Exercises: Representation on the Cartesian plane

In the following Cartesian plane, the positions of four objects in a room are marked. Write down the coordinates $(x, y)$ of each of the points represented: the remote control ($A$), the book ($B$), the keys ($C$) and the backpack ($D$).

Draw a set of coordinate axes in your notebook and identify which quadrant (first, second, third, or fourth) the following points are in without plotting them:

• $P(4, 7)$

• $Q(-3, 5)$

• $R(-2, -8)$

• $S(6, -1)$

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

We are designing a level for a video game on a grid. Three of the vertices of a rectangular room are the points $A(1, 2)$, $B(6, 2)$ and $C(6, 5)$.

1. a) Plot the points on a Cartesian plane.

1. b) What must the coordinates of the fourth vertex $D$ be so that the room is a perfect rectangle?

1. c) Join the points in order ($A$, $B$, $C$ and $D$) and check your answer.

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

Complete the following sentences about the Cartesian coordinate system:

1. a) The horizontal axis is called the axis of ________ and is represented by the letter $X$.
2. b) The vertical axis is called the axis of ________ and is represented by the letter $Y$.
3. c) The point of intersection of both axes is called the ________ and its coordinates are $(0, 0)$.

A delivery drone departs from the logistics center, located at the point $(0, 0)$. It makes two deliveries: the first at the point $A(0, 4)$ and the second at the point $B(5, 0)$.

1. a) State which axis each of the delivery points lies on.

1. b) If the drone flies in a straight line from point $A$ to point $B$, draw its path on a coordinate system (on squared paper, with one unit per square). Also mark the points $A$ and $B$.

Plot the following points on a grid and join them in alphabetical order. Finally, join the last point to the first. What geometric shape have you obtained?

• $A(-2, 2)$

• $B(2, 2)$

• $C(2, -2)$

• $D(-2, -2)$

Given the points $M(3, 2)$ and $N(3, -4)$:

1. a) Plot them on the Cartesian plane.

1. b) What do their coordinates have in common?

1. c) If you draw the line that passes through both, is it a horizontal or a vertical line?

Plot the point $A(-4, 3)$ in your notebook.

1. a) Write down the coordinates of a point $B$ that has the same x-coordinate as $A$, but its y-coordinate is the opposite.
2. b) Write down the coordinates of a point $C$ that has the same y-coordinate as $A$, but its x-coordinate is the opposite.
3. c) Plot all three points and observe the symmetry created with respect to the axes.

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

A group of archaeologists is mapping the ruins of an ancient Roman city. They have placed the center of an amphitheater at the origin of coordinates $O(0,0)$. Plot the following points of interest on a Cartesian plane (in your notebook, using grid paper) and determine which quadrant each one is in:

• Temple of Jupiter: $A(-4, 3)$

• Public baths: $B(5, -2)$

• North Gate: $C(0, 6)$

• Forum: $D(-3, -5)$

• Market: $E(6, 4)$

If they decided to build a fence joining points $E$, $B$ and $A$, what type of geometric figure would be formed?

Look at the points represented on the following marine navigation radar. Write down the coordinates of the ships identified by the letters $P$, $Q$, $R$ and $S$. Also indicate which ships have the same x-coordinate and which ones lie on one of the coordinate axes.

Draw a set of Cartesian axes and plot the points $A(2, 1)$ and $B(2, 5)$.

• Find the coordinates of a point $C$ such that triangle $ABC$ is an isosceles right-angled triangle (having two equal sides and an angle of $90^\circ$).

• Is there more than one possible solution? Write down the coordinates of all the ones you find.

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

In a strategy video game, the base of operations is at point $M(-3, 2)$. The programmer wants to place three guard posts in symmetrical positions:

• Post $P_1$ is the reflection of $M$ across the x-axis.

• Post $P_2$ is the reflection of $M$ across the y-axis.

• Post $P_3$ is the reflection of $M$ in the origin.

Plot the four points on the Cartesian plane and write down their coordinates.

A delivery drone follows a linear route. It takes off from the warehouse at point $S(-5, -2)$ and must pass through points $T(-2, 1)$ and $U(1, 4)$ before reaching its destination.

1. a) Plot the points $S$, $T$ and $U$ on a Cartesian plane (on grid paper). Are they collinear?

1. b) If the drone continues along its path in the same direction, will it pass through point $V(4, 7)$? Justify your answer graphically: plot $V$ as well and the line joining the points.

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

Draw a Cartesian plane and plot the following points that form the vertices of a rectangle: $A(-4, -1)$, $B(3, -1)$ and $C(3, 4)$.

• Find the coordinates of the fourth vertex $D$ to complete the rectangle.

• Calculate the perimetre and the area of this rectangle by counting the grid units.

• In which quadrant is most of the area of the rectangle located?

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

Identify which of the following points satisfy the condition that their y-coordinate is double their x-coordinate ($y = 2x$):

• $A(1, 2)$

• $B(-2, -4)$

• $C(3; 1,5)$

• $D(-1, 2)$

• $E(0, 0)$

Plot them on a Cartesian plane. What do you observe when you join these points?

At a weather station, the variation in temperature over the course of a morning has been recorded. The data obtained are:

• At $0$ hours (midnight): $-2\,^\circ C$

• At $2$ hours: $-1\,^\circ C$

• At $4$ hours: $2\,^\circ C$

• At $6$ hours: $4\,^\circ C$

In the figure you have the axes with their scale (on the $X$ axis, each unit is $1$ hour; on the $Y$ axis, each unit is $1\,^\circ C$). Copy it into your notebook, preferably on graph paper, and solve:

1. a) Represent these data as points $(x, y)$ on the Cartesian plane, where the $X$ axis represents time and the $Y$ axis represents temperature.

1. b) Join the points with line segments to draw the graph.

1. c) What is the meaning of the point where the graph crosses the x-axis? Give its coordinates approximately.

1. d) What is the meaning of the point where the graph crosses the y-axis? Give its coordinates.

The design of a new logo for a sportswear brand is based on a rhombus $ABCD$ (with the vertices in that order). Three of its vertices are known: $A(-2, 1)$, $B(1, 4)$ and $C(4, 1)$, which appear in the figure.

a) Determine the coordinates of the fourth vertex, $D$, so that the figure $ABCD$ is a rhombus, and plot it on the Cartesian plane.

b) Calculate the coordinates of the center of the rhombus (the point where its diagonals intersect).

c) If we wanted to double the size of the logo while keeping vertex $A$ in place (that is, multiplying by 2 the distances from $A$ to each vertex), what would be the new coordinates of points $B$ and $C$?

An electric scooter rental company uses a stepped tariff to avoid trips that are too short. The cost $y$ (in euros) depends on the time $x$ (in minutes) according to these rules:

• If the trip lasts between $0$ and $10$ minutes (inclusive), the cost is $2$ €.

• If it lasts more than $10$ and up to $20$ minutes, the cost is $4$ €.

• From $20$ minutes onwards, the cost is a fixed $4$ € plus $0,10$ € for each additional minute past $20$.

a) Draw a graph of this function on the Cartesian plane for a time from $0$ to $40$ minutes.

b) State whether the function is continuous or discontinuous. If it is discontinuous, indicate the jump points.

c) How much will a person pay who uses the scooter for $25$ minutes? And someone who uses it for $10$ minutes and one second?

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

A surveillance drone completes a round trip from its base. Its distance from the base (in metres) with respect to time (in seconds) is given by the following graph:

a) Determine the domain and range of this function according to the graph.

b) Identify the time intervals in which the drone moves away from the base, gets closer, or remains stationary.

c) What is the speed of the drone (in m/s) during the first $10$ seconds? Remember that $v = \frac{distancia}{tiempo}$.

In a strategy video game, the area of influence of a defence tower is bounded by the points $(x, y)$ that satisfy the following condition: the modulus of their x-coordinate plus the modulus of their y-coordinate must be less than or equal to $4$, that is, $|x| + |y| \leq 4$.

a) Plot on the Cartesian plane all the points that form the boundary of this region (where $|x| + |y| = 4$). What geometric shape do you obtain?

b) Check whether the points $P(2, 2)$, $Q(-3, 2)$ and $R(0, -4)$ are inside, outside, or on the boundary of the area of influence.

c) Calculate the area of this region in square units.

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

A biologist measures the depth of a lake at different points along a straight line that crosses from one shore to the other. The starting shore is at $x=0$ and the final shore is at $x=10$. The depth $y$ (in meters, expressed as negative values to indicate depth) follows the function $y = x^2 - 10x$.

a) Make a table of values for $x \in \{0, 2, 4, 5, 6, 8, 10\}$.

b) Plot the points on the Cartesian plane and join them with a smooth curve.

c) At which point is the maximum depth of the lake reached, and what is that value?

d) What does the fact that the value of $y$ is $0$ at $x=0$ and $x=10$ represent?

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

In an art design class, students must create shapes using symmetry. A triangle is drawn with vertices $A(1, 2)$, $B(4, 2)$ and $C(2, 5)$.

a) Plot the original triangle.

b) Draw the reflection of the triangle across the $Y$-axis. Write down the coordinates of the new vertices $A'$, $B'$ and $C'$.

c) Draw the reflection of the original triangle about the origin $O(0,0)$. Write down the coordinates of these new vertices $A''$, $B''$ and $C''$.

d) How have the signs of the coordinates changed in each case?

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

Two ships, the Alfa and the Beta, are travelling across the ocean. Their positions on the Cartesian plane (where each unit on the vertical axis represents $1$ nautical mile) are given by the following functions with respect to time $t$ (in hours):

• Ship Alfa: $y = 2t - 4$

• Ship Beta: $y = -t + 5$

Use the axes in the figure (copy them into your notebook) to solve the parts.

a) Plot both paths on the same coordinate axes for values of $t$ between $0$ and $6$.

b) At what point on the map will the two ships cross? Find the coordinates $(t, y)$ of that point visually from the graph.

c) At what time ($t$) will that meeting take place?

Analyse the following two graphs, $G_1$ and $G_2$, drawn on the same set of axes, and answer with reasoning. The graph $G_1$ is the line $y=0{,}5x+4$ and the graph $G_2$ is a circle centred at the origin with radius $3$.

a) Which of them represent a function? Justify your answer using the concept of "uniqueness of image".

b) For those that are functions, state whether they are increasing or decreasing over their whole domain.

c) Look at the points of $G_2$ with $x=0$: which values of $y$ correspond to them? What do you conclude about whether $G_2$ satisfies uniqueness of image? Also give another value of $x$, different from $0$, for which the same happens.