Exercises: Graph of a function

On the following coordinate map, the locations of several items in an adventure video game are marked. State the Cartesian coordinates of each item: the Chest ($C$), the Key ($L$), the Potion ($P$), and the Shield ($E$).

For a graph to represent a function, each value of the independent variable $x$ must correspond to a single value of the dependent variable $y$. Look at the following graphs and justify which of them is not a function.

A streaming platform charges a monthly flat fee of $5\ €$ plus $2\ €$ for each new release film you rent.

1. a) State which is the independent variable and which is the dependent variable.
2. b) If we let $x$ be the number of films and $y$ be the total cost, complete the following table of values:

Films ($x$)	0	1	2	3
Cost ($y$)

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?

Time ($s$)	1	2	3	4	5
Distance ($m$)	4	8	12	16	20

The following graph shows the temperature of a can of soft drink from the time it is taken out of the fridge and left on the kitchen table.

1. a) What was the temperature of the soft drink at the moment it was taken out ($t = 0$)?
2. b) What is its temperature after $20$ minutes?
3. c) Is the function increasing or decreasing? What does this mean in the context of the problem?

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?

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.

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?

Analyse the following three graphs. State with reasons which of them does not represent a function and explain why based on the definition of a function you have studied.

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?

A sound engineer measures the noise level (in decibels, dB) during a band's rehearsal. The graph shows the recorded levels over a period of $10$ minutes.

1. a) State the time intervals during which the noise increases (increasing function).
2. b) Indicate the local maxima and local minima of the function. What do these points mean in the context of the rehearsal?
3. c) Is it a continuous function? Justify your answer.

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?

An online film rental site offers a pricing scheme where the first three days cost $4$ € in total, and from the fourth day onwards, the price increases by $1,5$ € for each additional day of delay.

1. a) Complete the following table of values:

Days ($x$)	1	2	3	4	5	6
Price ($y$)

1. b) Represent the function graphically.
2. c) Is it a continuous or discontinuous function? Explain whether it makes sense to join the points on the graph with a continuous line in this real-life context.

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

Two friends, Marcos and Elena, are saving to buy a console. The graph shows the money each person has at the end of each week.

1. a) Who started with more money saved?
2. b) In which week do their savings become equal? How much money do they have at that moment?
3. c) State during which periods Elena saves faster than Marcos (where her graph is steeper).
4. d) What is the domain of both functions according to the graph?

A container with water is heated and its temperature is recorded every minute. When it reaches $100^{\circ}C$, the flame is kept constant for $3$ minutes and then turned off, letting the water cool down slowly.

1. a) Draw a qualitative graph that represents this situation (time on the $X$ axis, temperature on the $Y$ axis).

1. b) On your graph, identify the increasing zone, the constant function zone and the decreasing zone.

1. c) If the room temperature is $20^{\circ}C$, what value will the water temperature approach after a long time?

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

A Twitch content creator analyzes the evolution of their subscribers during a special 10-hour event. The function that describes the number of subscribers $S$ as a function of time $t$ (in hours) follows this behavior:

• During the first 2 hours, the growth is constant and linear, going from 100 to 500 subscribers.

• Between hour 2 and hour 5, the number of subscribers remains stable.

• From hour 5 onwards, due to a "raid" from another channel, the subscribers increase following the expression $S(t) = 100t$ until hour 8.

• In the last 2 hours, the channel loses subscribers at a constant rate until ending with 600.

Sketch the graph of the function (put time $t$ on the horizontal axis, from 0 to 10 hours, and the number of subscribers $S$ on the vertical axis), determine the domain and the range in this context, and identify the intervals of increase and decrease.

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

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.

A surveillance drone performs a reconnaissance flight. The graph shows its height $h$ (in metres) against time $t$ (in minutes). The drone takes off, rises to 40 m, moves horizontally for a period of time, rises again to 60 m to avoid an obstacle, drops rapidly to 20 m to observe something in detail, and finally lands.

From the graph that you must sketch yourself using these data:

1. a) What is the rate of change (gradient) during the rapid descent if it occurs between minute 10 and minute 12?
2. b) State the time interval where the function is constant.
3. c) If the drone has a battery life of 15 minutes, what is the actual domain of the function if landing finishes right at that limit?

The temperature of a chemical reaction is studied in a laboratory. The temperature $T$ (in $^\circ C$) follows the function $T(x) = |x - 5| + 10$, where $x$ is the time in minutes since a reagent was added.

1. a) Construct a table of values for $x$ from 0 to 10 minutes.
2. b) Draw the graph. You will notice that it has a "V" shape.
3. c) In which minute is the minimum temperature reached? What is this value?
4. d) Determine the interval where the function is decreasing and explain what it means physically in the experiment.

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

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)?

A 1000-litre cylindrical tank is filled using a hose that provides a constant flow rate of 50 litres per minute. However, the tank has a leak at the base losing 10 litres per minute.

1. a) Write down the algebraic expression of the function relating the volume of water $V$ (in litres) to the time $t$ (in minutes), assuming that initially there are 100 litres.
2. b) Draw the graph. What type of function is it (linear or affine)?
3. c) How long will it take to fill completely? State this value as the upper limit of the domain of the function in this practical situation.

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

Look at the following four graphs. One of them represents the height of a ball that is dropped from a certain height and bounces on the ground, another the filling with water, at a constant flow rate, of a cone-shaped glass (with the vertex at the bottom), another the speed of a car that brakes at a traffic light until it stops, and the last one is not a function. In the real situations, $x$ is time.

1. a) Identify each one, justifying your answer based on the shape of the curve (growth, curvature, continuity).

1. b) Explain why one of the graphs cannot be considered a function.

Two siblings, Iratxe and Ionut, leave home to go to school, which is 1 km away.

• Ionut leaves at 8:00 and walks at a constant speed, taking 20 minutes to arrive.

• Iratxe leaves 5 minutes later, runs for the first 500 metres (taking 4 minutes), and then walks the rest of the way at the same speed as Ionut.

1. a) On the same graph, represent the distance from home for both of them as a function of the time elapsed since 8:00.
2. b) At what time and at what distance from home does Iratxe overtake Ionut? Find the point of intersection of the two graphs.
3. c) Who arrives at school first, and by how much time difference?