Exercises: Drawing a function step by step

A food delivery rider earns a base pay of $4$ € per hour worked. Graph the linear function $f(x) = 4x$, where $x$ is the number of hours and $f(x)$ the money earned in euros. To draw it, follow these steps:

1. Make a table of values by calculating the images for $x = 0, 1, 2$ and $3$.

2. Plot the points on the Cartesian plane.

3. Join the points with a ray (bearing in mind that hours cannot be negative).

We give you the axes already set up so you can place the points:

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

Graph the quadratic function $f(x) = x^2 - 4$. Follow these steps in order:

1. Identify the points where the graph crosses the $X$-axis by solving the equation $x^2 - 4 = 0$.

2. Find the point where the graph crosses the $Y$-axis by calculating $f(0)$.

3. Determine the coordinates of the vertex $V(x_v, y_v)$.

4. Draw the parabola by joining the points above. You can use the axes in the figure, where each division equals 1 unit on both axes.

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

A streaming platform offers a flat rate of $12$ € per month, no matter how many movies you watch. Graph the function $f(x) = 12$, where $x$ is the number of movies watched in a month and $f(x)$ is the monthly cost in euros.

1. a) Explain what type of function it is (linear, quadratic or constant).

1. b) Determine the value of the function for $x = 0$, $x = 10$ and $x = 20$.

1. c) Draw the graph on a coordinate system: put the number of movies ($x$) on the horizontal axis and the cost in euros ($y$) on the vertical axis, and choose a suitable scale for each axis.

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

Draw the piecewise function step by step:

$f(x) = \begin{cases} x + 2 & \text{if } x < 1 \\ 4 & \text{if } x \ge 1 \end{cases}$

1. Calculate the values of the first section for $x = -1, x = 0$ and the value it approaches as $x = 1$.

2. Draw the second section knowing that it is a constant function for values greater than or equal to $1$.

3. Pay attention to the joining point at $x = 1$ to see if there is a jump or if the function is continuous.

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

You are going to graph the absolute value function $f(x) = |x - 3|$ (in your notebook, if you like, as a support). Follow these steps:

1. First graph the auxiliary line $y = x - 3$ as a dashed line.

2. Identify the point where the line crosses the $X$ axis (it is the point where the direction of the graph changes, the vertex).

3. "Reflect" the part of the line that lies below the $X$ axis onto the positive side of the $Y$ axis.

4. Highlight the final stroke of the function $f(x)$.

Now answer based on your graph:

1. a) What is the abscissa $x$ of the point where the line $y = x - 3$ crosses the $X$ axis?

1. b) What value does the auxiliary line take at $x = 0$? And what value does $f(0)$ take after reflecting?

1. c) Calculate $f(5)$.

1. d) For $x < 3$, the graph of $f$ coincides with the line $y = -x + 3$. What is its slope?

It will be assessed that the auxiliary line, the crossing point, the reflection and the final stroke are consistent with each other, and that the values you give in the parts match your graph.

Graph the inverse proportionality function $f(x) = \dfrac{4}{x}$.

1. a) Explain why the value $x = 0$ does not belong to the domain.

1. b) Calculate the images for $x = 1, 2, 4$ and their opposites $x = -1, -2, -4$.

1. c) Plot the points obtained on a coordinate system and draw the two branches of the hyperbola, taking into account that the axes are asymptotes.

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

Draw the parabola $f(x) = -x^2 + 1$. Follow these steps to avoid mistakes:

1. Observe the sign of the coefficient of $x^2$ and decide whether the branches open upwards or downwards.

2. Calculate the vertex. Since there is no term in $x$, what is its x-coordinate $x_v$?

3. Find the intercepts with the $X$ axis.

4. Draw the curve symmetrically with respect to the $Y$ axis.

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

An electric scooter rental company charges $1$ € to unlock the scooter and $0,50$ € for each minute of use. Graph the function $f(x) = 0,5x + 1$, where $x$ is the number of minutes and $f(x)$ is the cost in euros.

1. a) Calculate the cost for $0$, $5$, $10$ and $20$ minutes.

1. b) Identify the y-intercept and explain what it means in this real-life context.

1. c) Draw the resulting graph. You can use these axes with their grid as a guide (each square is $5$ minutes horizontally and $5$ € vertically).

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

To graphically represent the absolute value function $f(x) = |2x - 6|$, Marcos performs the following procedure:\n1. He calculates the intercepts of the inner function $g(x) = 2x - 6$ with the coordinate axes: $(0, -6)$ and $(3, 0)$.\n2. Since he knows that the absolute value cannot yield negative results, he changes the sign of the $y$-coordinate of the $y$-intercept, obtaining the point $(0, 6)$.\n3. Using a ruler, he joins the points $(0, 6)$ and $(3, 0)$ and extends this same straight line indefinitely to the right.\n\nAnswer the following questions with reasoning:\na) Identify the two conceptual errors Marcos made when drawing the final graph.\nb) Explain why the graph of a function of the form $f(x) = |mx + n|$ with $m \ne 0$ can never be a single indefinite straight line.\nc) Describe the correct procedure to follow to draw $f(x)$ step by step starting from the auxiliary line $g(x)$, stating which part remains unchanged and which part is transformed.

A telecommunications company in Spain markets a data plan whose monthly rate in euros corresponds to the linear function $C(x) = 1{,}50x + 12$, where $x$ is the number of additional gigabytes (GB) consumed during the month. Two students debate what the best strategy is to represent it on the Cartesian plane:\n- Alba suggests making an exhaustive table of values calculating the outputs for $x = 0, 1, 2, 3, 4, 5$, and $6$, plotting all the points, and joining them with a ruler.\n- Hugo states that it is enough to calculate the $y$-intercept $(0, C(0))$ and a single, well-chosen additional point, such as $(4, C(4))$, draw the line that passes through them, and use a third point, for example $(8, C(8))$, solely to verify that the ruler is properly aligned.\n\na) Based on the principles of plane geometry regarding the determination of a line, justify why Hugo's proposal is mathematically sufficient.\nb) Why is Hugo's choice of $x = 4$ and $x = 8$ more sensible and accurate for drawing by hand on a grid than choosing $x = 1$ or $x = 3$?\nc) Taking into account the real-world context of data consumption, indicate on what interval of the $x$-axis the graph should effectively be drawn and whether it is appropriate to extend it toward negative values of $x$.

A basketball player shoots a free throw whose trajectory follows the function $h(t) = -t^2 + 4t + 2$, where $h$ is the height in metres and $t$ is the time in seconds from release. Plot the trajectory of the ball by following these steps:

1. Identify the type of function and determine the direction in which its branches open.

2. Calculate the coordinates of the vertex of the parabola.

3. Find the intercepts with the x-axis ($OX$) and with the y-axis ($OY$).

4. Establish the physical domain of the function taking into account the real-world context (time cannot be negative and the ball stops when touching the ground).

5. Draw the graph using the above data.

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

In the sports shoe market, the demand for a specific model is governed by the rational function $q(p) = \frac{200}{p-10}$, where $q$ is the number of pairs demanded and $p$ is the price in euros ($p > 10$). Study the demand curve by carrying out the following analysis (you can make a sketch in your notebook to guide you):

1. Determine the domain of the function in the economic context.

2. Find the equations of the vertical and horizontal asymptotes.

3. Obtain the points where the graph meets the axes, if any.

4. Make a small table of values for significant prices (for example, $p=20$, $p=50$ and $p=110$).

5. Describe what the graph of the function looks like (position relative to the asymptotes and shape of the curve) and state clearly the trend of demand as the price rises.

A courier company charges its customers according to the weight of the parcels. The rate is defined by the following piecewise function:

$f(x) = \begin{cases} 2x + 5 & \text{if } 0 \le x < 5 \\ 15 & \text{if } 5 \le x \le 10 \\ x + 5 & \text{if } x > 10 \end{cases}$

where $x$ is the weight in kg and $f(x)$ the price in euros. Graph the function step by step (you can use the axes in the figure as a guide, with 1 kg = 30 units horizontally and 1 € = 12 units vertically, or draw your own on paper):

1. Identify the type of function in each of the three intervals.

2. Calculate the values of the function at the break points ($x=5$ and $x=10$).

3. Determine whether the function is continuous at those break points by calculating the one-sided limits.

4. Draw each piece of the function in its corresponding interval, indicating with a filled or open dot whether the endpoint of each piece belongs to the function or not.

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

We want to analyze the evolution of the balance of a youth account that has had sudden movements. The function that describes the estimated balance is $f(x) = |x^2 - 4x|$, where $x$ represents the months elapsed. Draw the graph following this procedure:

1. First, plot the auxiliary parabolic function $g(x) = x^2 - 4x$, finding its vertex and its intersections with the $OX$ axis.

2. Identify the intervals where the value of $g(x)$ is negative.

3. Apply the definition of absolute value to "reflect" the negative part of the graph over the $OX$ axis.

4. Draw the final graph of $f(x)$, making sure there are no values below the x-axis.

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

The number of users of a new social network grows exponentially according to the function $U(t) = 2 \cdot 1,5^t$, where $U$ is the number of users (in thousands) and $t$ is the time in months. To graph this growth, follow these steps:

1. a) Identify the initial number of users (for $t=0$).

1. b) Determine the type of growth (is it increasing or decreasing?) by looking at the base of the power.

1. c) State the equation of the horizontal asymptote of the function.

1. d) Calculate the values for $t=1$, $t=2$ and $t=3$.

1. e) On Cartesian axes (time $t$ in months on the horizontal axis and users $U$ in thousands on the vertical axis), plot the points obtained and the asymptote, and join them with a smooth curve that represents the evolution of the community.

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

An archaeological study uses the function $f(x) = \sqrt{x+4} - 2$ to model the depth of an excavation in meters, where $x$ is the horizontal distance from the center of the site. Graph the function by following these steps:

1. a) Calculate the domain of the function.

1. b) Find the starting point of the curve (where the domain begins).

1. c) Find the points where the graph meets the coordinate axes.

1. d) Determine whether the function is increasing or decreasing on its domain.

1. e) Sketch the graph in your notebook, using a table of values, and describe the characteristic curvature of root functions.

The sound intensity (on a simplified logarithmic scale) perceived by a microphone is given by $S(x) = \log_2(x) + 3$, where $x$ is the power of the sound source. Graph the function $S(x)$ by doing the following:

1. a) Establish the domain of the function.

1. b) Identify the vertical asymptote.

1. c) Calculate the point where the intensity is zero (point of intersection with the $OX$ axis).

1. d) Obtain the value of the function for key powers such as $x=1$, $x=2$ and $x=4$.

1. e) Draw the graph showing the logarithmic behavior, indicating on it the asymptote, the point of intersection with $OX$ and the points from part d).

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

The average cost of manufacturing a run of customized T-shirts is given by the function $C(x) = \frac{4x + 100}{x}$, where $x$ is the number of T-shirts produced ($x > 0$) and $C(x)$ is the cost per unit in euros. Graph this rational function by following these steps:

1. a) Simplify the expression of the function by dividing each term of the numerator by the denominator.

1. b) Identify the vertical asymptote and the horizontal asymptote. Explain the economic meaning of the horizontal asymptote.

1. c) Determine whether the function is increasing or decreasing.

1. d) Calculate the average cost for $x=10$, $x=20$ and $x=50$.

1. e) Draw the average cost curve, marking the asymptotes and the points calculated in the previous part.

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

An agricultural cooperative in Valencia markets seasonal citrus fruits. The farm's weekly net profit (in hundreds of euros) depends on the selling price per kilogram $x$ (in euros) according to the quadratic function $B(x) = -2x^2 + 12x - 10$, defined for $x \in [1, 5]$. Before plotting the graph, you must plan the drawing:\n\n1. Estimate: Knowing that the intercepts with the horizontal axis represent the prices that yield neither profit nor loss ($B(x) = 0$), and mentally checking that $B(1) = 0$ and $B(5) = 0$:\n - Deduce, relying exclusively on the symmetry property of parabolas and without yet applying any algebraic formula, at what exact price $x$ the maximum weekly profit will be achieved.\n - Reason whether the branches of the parabola open upward or downward based on the sign of the leading coefficient, and estimate whether the vertex will lie above or below the horizontal axis.\n2. Verify: Apply the analytical formula for the $x$-coordinate of the vertex $x_v = -\dfrac{b}{2a}$ and find its $y$-coordinate $B(x_v)$. Check whether the result matches your previous estimate.\n3. Plotting plan: Explain step by step how you would sketch the parabola on the interval $[1, 5]$ using only the vertex, the $x$-intercepts, and the vertical axis of symmetry, justifying why it is not necessary to calculate a table of values with intermediate decimal numbers.

A short-stay car park in the centre of Málaga applies the following charging conditions for stays of up to 12 hours:\n- During the first 60 minutes ($0 \le t \le 1$ hour), there is a promotional rate of $2$ € per hour (or proportional fraction).\n- Between the first and the fifth hour ($1 < t \le 5$), the cost corresponds to the $2$ € for the first hour plus $3$ € for each additional hour: $C(t) = 2 + 3(t - 1)$.\n- From the fifth hour up to a maximum of 12 hours ($5 < t \le 12$), a long-stay flat rate set at $14$ € applies.\n\n1. Write the simplified analytical expression of the piecewise function $C(t)$ for its entire domain $[0, 12]$.\n2. Analyse the transition points between segments ($t = 1$ and $t = 5$): calculate the value that the function yields just to the left and to the right of each of them to determine whether the graph is continuous or has any jump.\n3. Detail the step-by-step instructions to draw the graph with complete accuracy: state which key points delimit each segment, what type of geometric line corresponds to each interval, and how the endpoints should be represented to reflect whether a boundary point is included or excluded.

A financial analyst is studying the evolution of the return on a technology investment over 10 years. The function that describes the return $R(t)$ as a function of time $t$ (in years) is given by the following piecewise expression:

$R(t) = \begin{cases} \dfrac{2t - 4}{t + 1} & \text{if } 0 \le t < 4 \\[2mm] |t^2 - 10t + 21| & \text{if } 4 \le t \le 10 \end{cases}$

Draw the graph of the function following these steps:

1. a) Determine the domain of definition in the context of the problem and study whether the function is continuous at $t = 4$ (calculate the left-hand limit and the value $R(4)$).

1. b) Find the points where it crosses the coordinate axes.

1. c) Calculate the asymptotes of the rational branch and decide which ones affect the graph in the interval $0 \le t < 4$.

1. d) Study the absolute value in the second piece: determine for which values of $t$ the expression $t^2 - 10t + 21$ equals zero, write $R(t)$ without absolute value on $[4, 7]$ and on $[7, 10]$, and find the point where the curve touches the $X$ axis forming a peak, as well as the local maximum on $[4, 7]$.

1. e) Graph the function accurately, indicating with an open or closed dot whatever is appropriate at $t = 4$.

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

An artisan bakery studies its production costs. The unit cost $C(x)$ (in euros) depends on the number of daily units produced $x$ and follows the function:

$C(x) = \frac{3x^2 + 12}{x - 1}$

Draw a detailed graph of the function for $x > 1$ by carrying out the following analysis:

1. Identify the domain and explain why it makes no sense to analyse values $x \le 1$.

2. Find the vertical asymptote and study the position of the curve relative to it using one-sided limits.

3. Determine whether an oblique asymptote exists using polynomial division and draw it as a construction line.

4. Calculate the intercepts (if any) and use a small table of strategic values to define the curvature.

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

The temperature of a new graphics processor is being monitored during a stress test. The temperature $T$ (in $^\circ C$) after $x$ minutes is modeled by the function:

$T(x) = 25 + \dfrac{60x^2}{x^2 + 9}$

To draw the graph that will represent the technical report, carry out the following steps:

1. a) Find the value of the temperature at the initial moment.

1. b) Calculate the horizontal asymptote and explain its physical meaning (limit saturation temperature).

1. c) Determine whether the function has intersection points with the x-axis and justify the result based on the range of the function.

1. d) Study the symmetry of the function and explain how it helps you draw it for hypothetically negative time values.

1. e) Draw the smoothed curve on your axes (you can use a table of values) clearly indicating the asymptote.

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

Given the rational function $f(x) = \frac{x-2}{x^2-4}$, draw its graph following a comprehensive analysis:

1. Determine the domain. Be careful! Identify the type of discontinuity at each point where the denominator equals zero.

2. Simplify the expression of the function and explain what happens graphically at $x = 2$ (is there an asymptote or a "hole"?).

3. Find the vertical and horizontal asymptotes of the simplified function.

4. Study the sign of the function over the intervals defined by the roots and asymptotes.

5. Draw the graph, representing the removable discontinuity as a small empty circle.

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

Graph the function $f(x) = \sqrt{|x| - 3}$. To do so, follow these logical steps:

1. Determine the domain of the function by solving the inequality $|x| - 3 \ge 0$.

2. Analyse the symmetry of the function. Is it even or odd? How does this affect the construction of the sketch?

3. Study the increase in the interval $(3, +\infty)$.

4. First plot the right side of the graph and then use symmetry to complete the left side.

5. State the intercepts with the $X$-axis.

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

The net profit $B(x)$ of a mobile app startup (in thousands of euros) follows the model:

$B(x) = \begin{cases} x^2 + 4x & \text{if } x < 0 \\ 5 \cdot \ln(x + 1) & \text{if } x \ge 0 \end{cases}$

Where $x$ represents the advertising investment balance (negative is debt, positive is net investment). Draw the graph step by step:

1. Study the continuity at the critical point $x = 0$.

2. For the left-hand piece ($x < 0$), find the vertex of the parabola and its points of intersection with the axes.

3. For the right-hand piece ($x \ge 0$), study the growth and find the point where it crosses the $Y$ axis. Also calculate at least one more point (for example, for $x = 4$) to help you draw the curve.

4. Determine whether the function has any asymptotes.

5. Draw both branches on the same axes and mark the joining point.

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

We wish to sketch the function $f(x) = \frac{|x^2 - 1|}{x}$. Draw the graph following these guidelines:

1. Rewrite the function as a piecewise function by removing the absolute value.

2. Determine the domain and vertical asymptotes.

3. Find the oblique asymptotes for the corresponding branches.

4. Study the symmetry of the function to simplify sketching.

5. Identify the "change of curvature" points where the absolute value acts and draw the final graph.

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

Draw the graph of the function $f(x) = 2 + e^{1/(x-2)}$ step by step. These types of functions are common in growth studies with critical barriers.

1. Find the domain and the vertical asymptote.

2. Calculate the one-sided limits as $x \to 2^-$ and $x \to 2^+$. What value does the function tend to from the left of the asymptote? And from the right?

3. Determine the horizontal asymptote by calculating the limit as $x \to \pm\infty$.

4. Check whether the function intersects the $X$-axis or the $Y$-axis.

5. Use all the above information to sketch the graph, paying special attention to the discontinuity at $x = 2$.

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

In an energy audit on the climate control of a corporate headquarters, the temperature deviation from the comfort level is modelled using a function of the form $f(x) = |ax^2 + bx + c|$, where $x \in [0, 8]$ represents the elapsed hours of the workday. Due to a computer error, the coefficients $a$, $b$, and $c$ have been erased from the report, but the following visual features of the graph remain:\n- The curve touches the horizontal axis ($y = 0$) at exactly two instants: $x = 2$ and $x = 6$, where the graph 'bounces', forming two sharp corners.\n- Between both values, the curve rises forming a symmetric continuous arc whose highest point is at $(4, 8)$.\n- At the boundaries of the domain, the function reaches the bounds $f(0) = 24$ and $f(8) = 24$.\n\n1. Formulate the inverse problem: Reconstruct the inner parabola $g(x) = ax^2 + bx + c$ before applying the absolute value. To do this, deduce with reasoning what the coordinates of the vertex of $g(x)$ must have been and in which direction its original branches must have opened.\n2. Analytically calculate the values of the parameters $a$, $b$, and $c$.\n3. Describe the protocol for drawing the final graph of $f(x)$ from the auxiliary parabola $g(x)$, stating which interval of the $x$-axis undergoes a specular reflection with respect to the $x$-axis, and expressing $f(x)$ as a piecewise function without absolute value.

The environmental department of an association of municipalities in Spain analyses the average annual cost per resident of the new selective waste collection service. This unit cost (in euros per resident) is governed by the rational function:\n$$\bar{C}(x) = \frac{180.000 + 45x}{x} = \frac{180.000}{x} + 45$$\nwhere $x$ is the number of registered residents who sign up for the system ($x > 0$).\n\n1. Asymptotic analysis and theoretical justification:\n - Explain why the vertical line $x = 0$ acts as a vertical asymptote of the function and what socioeconomic meaning the behaviour of the curve has when community participation is practically zero ($x \to 0^+$).\n - Reason why the horizontal line $y = 45$ is a horizontal asymptote as the number of participants grows indefinitely ($x \to +\infty$). Prove analytically whether the curve ever touches or intersects its horizontal asymptote at any point with $x > 0$ or whether the average cost always remains strictly above $45$ €.\n2. Critique of representation methods: A technical assistant suggests representing the function by plotting an elementary table of values with $x = 1, 2, 3, 4$, and $5$ participants. Argue in detail why this common table-of-values method is terrible for capturing the global shape of the hyperbola on real-scale Cartesian axes.\n3. Criterion-based drawing strategy: Propose a suitable scale for the axes and select four strategic values of $x$ (justifying your choice) that allow you to accurately capture the curvature of the function and its progressive approach to the horizontal asymptote.