Exercises: Table and graph of a function

Given the linear function $f(x) = 2x - 3$:

1. a) Complete a table of values for $x \in \{-2, 0, 1, 3\}$.
2. b) Draw the graph of the function in the Cartesian plane using the points obtained.

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

Consider the quadratic function $f(x) = x^2 - 4$:

1. a) Calculate the images for $x = -3, -2, 0, 2, 3$ and organise the results in a table.
2. b) Draw the graph of the function and identify the coordinates of its vertex.

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

Observe the following graph of a function $f(x)$ defined by straight-line segments:

From the graph, determine:

1. a) The domain and range of the function.
2. b) The value of $f(-2)$ and $f(1)$.

Given the radical function $f(x) = \sqrt{x+2}$:

1. a) Determine its domain analytically.
2. b) Complete a table of values for $x \in \{-2, -1, 2, 7\}$ and use it to sketch the graph.

Calculate analytically the points of intersection with the x-axis and with the y-axis of the rational function $f(x) = \frac{3x+6}{x-1}$.

Given the modulus function $f(x) = |2x - 5|$:

1. a) Evaluate the function at the points $x = 0; 2; 2,5; 3; 5$ and construct a table of values.
2. b) Draw the graph of the function.

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

Identify whether the following table of values corresponds to a linear function or a quadratic function by calculating the differences between consecutive values of $y$. Then, find its algebraic expression $y = f(x)$.

The cost $C(t)$ in euros of leaving a car in a car park is given by the function $C(t) = 2 + 1,5t$, where $t$ is the time in hours.

1. a) Make a table of values for the first 4 hours ($t=0, 1, 2, 3, 4$).
2. b) Draw the graph of the function in that interval.

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

Marcos has to plot the piecewise function $f(x) = \begin{cases} 2x + 1 & \text{if } x \le 2 \\ x^2 - 3 & \text{if } x > 2 \end{cases}$ and decides to calculate the value at the boundary $x = 2$ for his table of values. In his notebook, he writes: "Since $x = 2$ is the transition point, I substitute into both branches: $f(2) = 2 \cdot 2 + 1 = 5$ and $f(2) = 2^2 - 3 = 1$; therefore, both the point $(2, 5)$ and the point $(2, 1)$ belong to the graph".\na) Identify the conceptual error that Marcos makes and explain why his statement contradicts the mathematical definition of a function.\nb) State the only correct value for $f(2)$, justifying which branch corresponds according to the given condition.\nc) To analyze the behavior of the graph just after the boundary without making Marcos's mistake, what value of $x$ would you propose for the table, and what value of the function is obtained?

Given the rational function $f(x) = \dfrac{3}{x - 4}$:\na) Before calculating any exact value, estimate what sign and what order of magnitude (very close to zero or very large in absolute value) you expect $f(x)$ to take when evaluated at values very close to $4$, such as $x = 3,9$ and $x = 4,1$. Justify your prediction using the signs of the numerator and the denominator.\nb) Verify your estimation by calculating the values of the function for $x = 3,9$; $x = 3,99$; $x = 4,01$, and $x = 4,1$, and arrange them in a table of values.\nc) From the numerical results obtained, interpret what happens geometrically at the line $x = 4$ and reason whether it is a vertical asymptote.

The temperature $T$ (in $^\circ\text{C}$) of an industrial oven from when it is turned on is given by the function $T(t) = 20 + 15t$, where $t$ is measured in minutes.

1. a) What is the initial temperature of the oven?
2. b) Construct a table of values for $t = 0, 5, 10, 15$.
3. c) How much time must elapse for the oven to reach $200^\circ\text{C}$?

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

Given the piecewise function:

$f(x) = \begin{cases} -x + 2 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \ge 1 \end{cases}$

1. a) Construct a table of values by evaluating the function at $x = -2; 0; 0,99; 1; 2; 3$.
2. b) Draw the graph of the function and state whether it is continuous at $x = 1$.

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

Consider the quadratic function $f(x) = -x^2 + 6x - 5$.

1. a) Calculate analytically the coordinates of its vertex and its axis intercepts.
2. b) Create a table of values taking two points to the left of the vertex and two to the right.
3. c) Draw the graph of the parabola.

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

The following graph shows the evolution of the temperature $T$ (in $^\circ\text{C}$) of a cup of coffee as a function of time $t$ (in minutes) since it is served.

From the graph:

1. a) State the initial temperature of the coffee.
2. b) What value does the temperature approach as a large amount of time elapses? What physical meaning does the horizontal asymptote have?
3. c) Is the function increasing or decreasing? Justify your answer.

Given the rational function $f(x) = \frac{4}{x-2}$:

1. a) Determine its domain.
2. b) Construct a table of values evaluating the function at points close to $x = 2$ from the left and from the right ($x = 1,9; 1,99; 2,1; 2,01$).
3. c) Based on the results obtained, what graphical feature does the function display at $x = 2$?

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

In a laboratory, the growth of a bacterial population is studied. The number of bacteria $N(t)$ in thousands is given by the function $N(t) = 100 \cdot 2^t$, where $t$ is the time in hours.

1. a) Construct a table of values for the first 4 hours ($t = 0, 1, 2, 3, 4$).
2. b) Draw the graph of the function. What type of growth is observed?

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

Consider the function $f(x) = \sqrt{9-x^2}$.

1. a) Calculate its domain of definition analytically.
2. b) Make a table of values for $x \in \{-3, -2, 0, 2, 3\}$.
3. c) Draw the graph of the function and state its range.

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

Given the absolute value function $f(x) = |x^2 - 4|$:

1. a) Express the piecewise function by removing the absolute value.
2. b) Find the axis intercepts.
3. c) Construct a table of values in the interval $[-3, 3]$ and draw its graph.

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

In a Spanish town, the monthly water bill is structured into two consumption tiers to encourage water conservation:\n- Tier 1: If the monthly consumption $x$ does not exceed $15\text{ m}^3$, a fixed fee of $6\text{ \euro}$ is applied plus $0,50\text{ \euro}$ for each cubic meter consumed.\n- Tier 2: If consumption exceeds $15\text{ m}^3$, the first $15\text{ m}^3$ are billed according to Tier 1, and each additional cubic meter exceeding $15$ is charged at $1,20\text{ \euro}$.\na) Write the analytic expression of the total cost $C(x)$ (in euros) as a function of the cubic meters $x$ consumed ($x \ge 0$) as a piecewise-defined function.\nb) Create a strategic table of values that includes zero consumption ($x = 0$), the boundary point where the tier changes, and one value on each side of the change ($x = 14$ and $x = 16$).\nc) Determine whether the bill experiences an abrupt jump when going from $15\text{ m}^3$ to a slightly higher consumption or if the cost varies continuously. Justify your answer using the values from your table.

The goal is to design the analytic formula of a rational function of the form $f(x) = \dfrac{ax + b}{x - c}$ so that it satisfies the following characteristics observed in its key table of values:\n1. Its graph intersects the x-axis only at the point $(4, 0)$.\n2. Its graph intersects the y-axis at the point $(0, -2)$.\n3. The function is not defined at $x = 2$, having a vertical asymptote at that line.\na) Reasonably determine the values of the real constants $a$, $b$, and $c$.\nb) With the obtained function, create a strategic table of values that includes the intercepts with the axes and two values located immediately before and after the forbidden value ($x = 1,9$ and $x = 2,1$).\nc) Explain what happens to the values of $f(x)$ as the variable $x$ takes increasingly large positive values ($x = 100$, $x = 1000$) and deduce the equation of its horizontal asymptote.

The monthly profit $B(x)$, in euros, of an electronic components factory as a function of the number of units produced $x$ is modelled by the function $B(x) = -5x^2 + 300x - 4000$.

1. a) Find the break-even points (production for which the profit is zero).
2. b) Generate a strategic table of values including the break-even points and the vertex.
3. c) How many units must be produced to maximise the profit, and what is this maximum profit?

A quadratic function of the form $f(x) = ax^2 + bx + c$ passes through the points $(0, 3)$, $(1, 0)$ and $(2, -1)$.

1. a) Set up and solve the system of equations to find the values of $a, b$ and $c$.
2. b) Using the algebraic expression obtained, calculate the vertex and construct a table of values to plot the parabola.

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

Given the piecewise function depending on a parameter $k$:

$f(x) = \begin{cases} x^2 - k & \text{if } x < 2 \\ 2x + 1 & \text{if } x \ge 2 \end{cases}$

1. a) Find the value of $k$ so that the two branches of the function "connect" at the point $x = 2$ (that is, so that the function is continuous).
2. b) Using the value of $k$ found, construct a table of values in the interval $[0, 4]$ and sketch the graph of the function.

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

The mass $M(t)$ in grams of a radioactive isotope decreases over time $t$ (in days) according to the exponential function $M(t) = M_0 \cdot (0,5)^{\frac{t}{T}}$, where $M_0$ is the initial mass and $T$ is the half-life.

Given that at time $t=0$ there were $200\text{ g}$ of substance and after $60\text{ days}$ there are $50\text{ g}$ remaining:

1. a) Calculate the values of the parameters $M_0$ and $T$.
2. b) Construct a table of values for $t = 0, 30, 60, 90, 120$ and sketch the graph.

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

Observe the following graph of a rational function of the form $f(x) = \frac{ax+b}{cx+d}$:

From the information provided by the graph (asymptotes and axis intercepts):

1. a) Determine the domain and range of the function.
2. b) Deduce the algebraic expression of the function $f(x)$.

Given the logarithmic function $f(x) = \log_2(x+4)$:

1. a) Determine analytically its domain and the equation of its vertical asymptote.
2. b) Construct a table of values by choosing strategic values of $x$ that result in exact powers of base 2 for the argument of the logarithm.
3. c) Plot the graph of the function.

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

It is desired to build a rectangular pen taking advantage of an existing wall for one of its sides. There are $20\text{ m}$ of wire fencing available for the other three sides.

1. a) Express the area $A(x)$ of the pen as a function of the length $x$ of one of the sides perpendicular to the wall.
2. b) Determine the valid domain of the function $A(x)$ in the context of the problem.
3. c) Construct a table of values and use it to sketch the graph and find the dimensions that maximise the area.

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

The following figure shows the graph of the exponential function $f(x) = 2^x$ and the line $y = x$.

1. a) From the highlighted points on the blue curve of $f(x)$, construct a table of values for its inverse function $f^{-1}(x)$.
2. b) Draw the graph of the inverse function in your exercise book using symmetry with respect to the line $y=x$.
3. c) Write the algebraic expression for $f^{-1}(x)$.

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

To plot the graph of the function $f(x) = \dfrac{x^2 - 4x + 3}{x - 1}$, two students propose different methods:\n- Lucía's method: Directly build a table with seven consecutive integers: $x \in \{-1, 0, 1, 2, 3, 4, 5\}$, substituting directly into the given formula.\n- David's method: Before calculating points, study the domain of the function, factor the numerator to see if it simplifies with the denominator for $x \ne 1$, and then create a strategic table with intercepts and values close to the discontinuities.\na) If Lucía's method is followed, what insurmountable problem arises exactly at $x = 1$? Could this lead to the erroneous conclusion that the line $x = 1$ is a vertical asymptote?\nb) Carry out David's method: simplify the expression of $f(x)$ for all $x \ne 1$ and calculate the values of $f(x)$ for $x = 0,99$ and $x = 1,01$. Do these values grow without bound like at an asymptote, or do they approach a finite number?\nc) Compare both methods and justify which one is more reliable for avoiding false graphical interpretations. Precisely describe what geometric feature appears on the graph at the point with abscissa $x = 1$ (indicating its coordinates).

During an atmospheric pollution episode in a city, the nitrogen dioxide concentration $C(t)$ (in $\mu\text{g}/\text{m}^3$) during the first $24$ hours is modeled by a piecewise function:\n- Initial segment ($0 \le t \le 6$): The concentration follows a parabola $C(t) = -t^2 + bt + c$, whose vertex (maximum of the segment) is reached exactly at $t = 4\text{ h}$ with a level of $36\text{ }\mu\text{g}/\text{m}^3$.\n- Later segment ($6 < t \le 24$): Traffic restriction measures are activated and the concentration decreases according to the rational function $C(t) = \dfrac{k}{t - 2} + 12$.\na) Determine the constants $b$ and $c$ from the information about the vertex and find the concentration at the initial instant $t = 0$.\nb) Calculate the value of the parameter $k$ so that the concentration does not undergo a discontinuity at the transition between segments ($t = 6$).\nc) Design a strategic table of values for the interval $[0, 24]$ containing: the start, the vertical axis intercept, the vertex, the joining point, a subsequent point ($t = 12$), and the final endpoint ($t = 24$). From the table, reason what minimum value the concentration would tend toward if the restrictions were extended indefinitely over time.