generate
$g(x)=-2f\left(\frac12(x-4)\right)+5$

A repeating machine motion is based on the graph of $y=f(t)$. One cycle of $f$ has a maximum at $(1,4)$, a minimum at $(3,-2)$ and the next maximum at $(5,4)$. A modified motion is
$G(t)=-\frac12f(2(t-3))+1$

A digital elevation profile $y=f(x)$ has domain $[-5,7]$ and range $[-3,9]$. It begins at $(-5,4)$ and ends at $(7,1)$. A transformed profile is defined by
$g(x)=3-f\left(\frac{6-x}{2}\right)$

For a manufacturer, $C=f(n)$ gives the daily operating cost $C$, in euros, when $n$ items are produced. After equipment is replaced, the cost is modelled by
$G(n)=0.8f\left(\frac{n-120}{1.5}\right)+600$
The original graph contains the point $(300,5400)$.

Let

and

The graph of $f$ has a local maximum at $(-1,2)$ and a local minimum at $(1,-2)$.

An animation is described by $y=f(t)$ for $0\leq t\leq12$. Let $T$ be a translation $3$ seconds to the right and let $S$ be a horizontal stretch with scale factor $1.5$.