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 meters 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 pesos, is modeled by the function $B(x) = -40x^2 + 240x - 200$, 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 corresponds to a horizontal shift of the profit curve, and also incurs an increase in fixed costs of $60$ thousand pesos 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 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 function $C(x)$ represents the cost in pesos of producing $x$ electronic components. Due to a technological improvement on the assembly line, three times as many components can now be produced for exactly the same total cost. Express the new cost function $N(x)$ in terms of the original function $C(x)$.

Observe the graph of the function $f(x)$ defined on the interval $[-4, 4]$. Determine the domain and 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, giving reasons and detailing the effects of each applied transformation, the domain and range of the function $h(x)$.

In an acoustics study, a base sound wave is modelled 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 time delay in emission of $\frac{\pi}{4}$ seconds is introduced. Write the algebraic expression for the new sound wave $W(t)$ and calculate its new period.

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 $g(x)$.

The efficiency of a solar panel throughout the day is given by a function $E(t)$. A new panel model is installed that starts generating energy $2$ hours earlier and, due to a special graphene coating, generates $15\%$ more energy at each equivalent moment. Express the new efficiency function $N(t)$ in terms of $E(t)$. If the original panel reached its maximum efficiency at 14:00, at what time will the new panel reach it?

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.

A bacterial culture grows according to the function $N(t) = 200 \times 3^{t/2}$, where $t$ is the time in hours. A second culture $M(t)$ starts with one quarter of the initial population of the first, but its growth rate is identical and measurement started $4$ hours earlier. Express $M(t)$ in terms of $N(t)$ and simplify the resulting expression to check whether 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 geological requirements of the terrain, he needs to propose a new profile $g(x) = a \times f(b \times x - c) + d$ that simultaneously satisfies the following conditions:

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

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

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

• The minimum height of the new profile must be located at exactly $10$ meters above ground level.

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

A parabola has its vertex at the origin and passes through the point $(2, 4)$. The following transformations are applied 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 algebraic expression for the final function $g(x)$ and determine the exact coordinates of its new vertex and its axis intercepts.

A programmed path 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 path $E(x)$ 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 $1$).

Let $f(x) = \ln(x)$ be a function. A change of reference frame in the Cartesian plane is performed such 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 length on the y-axis is halved. Write the equation of the curve in the new coordinate system.

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 $C(4, 2)$.

Let $f(x)$ be an odd and 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 fundamental period.

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

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.

Match each of the algebraic expressions with the corresponding panel in the figure, given that all of them derive from transformations involving absolute values applied to the base function $y = x - 3$.

1. $y = |x| - 3$

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

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

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

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.