Representation on the plane
On a map app on your phone, in a spreadsheet or on a game of Battleships, every square is located with two pieces of information: column and row. Picture a Luas stop on a street map of Dublin, given as "3 across, 2 up". The co-ordinate plane does the same with two numbers, and it is where all functions are drawn. Later we will use it to find the route of a hiker in the Wicklow Mountains.
Axes and coordinates
Drag the point or tap anywhere on the plane:
In the map app on your mobile, your local park is shown on a grid using a Cartesian coordinate system. Look at the position of the different elements and write down the…
In the map app on your mobile, your local park is shown on a grid using a Cartesian coordinate system. Look at the position of the different elements and write down the coordinates of the points where they are located:
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The fountain ($A$)
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The bench ($B$)
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The bin ($C$)
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The lamppost ($D$)
Draw a Cartesian coordinate system in your notebook and plot the following points, which form the vertices of a safe landing zone: $A(2, 3)$ $B(-2, 3)$ $C(-2, -3)$…
Draw a Cartesian coordinate system in your notebook and plot the following points, which form the vertices of a safe landing zone:
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$A(2, 3)$
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$B(-2, 3)$
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$C(-2, -3)$
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$D(2, -3)$
What geometric figure do you get when you join the points in order?
You can do it on paper and send a photo with your answer.
Identify the name of the elements indicated on this Cartesian reference system: 1. The horizontal line (... axis) 2. The vertical line (... axis) 3. Point $(0,0)$ where…
Identify the name of the elements indicated on this Cartesian reference system:
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The horizontal line (... axis)
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The vertical line (... axis)
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Point $(0,0)$ where they intersect.
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The four regions into which the plane is divided.
In a strategy video game, the main base is located at the origin of coordinates $O(0, 0)$. The radar detects three enemy targets at the following positions: target $A$…
In a strategy video game, the main base is located at the origin of coordinates $O(0, 0)$. The radar detects three enemy targets at the following positions: target $A$ at $(-4, 3)$, target $B$ at $(2, 5)$ and target $C$ at $(3, -2)$.
Represent the points on the Cartesian plane and state in which quadrant each of the enemy targets is located.
You can do it on paper and send a photo with your answer.
Quadrants and points on the axes
The axes divide the plane into four quadrants, numbered anticlockwise (1st to 4th quadrant). The signs of the coordinates tell you which quadrant a point is in, without drawing it:
Without plotting them, state which quadrant the following points lie in, which represent the position of several players on a video game server: Player 1: $P_1(5, -2)$…
Without plotting them, state which quadrant the following points lie in, which represent the position of several players on a video game server:
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Player 1: $P_1(5, -2)$
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Player 2: $P_2(-3, -8)$
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Player 3: $P_3(-1, 4)$
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Player 4: $P_4(2, 6)$
State the coordinates of the following special points located on the coordinate axes and explain why their position is characteristic:
State the coordinates of the following special points located on the coordinate axes and explain why their position is characteristic:
Without plotting the points, state which quadrant or axis the following points lie on based on the signs of their coordinates. Justify your answer by explaining whether…
Without plotting the points, state which quadrant or axis the following points lie on based on the signs of their coordinates. Justify your answer by explaining whether the x-coordinate (abscissa) or y-coordinate (ordinate) is positive, negative or zero:
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$A(0, -5)$
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$B(-2, -8)$
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$C(4, 0)$
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$D(-1, 7)$
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$E(3, -3)$
Calculating with co-ordinates
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Horizontal distance between two points with the same $y$: subtract the $x$-values (in absolute value). $A(3, 2)$ and $B(-1, 2)$: $|3 - (-1)| = 4$. The line through them is horizontal ($y = 2$).
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Vertical distance with the same $x$: subtract the $y$-values. The line through the points is vertical, e.g. $x = 3$.
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Any distance: Pythagoras with the two differences: $|AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
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Midpoint of a line segment $[AB]$: the mean of the co-ordinates, $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
Back to the start: the hiker's position, like any Luas stop on the map, is just an ordered pair, and distances come from subtracting its co-ordinates.
Three vertices of a rectangle are $A(-3, 1)$, $B(4, 1)$ and $C(4, -2)$. a) Plot the three points on a Cartesian plane. b) Find the coordinates of the fourth vertex $D$…
Three vertices of a rectangle are $A(-3, 1)$, $B(4, 1)$ and $C(4, -2)$.
- a) Plot the three points on a Cartesian plane.
- b) Find the coordinates of the fourth vertex $D$ to complete the rectangle.
- c) In which quadrant is the new point $D$ located?
Calculate the area of a triangle whose vertices are the points $A(-2, 1)$, $B(4, 1)$ and $C(1, 6)$. To solve it: 1. Plot the points on a Cartesian plane (it will help to…
Calculate the area of a triangle whose vertices are the points $A(-2, 1)$, $B(4, 1)$ and $C(1, 6)$.
To solve it:
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Plot the points on a Cartesian plane (it will help to use graph paper) and draw triangle $ABC$.
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Identify which side is the base of the triangle and how long it is (units on the $X$ axis).
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Identify the height corresponding to that base and how long it is (units on the $Y$ axis).
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Apply the formula for the area of a triangle: $Área = \frac{base \times altura}{2}$.
Write the result in square units.
Three of the vertices of a rectangle intended for a skating area are located at the points $P(-3, -2)$, $Q(4, -2)$ and $R(4, 3)$. The fourth vertex is called $S$. a)…
Three of the vertices of a rectangle intended for a skating area are located at the points $P(-3, -2)$, $Q(4, -2)$ and $R(4, 3)$. The fourth vertex is called $S$.
- a) Plot the points on the plane and draw the rectangle.
- b) Determine the coordinates of the fourth vertex $S$ so that the figure is a rectangle.
- c) Calculate the perimeter of the skating area knowing that each unit on the plane is equivalent to $2$ meters.
You can do it on paper and send a photo with your answer.
Reflections and translations of points
A point can be moved or flipped, and its co-ordinates change in a simple way:
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Reflection in the $x$-axis: $(x, y) \to (x, -y)$.
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Reflection in the $y$-axis: $(x, y) \to (-x, y)$.
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Reflection in the origin (central symmetry): $(x, y) \to (-x, -y)$.
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Reflection in the line $y = x$: $(x, y) \to (y, x)$.
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Reflection in any point $C$ (central symmetry about $C$): $C$ is the midpoint of $P$ and its image $P'$, so $P' = (2x_C - x_P,\ 2y_C - y_P)$.
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Translation by $a$ right and $b$ up: $(x, y) \to (x + a, y + b)$.
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Stretching: multiplying every $y$-co-ordinate by $k$ gives $(x, y) \to (x, ky)$. The figure is stretched vertically if $k > 1$ (or squashed if $0 < k < 1$), and its area is multiplied by $k$. Multiplying every $x$-co-ordinate by $k$ stretches it horizontally in the same way.
The logo of a new clothing brand is symmetrical with respect to the y-axis ($Eje\ Y$). The right side of the logo is defined by the points $A(0, 5)$, $B(2, 4)$ and…
The logo of a new clothing brand is symmetrical with respect to the y-axis ($Eje\ Y$). The right side of the logo is defined by the points $A(0, 5)$, $B(2, 4)$ and $C(3, 0)$.
- a) Find the coordinates of the symmetrical points $B'$ and $C'$ with respect to the $Y$ axis.
- b) Plot the five points on the plane and join them in order ($C'-B'-A-B-C$) to visualise the shape of the logo.
In the design of a circular logo, the centre of the circle is at point $C(1, 2)$. It is known that point $P(4, 6)$ lies on the edge of the logo. a) Plot the centre and…
In the design of a circular logo, the centre of the circle is at point $C(1, 2)$. It is known that point $P(4, 6)$ lies on the edge of the logo.
- a) Plot the centre and point $P$ on the plane.
- b) Find another point $Q$ that is symmetrical to point $P$ with respect to the centre $C$. What does line segment $PQ$ represent in the circle?
- c) If the logo is translated 4 units to the left and 2 units down, what will the new coordinates of the centre $C'$ and of point $P'$ be?
Three vertices of a parallelogram are the points $A(-4, 1)$, $B(2, 1)$ and $C(4, 5)$. Use your knowledge of plotting on the plane to carry out the following tasks: a)…
Three vertices of a parallelogram are the points $A(-4, 1)$, $B(2, 1)$ and $C(4, 5)$. Use your knowledge of plotting on the plane to carry out the following tasks:
- a) Plot the points on the Cartesian coordinate system and determine the coordinates of the fourth vertex $D$ so that the points, taken in alphabetical order $A$, $B$, $C$, $D$, form a parallelogram.
- b) Calculate the area of the parallelogram taking the grid square as the unit.
- c) If we reflect point $A$ through the origin of coordinates $(0,0)$, in which quadrant would the new point $A'$ be located and what would its coordinates be?
You can do it on paper and send a photo with your answer.
Plotting tables and lines
A table of values gives pairs $(x, y)$. Plot each pair and see what shape they make. For the rule $y = 2x$:
| $x$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|
| $y$ | $-2$ | $0$ | $2$ | $4$ |
The points lie on a straight line through the origin. A line crosses the $x$-axis where $y = 0$ and the $y$-axis where $x = 0$. To test a rule, substitute the co-ordinates of each point.
Finding a co-ordinate by solving an equation
Sometimes a co-ordinate is not given directly: it is the solution of an equation. Solve it first, then plot the point. The modulus (absolute value) $|a|$ of a number is its distance from $0$, so it is never negative: $|-8| = 8$ and $|8| = 8$.
Given the following table of values relating the input ($x$) to the output ($y$) of a function: | $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | | :--- | :--- | :--- | :--- |…
Given the following table of values relating the input ($x$) to the output ($y$) of a function:
| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|---|
| $y$ | $4$ | $1$ | $0$ | $1$ | $4$ |
- a) Plot the resulting points on a Cartesian coordinate system.
- b) What geometric shape do the points appear to form if we were to join them with a smooth curved line?
- c) State the coordinates of the point where the function reaches its minimum value.
You can do it on paper and send a photo with your answer.
Look at the following figure, which shows the line segment connecting the points $A(-4, 2)$ and $B(2, -1)$. 1. State the coordinates of the point where the segment…
Look at the following figure, which shows the line segment connecting the points $A(-4, 2)$ and $B(2, -1)$.
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State the coordinates of the point where the segment crosses the x-axis ($X$ axis).
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State the coordinates of the point where the segment crosses the y-axis ($Y$ axis). What is special about both points of intersection?
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Which of these expressions relates the coordinates of the points on the segment: $y = 2x$, $y = -\frac{x}{2}$ or $y = x - 2$? Verify your choice by substituting the coordinates of $A$ and $B$.
We are looking for a secret point $S(x, y)$ that satisfies the following logical conditions: 1. Its x-coordinate (abscissa) is the integer that solves the equation…
We are looking for a secret point $S(x, y)$ that satisfies the following logical conditions:
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Its x-coordinate (abscissa) is the integer that solves the equation $3(x - 2) = x + 4$.
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Its y-coordinate (ordinate) is the modulus of the solution to $\frac{y}{2} - 5 = -8$.
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The point belongs to a direct proportion function that passes through the point $(1, -2)$.
- a) Find the coordinates of point $S$ by solving conditions 1 and 2.
- b) Check whether the point $S$ found satisfies condition 3. Justify your answer by plotting the direct proportion function.
- c) State in which quadrant point $S$ is located.