Exercises: Sketching functions

Find the points of intersection of the graph of $f(x) = x^3 - 4x$ with the coordinate axes.

Determine the domain of the function $f(x) = \frac{x + 1}{x - 3}$ and state its vertical asymptote, studying the one-sided limits at the point that does not belong to the domain.

Check that the function $f(x) = \frac{2x + 1}{x - 4}$ has a horizontal asymptote and find its equation by studying the limit as $x \to \pm \infty$.

Check that the function $f(x) = x^3 - x$ is odd and explain the advantage of this symmetry when sketching its graph.

The figure shows the complete graph of a rational function, with its asymptotes shown as dashed lines. Reading from the graph, state: domain, axis intercepts, equations of the asymptotes and intervals of increase and decrease.

The figure shows the sign diagram of a quadratic polynomial function: its graph lies above the x-axis outside the interval $[1, 4]$ and below inside.

1. a) Write down a possible algebraic expression for $f$.

1. b) How many different quadratic polynomial functions satisfy this same sign diagram? Justify your answer.

Perform the division $\frac{x^2 + 1}{x}$ and check that the graph of $f(x) = \frac{x^2 + 1}{x}$ approaches the line $y = x$ as $x \to \pm \infty$, as illustrated in the figure. What is the name of this line?

Reason whether the following statement is true or false: "If a function has a horizontal asymptote, its graph can never cross that asymptote." If false, verify it using the function $f(x) = \frac{2x^2 + x}{x^2 + 1}$ by checking that its graph crosses its horizontal asymptote at some point.

Look at the following graph of a polynomial function. Identify the intervals on which the function is convex ($\cup$-shaped) and concave ($\cap$-shaped). Also point out the approximate coordinates of its point of inflection.

The following graph corresponds to the inverse proportion function $y = -\frac{2}{x}$. By observing the curve, determine the intervals of concavity and convexity. Does this function have any point of inflection in its domain?

State whether the following statement is true or false and justify your answer in detail:

"A point of inflection is a point on the graph where the function changes from increasing to decreasing, or vice versa."

Marcos has to find the vertical asymptotes of the rational function $f(x) = \frac{x^2 - x - 2}{x^2 - 4}$. His reasoning on the exam was as follows: «I set the denominator equal to zero: $x^2 - 4 = 0 \implies x = 2$ and $x = -2$. Therefore, the function has two vertical asymptotes: the lines $x = 2$ and $x = -2$».

1. a) Explain the conceptual error made by Marcos and why making the denominator zero is a necessary but not sufficient condition for the existence of a vertical asymptote.
2. b) Calculate the relevant limits ($\lim_{x \to 2} f(x)$ and $\lim_{x \to -2} f(x)$) to determine rigorously which of the two values actually corresponds to a vertical asymptote and what happens on the graph at the other point.
3. c) Write the correct conclusion, stating the equation of the only vertical asymptote and the coordinates of the «hole» or removable discontinuity present on the curve.

To study whether the graph of the function $f(x) = \frac{x^3}{x^2 + 1}$ has an oblique asymptote and to find its relative position with respect to it, two classmates propose different strategies:

• Lucía proposes calculating the general limits at infinity: $m = \lim_{x \to \pm\infty} \frac{f(x)}{x}$ and $n = \lim_{x \to \pm\infty} [f(x) - m \cdot x]$.

• Sergio proposes directly carrying out polynomial division between the numerator and the denominator to rewrite $f(x) = c(x) + \frac{r(x)}{d(x)}$.

1. a) Compare both approaches and justify which one is more direct and provides immediate information about the position of the curve relative to the asymptote.
2. b) Apply the most convenient method to obtain the equation of the oblique asymptote line $y = m \cdot x + n$ and determine whether the curve lies above or below it as $x \to +\infty$ and as $x \to -\infty$.

Analyse and sketch the function $f(x) = x^3 - 3x^2$ by following these steps:

1. a) Axis intercepts.

1. b) Derivative, stationary points and intervals of monotonicity.

1. c) Local maximums and minimums.

1. d) Draw the graph using all the previous information.

You can do it on paper and send a photo with your answer.

Analyse and sketch the biquadratic function $f(x) = x^4 - 2x^2$: symmetry, axis intercepts, intervals of monotonicity, local extrema and a sketch of the graph.

You can do it on paper and send a photo with your answer.

Analyse and sketch the rational function $f(x) = \frac{x - 1}{x + 2}$: domain, intercepts, vertical and horizontal asymptotes with their limits, monotonicity using the derivative and a sketch of the graph.

You can do it on paper and send a photo with your answer.

Analyse and sketch the function $f(x) = \frac{x^2}{x - 1}$: domain, intercepts, vertical asymptote, oblique asymptote (perform the division), intervals of monotonicity and local extrema, and a sketch of the graph.

You can do it on paper and send a photo with your answer.

The temperature of a glass of hot water is measured in a kitchen at $20\ \text{°C}$ and the data in the figure are obtained. The proposed model is $T(t) = 20 + \frac{80}{t + 1}$, where $t$ is in minutes.

1. a) Check that the model is consistent with the first data points in the figure.

1. b) Study the monotonicity of $T(t)$ for $t \geq 0$ using its derivative.

1. c) Calculate the horizontal asymptote and interpret its physical meaning.

1. d) Sketch the graph of the model for $t \geq 0$.

You can do it on paper and send a photo with your answer.

Analyse and sketch the function $f(x) = x \cdot (x - 2)^2$: axis intercepts (note the double root and what it implies for the graph at $x = 2$), intervals of monotonicity, local extrema and sketch.

You can do it on paper and send a photo with your answer.

A student claims that the graph of the function $f(x) = \frac{x^2 - 4}{x - 2}$ is the parabola shown in the figure. Explain the mistake made, simplify the expression for $f$ and describe what its graph actually looks like, stating what happens at $x = 2$.

Analyse and sketch the piecewise function

analysing: domain, continuity at the join, monotonicity in each branch, local extrema and a sketch of the complete graph.

You can do it on paper and send a photo with your answer.

The following graph represents the distance covered by a vehicle as a function of time over a journey. Analyse the curvature in the three marked sections (A, B and C) and relate whether the function is concave, convex or linear to the state of motion of the vehicle (accelerating, braking or constant speed).

During an epidemic, the total number of infected people is modelled by the curve shown in the figure. Identify the point of inflection of the graph. What name is given to this point in popular epidemiological language and what does it indicate about the daily rate of new infections?

Water is being poured at a constant rate into three glass containers of different shapes: a right cylinder, a cone with its vertex pointing down, and a cone with its vertex pointing up. Match each container with the shape of the graph of the water height as a function of time: convex ($\cup$), concave ($\cap$) or linear. Justify your answer by observing how the width of the container changes.

Given the piecewise function:

$f(x) = \begin{cases} -x^2 & \text{if } x < 0 \\ x^2 & \text{if } x \ge 0 \end{cases}$

Sketch the graph of the function. Analyse its concavity and convexity on the intervals $(-\infty, 0)$ and $(0, +\infty)$. Is the origin of coordinates a point of inflection? Justify your answer.

You can do it on paper and send a photo with your answer.

In economics, a factory's total production cost function usually has a concave shape ($\cap$) at first due to economies of scale (costs grow slower and slower), and convex ($\cup$) from a certain production level onwards due to saturation (costs grow faster and faster).

If the total cost function is $C(x) = x^3 - 60x^2 + 1500x$, where $x$ are the units produced, and it is known that the point of inflection separates these two zones, from what production level do costs start rising more rapidly? (Note: the point of inflection of a cubic function $ax^3+bx^2+cx+d$ is located at $x = -\frac{b}{3a}$).

Given the polynomial function $f(x) = x^3 - 6x^2 + 9x$, it is known from its graph that its point of inflection is located at exactly the coordinate $x$, which is the midpoint between its two local extrema (a local maximum at $x=1$ and a local minimum at $x=3$).

Calculate the exact coordinates of the point of inflection and determine the intervals of concavity ($\cap$) and convexity ($\cup$) of the function.

The function $f(x) = |x^2 - 4|$ represents the modulus of a parabola and its graph is shaped like a rounded "W".

Observe its graph and determine on which intervals the function is concave ($\cap$) and on which it is convex ($\cup$). Can the points $x=-2$ and $x=2$ (marked in red) be considered proper points of inflection? Justify your answer.

Consider the family of parametric quadratic functions given by the expression $f(x) = (a^2 - 9)x^2 + 5x - 2$, where $a$ is a real number.

1. a) For which exact values of the parameter $a$ is the function a convex parabola ($\cup$)?
2. b) For which values of $a$ is it a concave parabola ($\cap$)?
3. c) Is there any value of $a$ for which the graph of the function lacks curvature? In that case, what type of geometric function is obtained?

An initial principal of $1000\ \text{€}$ is invested in a fund at an annual compound interest rate of $5\%$. The function describing the accumulated capital over time $t$ (in years) is $C(t) = 1000 \cdot (1,05)^t$.

Knowing that exponential functions with a base greater than $1$ have a characteristic curvature, determine whether this function is concave ($\cap$) or convex ($\cup$). Will there come a point in the future when the rate of growth of the capital begins to decrease, thus creating a point of inflection? Justify your answer.

Consider the rational function $f(x) = \frac{4x}{x^2 + 1}$.

1. a) Estimate without differentiating: Analyze its symmetry by calculating $f(-x)$, its sign for values of $x > 0$, the intercepts with the axes, and its asymptotic behavior as $x \to +\infty$. From these data alone, reason whether the function must necessarily reach a relative maximum for some positive value of $x$. If you also calculate that $f(1) = 2$, do you think it is reasonable that the maximum height exceeds the value $y = 3$? Justify your estimate.
2. b) Check analytically: Calculate the first derivative $f'(x)$, determine the stationary points, and check whether the exact value of the maximum matches qualitatively what you had deduced.
3. c) Use the symmetry property proven in the first section to deduce, without studying derivative signs again, the exact coordinates and nature of the corresponding relative extremum on the negative half-line.

A graphic design team needs to reproduce a symmetric curve that follows the rational model $f(x) = \frac{a \cdot x^2 + b}{x^2 - c}$, where $a$, $b$, and $c$ are real coefficients. The following conditions are established:

1. The graph has two symmetric vertical asymptotes at the lines $x = 2$ and $x = -2$.

2. At infinity, the curve approaches the horizontal line $y = 3$.

3. The curve intersects the vertical y-axis at the point $(0, -1)$.

1. a) Set up the corresponding equations and calculate the value of the parameters $a$, $b$, and $c$.
2. b) With the function now determined, find out with reasoned justification whether the graph ever intersects the horizontal X-axis.
3. c) Study the sign of the derivative $f'(x)$ and prove that the y-intercept is necessarily a relative extremum, indicating whether it is a maximum or a minimum.

Carry out a complete analysis and sketch the graph of $f(x) = \frac{x^2}{x - 2}$:

1. a) Domain, axis intercepts and symmetries.

1. b) Vertical, horizontal and oblique asymptotes, with the corresponding limits.

1. c) Intervals of monotonicity and local extrema.

1. d) Position of the curve relative to the oblique asymptote and final sketch.

Analyse and sketch the function $f(x) = x \cdot e^{-x}$: domain, intercepts, horizontal asymptote as $x \to +\infty$, behaviour as $x \to -\infty$, intervals of monotonicity, local extremum and a sketch of the graph.

You can do it on paper and send a photo with your answer.

Analyse and sketch the function $f(x) = x + \frac{1}{x}$: domain, odd symmetry, absence of x-intercepts (explain why), vertical and oblique asymptotes, intervals of monotonicity, local extrema with their values, and sketch.

You can do it on paper and send a photo with your answer.

We are looking for a rational function whose graph has vertical asymptote $x = 1$, horizontal asymptote $y = 2$ and passes through the point $(0, 1)$.

1. a) Propose an expression of the form $f(x) = \frac{ax + b}{x - 1}$ and determine $a$ and $b$ using the given conditions.

1. b) Check that the resulting function satisfies all three conditions.

1. c) Study its monotonicity and sketch its graph.

You can do it on paper and send a photo with your answer.

Analyse and sketch the function $f(x) = \frac{x}{\sqrt{x^2 - 4}}$: domain, symmetry, vertical and horizontal asymptotes (calculate the four required limits), monotonicity on each branch using the derivative, and a sketch of the graph.

You can do it on paper and send a photo with your answer.

After taking a medication on an empty stomach, its blood concentration, in $\text{mg/l}$, is modelled by $C(t) = \frac{4t}{t^2 + 1}$, where $t \geq 0$ is the time elapsed in hours.

1. a) Study the monotonicity of $C(t)$ and determine when the concentration is at a maximum and what that maximum value is.

1. b) Calculate the horizontal asymptote and interpret it in this context.

1. c) Determine during which time interval the concentration exceeds $1\ \text{mg/l}$.

1. d) Sketch the complete graph of the model.

You can do it on paper and send a photo with your answer.

Consider the function $f(x) = \frac{x^3 + 2x}{x^2 + 1}$.

1. a) Perform the division and check that $f(x) = x + \frac{x}{x^2 + 1}$, deducing that the line $y = x$ is an oblique asymptote.

1. b) Determine at which point the graph intersects its oblique asymptote, and study in which regions the curve lies above and below it.

1. c) Complete the analysis (symmetry, monotonicity) and draw the graph together with the asymptote.

You can do it on paper and send a photo with your answer.

Analyse and sketch the function $f(x) = \frac{\ln x}{x}$: domain, x-intercept, limits at the boundaries of the domain (specifically the vertical and horizontal asymptotes), intervals of monotonicity, local extremum and a sketch of the graph.

You can do it on paper and send a photo with your answer.

An engineer is designing the profile of a roller coaster. The first descending section follows the equation $y = -x^2 + 4$ for $x \le 1$. To smooth the transition to the climb, they decide to join, at $x = 1$, a second section shaped like the parabola $y = a(x - 2)^2 + b$ for $x > 1$.

For the join to be smooth, the whole function must be continuous and differentiable at $x = 1$ (no jumps or corners). Moreover, the curvature must change at that point: the first section is concave and the second must be convex, so that $x = 1$ is an inflection point.

1. a) Find the values of $a$ and $b$ by imposing continuity and differentiability at $x = 1$.

1. b) Check that with those values the change of curvature at $x = 1$ holds, by studying the sign of the second derivative of each section.

A fundamental geometric property of convex functions ($\cup$) is that the line segment joining any two points on its graph always lies above the curve.

Verify this property analytically for the function $f(x) = x^2$. To do so, find the equation of the secant line passing through the points on the graph with x-coordinates $x = -1$ and $x = 3$. Then, calculate the value of the line at $x = 1$ and check that it is strictly greater than $f(1)$.

Describe and draw in your notebook a sketch of a continuous function $f(x)$ defined on the domain $[-5, 5]$ that simultaneously satisfies all of the following conditions:

• It is an odd function (symmetric with respect to the origin).

• It has a relative maximum at $x = 2$ and a relative minimum at $x = -2$.

• It has exactly three inflection points located at $x = -3$, $x = 0$, and $x = 3$.

• It is convex ($\cup$) on the intervals $(-3, 0) \cup (3, 5]$.

• It is concave ($\cap$) on the intervals $[-5, -3) \cup (0, 3)$.

You can do it on paper and submit a photo of your answer.

During the spring thaw period in the Aragonese Pyrenees, the water inflow rate $Q(t)$ (in $\text{hm}^3/\text{day}$) to a reservoir in the Ebro basin over the course of a month ($t \in [0, 30]$ days elapsed) is modeled by the function:

1. a) Determine the initial flow rate at time $t = 0$ and find the limit value that the flow rate would approach if the period were extended indefinitely ($t \to +\infty$). Interpret the horizontal asymptote found in the physical context of the reservoir.
2. b) By analyzing $Q'(t)$, locate with mathematical precision the day on which the peak (maximum) flow rate is recorded and calculate its daily volume.
3. c) The dam's technical team needs to anticipate the moment when the flow rate decreases most rapidly (maximum rate of decrease) to adjust the floodgates. Calculate $Q''(t)$, locate the inflection point in the interval $[5, 30]$, and interpret its physical meaning.
4. d) Draw a detailed sketch of the graph of $Q(t)$ for $t \in [0, 30]$, marking the key points (start, asymptote, extrema, and curvature) obtained in the previous sections.

To define the longitudinal profile of an acceleration ramp on a test bench, the trajectory is modeled by a cubic polynomial $f(x) = a \cdot x^3 + b \cdot x^2 + c \cdot x + d$ that must meet the following design constraints:

1. The ramp starts at the origin of coordinates $(0, 0)$ with an initial slope of $45^\circ$ (the tangent line has slope $m = 1$).

2. It has its inflection point exactly at the point in the plane $(2, 6)$.

1. a) Set up the corresponding system of equations from $f(0)$, $f'(0)$, $f''(2)$, and $f(2)$, and obtain the numerical values of $a$, $b$, $c$, and $d$.
2. b) Calculate the exact coordinates of the relative extrema (maximum and minimum) of the function on the entire real line and classify them using the second derivative test.
3. c) Prove that every cubic function has point symmetry with respect to its inflection point: to do this, define the function shifted to the origin $g(x) = f(x + 2) - 6$ and prove formally that $g(x)$ is an odd function ($g(-x) = -g(x)$).