Exercises: Tables and graphs

In the world of video games, the Cartesian plane is used to position objects on the map. Look at the following screen from an adventure game where several objects appear, and write the coordinates $(x, y)$ of each one:

• The legendary Sword.

• The gold Chest.

• The health Potion.

• The exit Portal.

Analyse the signs of the coordinates. In which quadrant is the Sword? And the Potion?

A music app charges a fixed monthly subscription of $300$ pesos. Create a table of values relating the number of months (from month $1$ to month $5$) to the total cumulative cost paid by a user. Then, plot these points on a coordinate grid.

Note: Place time on the horizontal axis ($X$) and money on the vertical axis ($Y$).

Marta is charging her cell phone and has written down the battery percentage every $10$ minutes. The data obtained are as follows: $(0, 15)$, $(10, 25)$, $(20, 35)$, $(30, 45)$ and $(40, 55)$.

Plot these points on a graph and answer the following questions:

1. a) What battery percentage did Marta have when she started charging her phone?
2. b) Is the graph increasing or decreasing? Justify your answer.
3. c) If the charging rate stays the same, what percentage will it have after $50$ minutes?

During a basketball training session, the height reached by the ball in a free throw over time is recorded. Look at the graph and extract the data to complete the accompanying table of values.

Time ($s$)	$0$	$1$	$2$	$3$
Height ($m$)

Looking at the graph:

1. a) What is the maximum height reached by the ball?
2. b) In which second does it reach this maximum height?

Plot the following points on a Cartesian coordinate system and state which quadrant or axis each one is on:

• $A = (4, 3)$

• $B = (-2, 5)$

• $C = (-3, -4)$

• $D = (0, -2)$

• $E = (5, 0)$

• $F = (1, -3)$

Remember that if a coordinate is $0$, the point lies on one of the axes.

A group of friends rent electric scooters to go around the park. The price is $120$ pesos for every $15$ minutes of use.

1. a) Complete the following table of values:

Time (min)	$15$	$30$	$45$	$60$	$75$
Price ( pesos)	$120$

1. b) Plot the points on a graph.
2. c) Does it make sense to connect the points with a continuous line? Explain whether you could rent the scooter for an intermediate time (for example, $20$ minutes) and pay the proportional amount.

Look at the graph representing the temperature of a chocolate milkshake taken out of the fridge and left on the kitchen table.

Answer the following questions:

1. a) What was the temperature of the milkshake as soon as it was taken out of the fridge (minute $0$)?
2. b) What happens to the temperature over time?
3. c) From which minute does the temperature stop rising and stay constant? What is this final temperature?

Draw a Cartesian coordinate grid and plot the following points: $P_1 = (-3, 2)$, $P_2 = (3, 2)$ and $P_3 = (0, -3)$.

Join the points in order ($P_1$ to $P_2$, $P_2$ to $P_3$, and $P_3$ to $P_1$).

1. a) What geometric shape have you formed?
2. b) State the coordinates of the midpoint of the line segment joining $P_1$ and $P_2$.

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

The graph shows how a tomato plant in the school garden grew over $8$ weeks.

1. a) How tall was the plant in week $5$?
2. b) In which week did it exceed $30$ cm for the first time?
3. c) Between which two consecutive weeks did it grow the most? How much did it grow?
4. d) Do you think that in week $12$ it will measure $50$ cm? Reason your answer by observing the shape of the graph.

The table shows life expectancy at birth in Spain by decade (rounded official data):

Year	$1970$	$1980$	$1990$	$2000$	$2010$	$2020$
Life expectancy (years)	$72$	$75.4$	$76.8$	$79.1$	$81.6$	$82.3$

1. a) By how much did life expectancy increase between $1970$ and $2020$?
2. b) In which decade did it increase the most?
3. c) Do you think that in the year $2050$ life expectancy will be $100$ years? Use the data in the table to reason your answer.

A group of friends is designing a new logo for their eSports team on grid paper. To make the design symmetrical, they need to place the vertices at specific points of the Cartesian plane.

Plot the following points in your notebook and join them in alphabetical order: $A(0, 6)$, $B(2, 2)$, $C(6, 2)$, $D(3, -1)$, $E(4, -5)$, $F(0, -3)$, $G(-4, -5)$, $H(-3, -1)$, $I(-6, 2)$ and $J(-2, 2)$. To close the figure, join point $J$ to point $A$.

1. a) What approximate geometric shape does the logo represent?

1. b) State which quadrant point $I$ is in.

The following graph shows the battery percentage of a cell phone throughout a school morning. Look closely and answer:

1. a) What percentage of battery did the phone have at 08:00?
2. b) During recess (from 11:00 to 11:30), the student did not use the phone. How is this reflected in the graph?
3. c) In which time interval did the battery drop most quickly? What do you think could cause this?
4. d) If the phone turns off when it reaches $0\%$, how much battery did it have left at 14:00?

A movie streaming platform offers a basic plan costing $8$ pesos a month, plus a surcharge of $1.50$ pesos for each new release rented.

1. a) Complete the following table of values relating the number of new releases rented ($x$) to the total monthly cost ($y$):

New releases ($x$)	0	1	2	4	6
Cost pesos ($y$)

1. b) Write the algebraic formula used to calculate the total monthly cost.
2. c) Plot the points on a coordinate system. Does it make sense to connect the points with a continuous line? Reason your answer.

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

In a study on the performance of "gaming" computers, the temperature of the processor ($^{\circ}C$) was measured during the first $10$ minutes of an online game. The data obtained are: $M(0, 30)$, $A(2, 45)$, $B(4, 60)$, $C(6, 60)$, $D(8, 75)$ and $E(10, 80)$.

1. a) Draw the graph of the temperatures, placing time on the x-axis and temperature on the y-axis.

1. b) State whether the function is increasing, decreasing or constant in the time interval from $4$ to $6$ minutes.

1. c) What is the maximum temperature reached and when does it occur?

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

The following graph compares the price of different wireless headphone models with their battery life (hours of battery).

1. a) State the coordinates of points $P_1$, $P_2$ and $P_3$.
2. b) Which model offers the most hours of battery life? How much does it cost?
3. c) If you have a maximum budget of $60$ pesos, which model with the greatest battery life can you buy?
4. d) Is there a direct proportional relationship between price and battery life? Justify your answer by observing the arrangement of the points.

A drone takes off from the ground and climbs vertically at a constant speed of $2$ meters per second for $10$ seconds. Then it stays stationary at that height for $5$ seconds to take a photo and, finally, descends to the ground in another $5$ seconds.

1. a) Identify the independent and dependent variables.
2. b) Construct a table of values relating time ($0, 5, 10, 15$ and $20$ seconds) to the drone's height.
3. c) Draw the corresponding graph. How high was the drone when it was taking the photo?

Analyze the following situations and determine in which of them a functional relationship exists. For those that are functions, state which is the independent variable ($x$) and which is the dependent variable ($y$):

1. The distance traveled by a bicycle traveling at $15$ km/h and the time elapsed.

2. The number of followers of an influencer and their hair color.

3. The weight of apples you buy at the grocery store and the money you pay for them.

4. A person's age and their height throughout their life.

Marta is training for a race. The graph shows the distance she covers after leaving her house.

1. a) How far from her house is she at $20$ minutes?
2. b) Marta stopped at a water fountain to drink. In which time interval did this happen? How long was she stopped for?
3. c) At what speed (in meters per minute) did she run during the first $10$ minutes? Help: $Velocidad = Distancia / Tiempo$.

The graph shows the temperature of a city throughout a day in July.

1. a) What was the maximum temperature and at what time was it reached? And the minimum recorded?
2. b) What was the temperature range for the day?
3. c) In which time intervals did the temperature rise? And in which did it fall?
4. d) At approximately what times was the temperature $20$ °C?

The graph shows the electricity consumption of a home, in kWh, throughout a weekday.

1. a) At what time of day does the peak consumption occur? What activities do you think explain it?
2. b) In which time interval is consumption lowest? Why?
3. c) Compare consumption at $8$ h with that at $20$ h: how many times greater is the second?
4. d) The family wants to lower their bill. Propose two specific habit changes based on what the graph shows and explain what effect they would have on it.

Two local gyms have the following rates: "Forja" gym charges a joining fee of $\text{RD\$}\ 1\,200$ plus $\text{RD\$}\ 900$ per month, and "Impulso" gym charges no joining fee but costs $\text{RD\$}\ 1\,500$ per month.

1. a) Make a table showing what a customer pays in total after $1$, $2$, $4$, $6$, $8$ and $12$ months at each gym.
2. b) Write the formula that gives the total cost as a function of the number of months $m$ for each gym.
3. c) If you plan to join for half a year, which gym is better for you? What if you join for a full year? Reason your answer using the calculations in the table.

For each of these studies, state which type of chart you think is most suitable (bar chart, line graph or circle graph) and justify your choice:

1. a) The change in temperature of a patient over three days.
2. b) The favorite sport of students at a high school.
3. c) The breakdown of a family's monthly budget between housing, food, transportation and leisure.
4. d) The number of daily visits to the school website over a month.

A group of scientists is studying the temperature at the Antarctic base "Esperanza". During a stormy day, they record the data in a coordinate system where the horizontal axis ($X$) represents the hours elapsed since midnight, and the vertical axis ($Y$) represents the temperature in degrees Celsius ($^\circ C$).

The temperature at 00:00 was $-4\,^\circ C$. During the first 4 hours, the temperature dropped steadily at a rate of $1\,^\circ C$ per hour. Then, it stayed stable for 2 hours. From hour 6, the sun came out and the temperature rose $2\,^\circ C$ every hour until 10:00.

1. a) Identify the independent and dependent variables.

1. b) Build a table of values that reflects the situation from 00:00 to 10:00.

1. c) Graph the situation. In which quadrants is the graph located?

1. d) State at what exact moment the temperature was $0\,^\circ C$.

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

Two e-scooter rental companies offer the following rates:

• Company A (Eco-Slide): Has no unlocking fee, but charges $\text{RD\$}\ 12$ for each minute of use.

• Company B (City-Roll): Charges $\text{RD\$}\ 60$ for unlocking and $\text{RD\$}\ 6$ for each minute of use.

• Express using a formula the relationship between time ($x$) and total cost ($y$) for both companies.

• Draw up a comparative table of values for $0, 5, 10, 15$ and $20$ minutes.

• Plot both functions on the same Cartesian graph.

• After how many minutes does Company B start to be cheaper? Justify your answer by observing the point where the graphs intersect.

A gamer is live streaming a game. The following graph shows the number of viewers over the 60 minutes that the stream lasted.

Analyse the graph and answer:

1. What is the domain of the function (total time)?

2. In which time intervals was the number of viewers increasing? In which was it decreasing?

3. State the maximum and minimum points (viewers and the minute in which they occur).

4. If the server pays them a bonus if they keep more than 500 people connected for at least 15 consecutive minutes, did they get the bonus?

We are designing a map for an adventure video game on a Cartesian plane. The main base is at the origin $O(0,0)$.

1. Plot the following points of interest:

2. The Dark Forest at $A(-5, 4)$.

3. The Dragon's Cave at $B(3, -6)$.

4. The Abandoned Castle at $C(-2, -3)$.

5. If a player moves from the base 4 units east and then 2 units north, what are their coordinates? Call it point $D$.

6. In which quadrant is each of the previous points located?

7. The "Danger Area" is a rectangle whose vertices are $P(-2, 2)$, $Q(2, 2)$, $R(2, -2)$ and $S(-2, -2)$. Is the base inside this area?

A cell phone battery behaves differently depending on usage. A user records that:

• At 100% charge (time 0), they start watching a TV series. The battery decreases by 15% each hour for 4 hours.

• Afterward, they put the phone into power saving mode and it only decreases by 5% per hour for the next 4 hours.

• Construct a table of values relating the time elapsed (hours) to the remaining battery percentage.

• Draw the graph. Is it a continuous function? Why?

• Does it make sense to connect the points in this situation? Explain whether the variable "time" is discrete or continuous.

• How much battery will be left after 6 hours of total use?

Look at the following table of values relating the depth of an exploration submarine to the pressure it withstands:

Depth ($m$)	$0$	$10$	$20$	$30$	$40$
Pressure ($atm$)	$1$	$2$	$3$	$4$	$5$

1. Plot the points on a Cartesian plane.

2. Are the points collinear? If so, draw the line connecting them.

3. Find the rule or formula to calculate the pressure ($y$) given the depth ($x$).

4. If the submarine can only withstand up to $12 atm$ of pressure, what is the maximum depth it can go down to?

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

Se ha realizado un experimento para ver cómo se enfría un chocolate caliente. Se han tomado medidas cada $2$ minutos:

Time (min)	$0$	$2$	$4$	$6$	$8$	$10$	$12$
Temp (°C)	$80$	$60$	$45$	$35$	$28$	$24$	$22$

1. a) Draw the graph in your notebook (time on the horizontal axis and temperature on the vertical axis). Is it a straight line? Explain why.

1. b) Is the function increasing or decreasing?

1. c) Look at the trend of the data: do you think the temperature will drop below $0\,^\circ C$ if we wait a long time? (Keep in mind that the room temperature is $20\,^\circ C$).

1. d) Look at the $2$-minute intervals in the table: in which of them did the temperature drop fastest?

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

Make up a story that corresponds exactly to the following distance-time graph, which represents a young woman's journey from her home to the library and her return. You must explain what happens in each section ($A, B, C, D$ and $E$), mentioning times, distances and possible reasons for stops or changes in speed.

The graph shows the distance from home of a group of cyclists during a $5$-hour trip.

1. a) How many kilometers did they travel in the first hour? And in the second?
2. b) How long were they stopped for? How can you tell from the graph?
3. c) What was the maximum distance they reached from home?
4. d) In the last section, they returned home in $1$ hour. How many kilometers did they cover in that hour? Was that the fastest section of the trip? Explain your answer by comparing the slopes of the sections.

For each of the following situations, describe what the graph representing it would look like (increasing, decreasing, constant, section by section...) and justify your choice:

1. a) A person's height from birth up to $20$ years old.
2. b) The price of a school canteen menu throughout a school year in which it has not changed.
3. c) The water level of a reservoir during a very dry summer, and then during a rainy autumn.
4. d) The number of passengers on a city bus from when it leaves the first stop until it reaches the end of the line.