Function transformations

Given the function $f(x) = x^2 - 3x$, write the algebraic expression for the function $g(x)$ obtained by translating the graph of $f(x)$ four units to the left and two units upwards.

A hot-air balloon descends at a constant speed. Its altitude in metres is given by $A(t) = 800 - 50t$, where $t$ is the time in minutes. A second balloon begins its descent $10$ minutes later with exactly the same flight profile. Write the expression for the altitude of the second balloon $B(t)$ as a function of $A(t)$ and expand its final algebraic expression.

The graph of a function $f(x)$ passes through the point $P(-3, 5)$. Determine the coordinates of the corresponding point on the graphs of the following transformed functions:

• $g(x) = -f(x)$

• $h(x) = f(-x)$

• $i(x) = 2f(x)$

• $j(x) = f(x) - 4$

Given the basic function $f(x) = \frac{1}{x}$, state which geometric transformations (translations, stretches, reflections) must be applied successively, and in what order, to obtain the function $g(x) = -\frac{3}{x} + 2$.

The graph of the function $f(x)$ consists of two line segments joining the points $(-4, -2)$, $(0, 4)$ and $(4, -2)$. A new function is defined by the transformation $g(x) = -f\left(\frac{x}{2}\right) + 3$.

• Describe the ordered sequence of geometric transformations applied to $f(x)$.

• Calculate the exact coordinates of the new vertices of the graph of $g(x)$.

Given the basic function $f(x) = \sqrt{x}$, we want to obtain the function $g(x) = \sqrt{4x - 12} + 5$.

Explain in detail two different sequences of geometric transformations (altering the order of horizontal translations and stretches) that allow obtaining $g(x)$ from $f(x)$. Prove algebraically that both sequences lead to the same expression.

Express the function $g(x) = |-x^2 + 4|$ as a piecewise function by removing the absolute value and clearly stating the intervals where the original expression changes sign.

The figure shows the light curve of a variable star, modelled by the function $L(t) = 4 \sin\left(\frac{\pi}{6}t\right) + 10$, where $t$ is measured in days and $L$ in units of luminosity. A dense cloud of cosmic dust obscures the star such that the luminosity $O(t)$ observed from Earth is exactly the absolute value of the difference between $L(t)$ and its mean luminosity.

Determine the algebraic expression for $O(t)$, calculate its new period and find at which times $t \in [0, 12]$ the observed luminosity is maximum.