Transformations of Functions

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

A hot air balloon descends at a constant speed. Its altitude in feet 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 monthly profit of a small tech company, in thousands of dollars, is modeled by the function $B(x) = -2x^2 + 12x - 10$, where $x$ represents the month of the year ($x=1$ for January, etc.). The company decides to delay the launch of its marketing campaign by one month, which means a horizontal translation of the profit curve, and also assumes an increase in fixed costs of $3$ thousand dollars per month. Find the new expression for the profit function $N(x)$ and determine in which month the maximum profit will now be reached.

The function $C(x)$ represents the cost in dollars of producing $x$ electronic components. Due to a technological improvement in the assembly line, it is now possible to produce triple the number of components for exactly the same total cost. Express the new cost function $N(x)$ in terms of the original function $C(x)$.

Given the elementary 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 $g(x) = -\frac{3}{x} + 2$.

Given the elementary 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 the horizontal translations and stretches) that allow $g(x)$ to be obtained from $f(x)$. Prove algebraically that both sequences lead to the same expression.

A parabola has its vertex at the origin and passes through the point $(2, 4)$. The following transformations are applied to it sequentially:

1. A horizontal compression that multiplies the x-coordinates of its points by $\frac{1}{3}$.

2. A translation of $5$ units to the left.

3. A reflection in the x-axis.

4. A translation of $7$ units upwards.

Find the analytical expression of the final function $g(x)$ and determine the exact coordinates of its new vertex and its points of intersection with the axes.

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:

• $g(x) = -f(x)$

• $h(x) = f(-x)$

• $i(x) = 2f(x)$

• $j(x) = f(x) - 4$

The efficiency of a solar panel throughout the day is given by a function $E(t)$. A new model of panel is installed that starts generating energy $2$ hours earlier and, due to a special graphene coating, generates $15\%$ more energy at each equivalent instant. Express the new efficiency function $N(t)$ in terms of $E(t)$. If the original panel reached its maximum efficiency at 2:00 p.m., at what time will the new panel reach it?

The graph of the function $f(x)$ is formed by 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 $g(x)$.

Observe the graph of the function $f(x)$ defined in the interval $[-4, 4]$. Determine the domain and the range of the transformed function $g(x) = f(x-3) + 1$.

Given an arbitrary function $f(x)$ whose domain is the interval $[-6, 8]$ and whose range is the interval $[-4, 12]$, a new function is defined by the expression $h(x) = -2f\left(3 - \frac{x}{2}\right) + 5$.

Calculate, showing your reasoning and detailing the effects of each applied transformation, the domain and the range of the function $h(x)$.

In an acoustics study, a base sound wave is modeled by the function $S(t) = \sin(t)$. For an experiment, the original wave is amplified by multiplying its amplitude by $4$ and is made higher-pitched by doubling its frequency. In addition, a delay of $\frac{\pi}{4}$ seconds is introduced in the emission. Write the analytical expression of the new sound wave $W(t)$ and calculate its new period.

A culture of bacteria grows according to the function $N(t) = 200 \cdot 3^{t/2}$, where $t$ is the time in hours. A second culture $M(t)$ starts with a quarter of the initial population of the first, but its growth rate is identical and it started being measured $4$ hours earlier. Express $M(t)$ in terms of $N(t)$ and simplify the resulting expression to check if both populations will be equal at any point.

An engineer designs the profile of a suspension bridge using the curve $f(x) = e^{-x^2}$ as a model. Due to the geological requirements of the terrain, they need to propose a new profile $g(x) = a \cdot f(b \cdot x - c) + d$ that simultaneously meets the following conditions:

• It must be twice as wide as the original design.

• The center of symmetry of the bridge must be translated $4$ units to the right.

• The profile must be inverted (opening upwards instead of downwards), without changing its vertical scale.

• The minimum height of the new profile must be situated exactly $10$ feet above ground level.

Determine analytically the precise values of the parameters $a$, $b$, $c$ and $d$.

From the graph of the function $f(x)$ shown in the figure, describe the geometric steps necessary to obtain the graph of the function $g(x) = |f(x)|$ and calculate the coordinates of the new vertices or inflection points.

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

The figure shows the light curve of a variable star, modeled 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 in such a way that the observed luminosity $O(t)$ from Earth is exactly the modulus of the difference between $L(t)$ and its mean luminosity.

Determine the analytical expression of $O(t)$, calculate its new period and find at what instants $t \in [0, 12]$ the observed luminosity is maximum.

Match each of the analytical expressions with the corresponding panel in the figure, knowing that they all derive from modulus transformations applied to the base function $y = x - 3$.

1. $y = |x| - 3$

2. $y = ||x-3| - 2|$

3. $y = ||x| - 3|$

4. $y = |x-3| - 2$

Express the rational function $f(x) = \frac{2x + 7}{x + 1}$ in the form $f(x) = k + \frac{m}{x+1}$, by carrying out polynomial division. From this result, describe step by step the geometric transformations that must be applied to the rectangular hyperbola $y = \frac{1}{x}$ to obtain the graph of $f(x)$.

Find the values of the parameters $a$ and $b$ so that the graph of the rational function $f(x) = \frac{ax + 3}{x + b}$ is symmetrical about the point $C(4, 2)$.

Hint: Remember that a function $f(x)$ is symmetrical about a point $(h, k)$ if the function translated to the origin, $g(x) = f(x+h) - k$, turns out to be an odd function.

Let $f(x)$ be an odd, periodic real function with period $T=6$. From it, the transformed function $g(x) = f(x-2) + f(x+2)$ is defined.

Prove analytically whether the function $g(x)$ is even, odd or neither. Also determine whether $g(x)$ is periodic and, if so, calculate its minimum period.

Let the function be $f(x) = \ln(x)$. A change of reference system is made in the Cartesian plane so that the new origin is located at the point $(-3, 2)$ of the old system, the positive direction of the x-axis is reversed, and the unit of measurement of the y-axis is halved. Write the equation of the curve in the new coordinate system.

A programmed trajectory for a robotic arm is given by the function $y = \sqrt{R^2 - x^2}$, which describes a semicircle of radius $R$ centered at the origin. The design team wishes to program a new elliptical trajectory $E(x)$ that has its center at the point $(h, k)$, a horizontal semi-axis of length $a$ and a vertical semi-axis of length $b$ (representing only the upper half of the ellipse).

Express the final function $E(x)$, clearly indicating the composition of geometric transformations applied to the base function $y = \sqrt{1 - x^2}$ (semicircle of radius $1$).