Sketching 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.