The graph of a function

A drone takes off from the ground, rises to a certain height, remains stable to take some photos, and then descends to land again. Look at the graph and answer:

1. a) What maximum height did the drone reach?
2. b) How long was the drone flying at that maximum height?
3. c) In which time interval was the drone descending?

Look at the graph representing a cyclist's speed during the first $10$ minutes of their training session.

1. a) Identify the time intervals in which the function is increasing, decreasing or constant.
2. b) What was the maximum speed reached?
3. c) At which minute did they start braking until coming to a stop?

The following graph shows how the battery of Lucía's mobile phone changes from when she leaves home in the morning until she returns in the afternoon. Look at the graph and answer the following questions:

1. a) What percentage of battery did it have when she left home? And 6 hours after leaving?

1. b) State the time intervals in which the function is increasing and explain what you think happened during that period.

1. c) At what moment does it first reach its minimum value, and for how long does it stay at that value?

1. d) What are the maximum and minimum values the function reaches?

The function $f(t)$ represents the depth (in meters) at which an underwater exploration drone is located as a function of time (in minutes) since it dives. Its graph is the following:

1. a) Determine the domain and the range of the function from the graph.

1. b) For how long was the drone at a constant depth of $4$ meters?

1. c) At what moments in time is the drone at a depth of $2$ meters?

The trajectory of a basketball when thrown is given by the graph of a function relating the height (in metres) to the horizontal distance from the player (in metres).

1. a) Find the axis intercepts of the function with the $X$ and $Y$ axes.
2. b) What is the height of the ball at the moment it is thrown ($x=0$)?
3. c) If the basket is located at a height of $3$ metres, at what horizontal distance from the player is the basket located if the ball goes in right at that point?

Decide whether the following functions are continuous or discontinuous, giving a brief justification for your answer:

1. a) The depth of a swimming pool according to our distance from the edge (assuming the bottom is sloped).
2. b) The number of attendees at a music concert according to the time of day.
3. c) The cost of a phone call billed for each whole minute started.

An electric scooter rental company offers a rate for tourists. The cost $y$ in euros depends on the time $x$ in minutes. The graph of the function is discontinuous because the company charges by blocks of 30 full minutes or fraction. Each time $x$ belongs to a single block, and the right endpoint of each block is included in it:

• If $0 < x \le 30$ minutes: 5 € (flat rate for unlocking and initial use).

• If $30 < x \le 60$ minutes: 8 €.

• If $60 < x \le 90$ minutes: 11 €.

• And so on, increasing by 3 € in each block, up to a maximum of 3 hours (180 min).

For example, exactly 30 minutes cost 5 €, but 31 minutes cost 8 €.

1. a) Draw the "step" graph of the function for $0 < x \le 180$. Use closed and open points at the jumps to indicate which value is taken at each endpoint.

1. b) State the values of $x$ at which the graph is discontinuous.

1. c) Calculate how much a person who uses the scooter for 105 minutes would pay.

1. d) Is this relation a function? Justify whether a time value can have two different prices.

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

Plot the points from the following table on a coordinate system, which relates the time elapsed (in seconds) to the distance covered (in metres) by an electric scooter travelling at a constant speed. Does it make sense to connect the points with a continuous line?

The following graph represents the distance of a hurdler from the finish line during a 400-metre race.

Analyse the graph and answer:

1. a) At what moments does the runner stop or fall? How is this reflected on the graph?
2. b) Is the runner's speed constant throughout the race? Compare the gradients of the straight sections.
3. c) What is the meaning of the y-intercept (point of intersection with the $Y$ axis) and the x-intercept (point of intersection with the $X$ axis)?