Exercises: 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.

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:

• $g(x) = -f(x)$

• $h(x) = f(-x)$

• $i(x) = 2f(x)$

• $j(x) = f(x) - 4$

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.

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.

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 euros 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 $g(x) = f(x-3) + 1$.

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.

Marcos is studying how to transform the graph of the function $f(x) = x^2 - 4x$. His goal is to obtain the analytic expression of a new function $g(x)$ whose graph is the result of shifting that of $f(x)$ exactly $3$ units to the right and then reflecting it across the horizontal axis ($X$). Marcos presents the following solution in his notebook:

"To translate $3$ units to the right, I add $3$ inside the argument: $x \to x + 3$. To reflect across the $X$-axis, I change the sign of the independent variable: $x \to -x$. Therefore, substituting into the formula I get: $g(x) = (-x + 3)^2 - 4(-x + 3)$".

1. a) Identify and explain in detail the two conceptual errors made by Marcos when applying the transformations.
2. b) Show the correct procedure step by step and write the algebraic expression of $g(x)$ in its expanded and simplified form.

Starting from the elementary quadratic function $f(x) = x^2$, a transformed function $g(x) = a \cdot f(x - h) + k$ is defined, where $a$, $h$, and $k$ are real numbers with $a \ne 0$. It is known that the graph of $g(x)$ has its vertex at the point $V(-2, 5)$ and intersects the vertical axis ($Y$) at the point $(0, -3)$.

1. a) Deduce with reasoning, from the geometric properties of translations, the values of the parameters $h$ and $k$.
2. b) Set up an equation using the $Y$-intercept to determine the exact value of the parameter $a$.
3. c) Explain what three geometric transformations have been applied to $f(x)$ to obtain $g(x)$ (indicating order, magnitudes, and directions) and write the final algebraic expression of $g(x)$ in general polynomial form ($Ax^2 + Bx + C$).

The monthly profit of a small tech company, in thousands of euros, is modelled 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 corresponds to a horizontal shift of the profit curve, and also incurs an increase in fixed costs of $3$ thousand euros per month. Find the new expression for the profit function $N(x)$ and determine in which month the maximum profit will now be reached.

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.

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 $f(x)$.

A bacterial culture 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 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.

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

Consider the polynomial function $f(x) = x^2 - 2x - 8$ and the transformed function $g(x) = |f(x)| - 5$.

1. a) Estimate: Without solving any quadratic equation yet, calculate the coordinates of the vertex of $f(x)$ and its $X$-intercepts. From that geometric information, describe what effect the absolute value produces on the negative regions and how the subsequent vertical translation of $-5$ units affects it. Based on this, estimate with reasoning how many $X$-intercepts the function $g(x)$ will have and between which values on the real axis they should approximately lie.
2. b) Check: Set up and solve the equation $g(x) = 0$ algebraically by transforming it into two independent quadratic equations according to the definition of absolute value. Find all the exact intercepts and check whether the number of solutions and their positions agree with the estimation you made in the previous part.

According to the records of an AEMET weather station in the Castilian plateau during a clear spring day, the temperature $T(t)$ in degrees Celsius ($^{\circ}\text{C}$) between $6:00$ and $22:00$ hours ($t \in [6, 22]$) follows the quadratic model:

$T(t) = -0,125(t - 14)^2 + 18$

where $t$ indicates the solar hour of the day.

The next day, a cold front arrives, altering the thermal pattern according to three simultaneous effects on the graph:

1. The time of maximum insolation and peak temperature occurs $2$ hours earlier.

2. The diurnal temperature variation (difference between the maximum and minimum temperature) is halved, which is equivalent to vertically compressing the graph by a scale factor $a = 0,5$.

3. An additional uniform thermal drop of $3\,^{\circ}\text{C}$ occurs throughout the day due to the cold air mass.

1. a) Write the functional relationship between the new model $N(t)$ and the initial model $T(t)$ in the form $N(t) = a \cdot T(t - h) + k$, justifying in detail the value and sign of each of the parameters $a$, $h$, and $k$.
2. b) Expand and simplify the final analytic expression of $N(t)$ as a function of the time $t$.
3. c) Determine with reasoning at what time the maximum temperature will be reached on this new cold day and what its exact value will be.

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 $g(x) = a \cdot f(b \cdot x - c) + d$ that simultaneously satisfies the following conditions:

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

• The centre of symmetry of the bridge must be shifted $4$ 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 located at exactly $10$ metres above ground level.

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

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.

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 $h(x)$.

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.

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 $C(4, 2)$.

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.

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.

A programmed path for a robotic arm is given by the function $y = \sqrt{R^2 - x^2}$, which describes a semicircle of radius $R$ centred at the origin. The design team wishes to program a new elliptical path $E(x)$ that has its centre 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 $1$).

We wish to study the transformed function $g(x) = f(2x - 8)$ from a base function $f(x)$ whose graph is known. Two students discuss the correct procedure to construct the graph of $g(x)$ through elementary transformations on the variable $x$:

• Alba's method (factoring the horizontal scale): She writes $g(x) = f(2(x - 4))$ and states that one must first apply a horizontal compression by a factor of $\frac{1}{2}$ with respect to the $Y$-axis and then a horizontal translation of $4$ units to the right.

• Bruno's method (prior translation without factor): He states that one must first apply a horizontal translation of $8$ units to the right to obtain $f(x - 8)$, and on this graph apply a horizontal compression replacing $x$ with $2x$, which compresses the entire graph by half with respect to the $Y$-axis.

1. a) Check analytically whether both methods lead to the same result. To do this, take a generic point $(x_0, y_0)$ that belongs to the graph of $f$ (that is, $y_0 = f(x_0)$) and determine what $x$-coordinate corresponds to it in $g(x)$ according to each method so that $g(x) = y_0$ holds.
2. b) Critically analyze the two methods: why is Alba's method the one usually recommended in mathematics, and what is the typical error students often fall into when trying to apply Bruno's method without paying enough attention to the center of compression?
3. c) Knowing that the base function $f(x)$ has a relative minimum at the point $(6, -3)$ and intersects the $X$-axis at $x = 2$ and at $x = 10$, choose the method you consider most reliable to calculate the coordinates of the new relative minimum of $g(x)$ and its new $X$-intercepts, justifying each step.

Given the rational function $f(x) = \frac{2x - 5}{x - 1}$:

1. a) Perform the algebraic division to express $f(x)$ in the canonical form $f(x) = c + \frac{k}{x - h}$. From this expression, describe precisely the sequence of elementary geometric transformations (translations, dilations, and reflections) that transform the basic hyperbola $y = \frac{1}{x}$ into the graph of $f(x)$.
2. b) The family of transformed functions $g_m(x) = |f(x) + m|$ is defined, where $m \in \mathbb{R}$ is a constant. Knowing that $f(x)$ has a horizontal asymptote, deduce with reasoning for what values of the parameter $m$ the graph of $g_m(x)$ will have its horizontal asymptote located strictly above the horizontal line $y = 3$.
3. c) For the particular value $m = -1$, obtain the simplified expression inside the absolute value of $g_{-1}(x) = |f(x) - 1|$, calculate the coordinates of its $X$-intercept, and determine the exact interval of the real line on which the graph of $f(x) - 1$ has undergone a reflection across the $X$-axis as a result of the absolute value.