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The four quadrants of Cartesian planes

A Cartesian plane (or coordinate plane) is divided into four quadrants that are distinguished by their relation to the intersection of the x-axis and the y-axis. This intersection is called the origin and has the coordinates $(0, 0) .$

The scale of the x-axis is positive to the right of zero and is negative to the left of zero.

The scale of the y-axis is positive above zero and is negative below zero.

Explore the following carousel to access a description of each quadrant and its attributes.

Identifying and plotting coordinates

Let's practice identifying and plotting coordinates on a Cartesian plane.

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

Activity

Press ‘Suggested Answers’ to reveal the solution.

Suggested Answers

Solution to question one
The coordinates of points A, B, C, D, and E are:
A(2,7)	B(−4,6)	C(7,−5)	D(−3,−3)	E(0,2)

Solution to question two
Description A Cartesian plane with three points plotted. The scale of the x-axis starts at negative five and goes up by ones to five. The scale of the y-axis starts at negative five and goes up by ones to five. Point J is four units left and three units up from the origin. Point K is three units left and two units down from the origin. Point L is four units right and three units down from the origin.

Solution to question two

Description

A Cartesian plane with three points plotted. The scale of the x-axis starts at negative five and goes up by ones to five. The scale of the y-axis starts at negative five and goes up by ones to five. Point J is four units left and three units up from the origin. Point K is three units left and two units down from the origin. Point L is four units right and three units down from the origin.

Action

Identifying and describing transformations

Transformations are changes that are made to geometric shapes or points.

Description

A Cartesian plane with two triangles plotted, representing translation. The scale of the x-axis starts at negative four and goes up by two to ten. The scale of the y-axis starts at zero and goes up by two to ten. Triangle A B C has the coordinates A (0,6), B (1,1), and C (-2,3). Triangle A-prime B-prime C-prime has the coordinates A-prime (8,10), B-prime (9,5), and C-prime (6,7).

We can differentiate the original shape from the transformed shape by using prime markings (′) on the coordinates of the transformed shape. This notation resembles an apostrophe, but in mathematics we call them prime markings.

Whenever a shape is changed in geometry, we describe the transformation that has taken place. All transformations can be defined using a mapping rule that is applied to every point of a shape.

Translations on a Cartesian plane

Translations (or slides) move every point of a shape the same distance in the same direction, forming a congruent shape.

The size and orientation of the shape remains the same, but the shape's position changes.

The original shape transforms into a translated shape.

When we describe a translation, we state the direction of translation (horizontal and/or vertical), and the number of units to translate in each direction.

You can think of translations like sliding furniture from one side of a room to another. The size and orientation of the furniture remains the same, but the furniture's position changes.

Mapping rule for translations

Translations can be described as adding $a$ to each x-coordinate and adding $b$ to each y-coordinate, where $a$ represents the number of units to translate horizontally and $b$ represents the number of units to translate vertically.

The mapping rule for translations can be written as:

$(x, y) \to (x + a, y + b)$

To perform a translation, we apply this mapping rule to all the points of a shape.

Explore the following example of a translation.

Description

A Cartesian plane with two triangles plotted, representing translation. The scale of the x-axis starts at negative six and goes up by two to eight. The scale of the y-axis starts at negative six and goes up by two to four. Triangle A B C has the coordinates A (3,-1), B (7,-3), and C (1,-6). Triangle A-prime B-prime C-prime has the coordinates A-prime (-3,-1), B-prime (1,-1), and C-prime (-5,-4).

Let's determine the mapping rule for the translation from our example:

Triangle A B C has been translated (or slid) six units left and two units up, forming triangle A′B′C′.
The horizontal translation is six units left:	$a = - 6$
The vertical translation is two units up:	$b = + 2$
Use the mapping rule for translations:	$(x, y) \to$ $(x + a, y + b)$
Input 𝑎 and 𝑏 from this translation:	$(x, y) \to$ $(x - 6, y + 2)$
Six units are subtracted from each x-coordinate, and 2 units are added to each y-coordinate.
Therefore, the mapping rule for this translation is:	$(x, y) \to$ $(x - 6, y + 2)$

Now, let’s use our mapping rule to calculate the coordinates of point A′.

Six units are subtracted from each x-coordinate, and 2 units are added to each y-coordinate.
Point A coordinates:	$(3, - 1)$
Use the translation mapping rule we determined for the example:	$(x, y) \to$ $(x - 6, y + 2)$
Input the x and y coordinates of point A:	$(3, - 1) \to$ $(3 - 6, - 1 + 2)$
Therefore, the coordinates of point A′ are:	$(- 3, 1)$

Calculate the coordinates of points B′ and C′ by applying the translation mapping rule we determined for the example to points B and C. Record your answers using a method of your choice.

Refer to the example to check your answers. You can also press 'Suggested Answers' to reveal the solutions.

Suggested Answers

Solution for point B′
Point B coordinates:	$(7, - 3)$
Use the translation mapping rule we determined for the example:	$(x, y) \to$ $(x - 6, y + 2)$
Input the x and y coordinates of point B:	$(7, - 3) \to$ $(7 - 6, - 3 + 2)$
Therefore, the coordinates of point B′ are:	$(1, - 1)$

Solution for point C′
Point C coordinates:	$(1, - 6)$
Use the translation mapping rule we determined for the example:	$(x, y) \to$ $(x - 6, y + 2)$
Input the x and y coordinates of point C:	$(1, - 6) \to$ $(1 - 6, - 6 + 2)$
Therefore, the coordinates of point C′ are:	$(- 5, - 4)$

Compare the coordinates of triangle A B C and triangle A′B′C′ then select the correct answer:

Practicing translations

Review the following mapping rule for translations:

$(x, y) \to (x + a, y + b)$

Predicting translations

Examine the following Cartesian plane:

Description

A Cartesian plane with a square plotted. The scale of the x-axis starts at negative six and goes up by two to six. The scale of the y-axis starts at negative six and goes up by two to six. Square A B C D has the coordinates A (-3,-1), B (1,-1), C (1,-5), and D (-3,-5).

Notice that square A B C D is mostly plotted in quadrant three. If you were to translate square A B C D using the mapping rule:

$(x, y) \to (x - 1, y + 5)$

In which quadrant would square A′B′C′D′ be plotted?

Record your prediction using a method of your choice.

Press ‘Suggested Answer’ to reveal the solution.

Suggested Answer

Square A B C D will be translated one unit left and five units up.
One unit is subtracted from each x-coordinate, and five units are added to each y-coordinate.	$(x, y) \to$ $(x - 1, y + 5)$
Therefore, Square A′B′C′D′ will be plotted in quadrant two.

Performing translations

It’s time to perform a translation on a Cartesian plane.

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

You can also record a detailed description of your translation using a method of your choice.

Activity

Press ‘Suggested Answer’ to reveal the solution.

Suggested Answer

Square A B C D has been translated one unit left and five units up, forming triangle A′B′C′D′.

Description

A Cartesian plane with two squares plotted. The scale of the x-axis starts at negative six and goes up by two to six. The scale of the y-axis starts at negative six and goes up by two to six. Square A B C D has the coordinates A (-3,-1), B (1,-1), C (1,-5), and D (-3,-5). Square A-prime B-prime C-prime D-prime has the coordinates A-prime (-4.4), B-prime (0,4), C-prime (0,0), and D-prime (-4,0).

Translation mapping rule:	$(x, y) \to$ $(x - 1, y + 5)$
Translating point A:	$A (- 3, - 1) \to A′ (- 4, 4)$
Translating point B:	$B (1, - 1) \to B′ (0, 4)$
Translating point C:	$C (1, - 5) \to C′ (0, 0)$
Translating point D:	$D (- 3, - 5) \to D′ (- 4, 0)$

My Results

Did your translation look approximately like your prediction?

Record your response using a method of your choice.

Reflections on a Cartesian plane

Reflections (or flips) mirror a shape across a line of reflection, forming a congruent shape.

The size of the shape remains the same, but the shape's orientation and position change. The original shape transforms into a reflected shape.

When we describe a reflection, we state the line of reflection (the x-axis or the y-axis).

You can think of reflections like a flipped image on a body of water. The size of the objects in the image remains the same, but the objects’ orientation and position change.

Mapping rules for reflections

There are different mapping rules for reflections depending on whether the line of reflection is the x-axis or the y-axis.

Press each Reflection tab to access a description and mapping rule for that type of reflection.

Reflections across the x-axis can be described as keeping each x-coordinate the same, and adding a negative to each y-coordinate.

The mapping rule for reflections across the x-axis can be written as:

$(x, y) \to (x, - y)$

Reflections across the y-axis can be described as adding a negative to each x-coordinate, and keeping each y-coordinate the same.

The mapping rule for reflections across the y-axis can be written as:

$(x, y) \to (- x, y)$

To perform a reflection, we apply one of these mapping rules to all the points of a shape.

Explore the following example of a reflection.

Description

A Cartesian plane with two polygons plotted. The scale of the x-axis starts at negative six and goes up by one to six. The scale of the y-axis starts at negative six and goes up by one to six. Polygon A B C D E F has the coordinates A (-5,5), B(-3,5), C(-3,3), D(-1,3), E(-5,1), and F(-1,1). Polygon A-prime B-prime C-prime D-prime E-prime F-prime has the coordinates A-prime(-5,-5), B-prime(-3,-5), C-prime(-3,-3), D-prime(-1,-3), E-prime(-5,-1), and F-prime(-1,-1).

Let's determine the mapping rule for the reflection from our example:

Polygon A B C D E F has been reflected (or flipped) from quadrant two to quadrant three, forming polygon A′B′C′D′E′F′.
Quadrant two is above the x-axis and left of the y-axis. Quadrant three is below the x-axis and left of the y-axis.	Polygon A B C D E F has been reflected across the x-axis.
Keep each x-coordinate the same, and add a negative to each y-coordinate.
Therefore, the mapping rule for this reflection is:	$(x, y) \to$ $(x, - y)$

Now, let's use our mapping rule to calculate the coordinates of point A′

Point A coordinates:	$(- 5, 5)$
Use the reflection mapping rule we determined for the example:	$(x, y) \to$ $(x, - y)$
Input the x and y coordinates of point A:	$(- 5, 5) \to$ $(- 5, - 5)$
Therefore, the coordinates of point A′ are:	$(- 5, - 5)$

Calculate the coordinates of points B′, C′, D′, E′, and F′ by applying the reflection mapping rule we determined for the example to points B, C, D, E, and F.

Record your answers using a method of your choice.

Press ‘Suggested Answers’ to reveal the solution.

Suggested Answers

Solution for point B′
Point B′ coordinates:	$(- 3, 5)$
Use the reflection mapping rule we determined for the example:	$(x, y) \to$ $(x, - y)$
Input the x and y coordinates of point B:	$(- 3, 5) \to$ $(- 3, - 5)$
Therefore, the coordinates of point B′ are:	$(- 3, - 5)$

Solution for point C′
Point C coordinates:	$(- 3, 3)$
Use the reflection mapping rule we determined for the example:	$(x, y) \to$ $(x, - y)$
Input the x and y coordinates of point C:	$(- 3, 3) \to$ $(- 3, - 3)$
Therefore, the coordinates of point C′ are:	$(- 3, - 3)$

Solution for point D′
Point D coordinates:	$(- 1, 3)$
Use the reflection mapping rule we determined for the example:	$(x, y) \to$ $(x, - y)$
Input the x and y coordinates of point D:	$(- 1, 3) \to$ $(- 1, - 3)$
Therefore, the coordinates of point D′ are:	$(- 1, - 3)$

Solution for point E′
Point E coordinates:	$(- 5, 1)$
Use the reflection mapping rule we determined for the example:	$(x, y) \to$ $(x, - y)$
Input the x and y coordinates of point E:	$(- 5, 1) \to$ $(- 5, - 1)$
Therefore, the coordinates of point E′ are:	$(- 5, - 1)$

Solution for point F′
Point F coordinates:	$(- 1, 1)$
Use the reflection mapping rule we determined for the example:	$(x, y) \to$ $(x, - y)$
Input the x and y coordinates of point F:	$(- 1, 1) \to$ $(- 1, - 1)$
Therefore, the coordinates of point F′ are:	$(- 1, - 1)$

Compare the coordinates of polygon A B C D E F and polygon A′B′C′D′E′F′ then select the correct answer.

Practicing reflections

Review the following mapping rules for reflections:

Reflections across the x-axis:	$(x, y) \to (x, - y)$
Reflections across the y-axis:	$(x, y) \to (- x, y)$

Predicting reflections

Examine the following Cartesian plane:

Description

A Cartesian plane with a triangle plotted. The scale of the x-axis starts at negative six and goes up by one to six. The scale of the y-axis starts at negative six and goes up by one to six. Triangle Q R S has the coordinates Q (1,-2), R (3, -2), and S (1,-6).

Notice that triangle Q R S is plotted in quadrant four. If you were to reflect triangle Q R S across the x-axis, in which quadrant would triangle Q′R′S′ be plotted?

Record your prediction using a method of your choice.

Press ‘Suggested Answer’ to reveal the solution.

Suggested Answer

Triangle Q R S will be reflected vertically across the x-axis
Each x-coordinate stays the same, and a negative is added to each y-coordinate.	$(x, y) \to (x, - y)$
Therefore, triangle Q′R′S′ will be plotted in quadrant one.

Performing reflections

It's time to perform a reflection on a Cartesian plane.

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

Activity

Press ‘Suggested Answer’ to access the solution.

Suggested Answer

Triangle Q R S has been reflected across the x-axis, forming triangle Q′R′S′.

Description

A Cartesian plane with two triangles plotted. The scale of the x-axis starts at negative six and goes up by one to six. The scale of the y-axis starts at negative six and goes up by one to six. Triangle Q R S has the coordinates Q (1,-2), R (3,-2), and S (1,-6). Triangle Q-prime R-prime S-prime has the coordinates Q-prime (1,2), R-prime (3,2), and S-prime (1,6).

Reflection across the x-axis mapping rule:	$(x, y) \to (x, - y)$
Reflecting point Q:	$Q (1, - 2) \to Q' (1, 2)$
Reflecting point R:	$R (3, - 2) \to Q' (3, 2)$
Reflecting point S:	$S (1, - 6) \to S' (1, 6)$

My Results

Did your reflection look approximately like your prediction?

Record your response using a method of your choice.

Rotations on a Cartesian plane

Rotations (or turns) move a shape about a point of rotation, forming a congruent shape.

The size of the shape remains the same, but the shape's position and orientation change. The original shape transforms into a rotated shape.

When we describe a rotation, we state the point of rotation
$(x, y),$
the direction of rotation (Counterclockwise/CCW or clockwise/CW), and the degrees of rotation (90°, 180°, or 270°).

You can think of rotations like the turning of a Ferris wheel. The size of the cars remains the same, but the cars' position and orientation change.

Mapping rule for rotations about the origin

There are different mapping rules for rotations depending on whether the rotation is 90°, 180°, or 270°, and whether the direction of the rotation is counterclockwise or clockwise.

A counterclockwise (or positive) rotation will always have an equivalent clockwise (or negative) rotation that transforms the object to the same position.

Press the following tabs to access a description and mapping rule for each type of rotation.

90° counterclockwise rotations about the origin can be described as changing each x-coordinate to the y-coordinate and adding a negative and changing each y-coordinate to the x-coordinate.

The mapping rule for 90° counterclockwise rotations about the origin can be written as:

$(x, y) \to (- y, x)$

A 90° counterclockwise rotation is equal to a 270° clockwise rotation.

Description

A Cartesian plane with two points plotted. The scale of the x-axis starts at negative five and goes up by fives to five. The scale of the y-axis starts at negative five and goes up by fives to five. Point A has the coordinates (4,3). Point A-prime has the coordinates (-3,4). Point A has been rotated 90 degrees counter clockwise (or 270 degrees clockwise), forming point A-prime.

180° rotations about the origin can be described as adding a negative to each x-coordinate and to each y-coordinate.

The mapping rule for 180° rotations can be written as:

$(x, y) \to (- x, - y)$

A 180° counterclockwise rotation is equal to a 180° clockwise rotation.

Description

The scale of the x-axis starts at negative ten and goes up by fives to ten. The scale of the y-axis starts at negative ten and goes up by fives to ten. Point B has the coordinates (7,4). Point B-prime has the coordinates (-7,-4). Point B has been rotated 180 degrees counterclockwise (or 180 degrees clockwise), forming point B-prime.

270° counterclockwise rotations about the origin can be described as changing each x-coordinate to the y-coordinate and changing each y-coordinate to the x-coordinate and adding a negative.

The mapping rule for 270° counterclockwise rotations about the origin can be written as:

$(x, y) \to (y, - x)$

Description

A Cartesian plane with two points plotted. The scale of the x-axis starts at negative five and goes up by fives to five. The scale of the y-axis starts at negative five and goes up by fives to five. Point A has the coordinates (4,3). Point A-prime has the coordinates (3,-4). Point A has been rotated 270 degrees counterclockwise (or 90 degrees clockwise), forming point A-prime.

To perform a rotation about the origin, we apply one of these mapping rules to all the points of a shape.

Explore the following example of a rotation.

Description

A Cartesian plane with two polygons plotted. The scale of the x-axis starts at negative five and goes up by ones to six. The scale of the y-axis starts at negative five and goes up by ones to six. Polygon A B C D E has the coordinates A (-3,5), B (-4,4), C (-3,3), D (-1,2), and E (-1,4). Polygon A-prime B-prime C-prime D-prime E-prime has the coordinates A-prime (5,3), B-prime (4,4), C-prime (3,3), D-prime (2,1), and E-prime (4,1).

Let's determine the mapping rule for the rotation from our example.

Polygon A B C D E has been rotated (or turned) from quadrant two to quadrant one, forming polygon A′B′C′D′E′.
Quadrant two is above the x-axis and left of the y-axis. Quadrant one is above the x-axis and right of the y-axis.	Polygon A B C D E has been rotated 270° counterclockwise (or 90° clockwise) about the origin (0, 0).
Change each x-coordinate to the y-coordinate, and change each y-coordinate to the x-coordinate and add a negative.
Therefore, the mapping rule for this reflection is:	$(x, y) \to (y, - x)$

Now, let's use our mapping rule to calculate the coordinates of point A′

Point A coordinates:	$(- 3, 5)$
Use the rotation mapping rule we determined for the example:	$(x, y) \to$ $(y, - x)$
Input the x and y coordinates of point A:	$(- 3, 5) \to$ $(5, - (- 3))$
Therefore, the coordinates of point A′ are:	$(5, 3)$

Calculate the coordinates of point B′, C′, D′, and E′ by applying the rotation mapping rule we determined for the example to points B, C, D, and E. Record your answers using a method of your choice.

Refer to the example to check your answers. Press ‘Suggested Answers’ to access the solutions.

Suggested Answers

Solution for point B′
Point B coordinates:	$(- 4, 4)$
Use the rotation mapping rule we determined for the example:	$(x, y) \to$ $(y, - x)$
Input the x and y coordinates of point B:	$(- 4, 4) \to$ $(4, - (- 4))$
Therefore, the coordinates of point B′ are:	$(4, 4)$

Solution for point C′
Point C coordinates:	$(- 3, 3)$
Use the rotation mapping rule we determined for the example:	$(x, y) \to$ $(y, - x)$
Input the x and y coordinates of point C:	$(- 3, 3) \to$ $(3, - (- 3))$
Therefore, the coordinates of point C′ are:	$(3, 3)$

Solution for point D′
Point D coordinates:	$(- 1, 2)$
Use the rotation mapping rule we determined for the example:	$(x, y) \to$ $(y, - x)$
Input the x and y coordinates of point D:	$(- 1, 2) \to$ $(2, - (- 1))$
Therefore, the coordinates of point D′ are:	$(2, 1)$

Solution for point E′
Point E coordinates:	$(- 1, 4)$
Use the rotation mapping rule we determined for the example:	$(x, y) \to$ $(y, - x)$
Input the x and y coordinates of point E:	$(- 1, 4) \to$ $(4, - (- 1))$
Therefore, the coordinates of point E′ are:	$(4, 1)$

Compare the coordinates of polygon A B C D E and polygon, A′B′C′D′E′ then select the correct answer:

Practicing rotations

Review the following mapping rules for rotations:

90° CCW (or 270° CW) rotations about the origin:	$(x, y) \to$ $(- y, x)$
180° rotations about the origin:	$(x, y) \to$ $(- x, - y)$
270° CCW (or 90° CW) rotations about the origin:	$(x, y) \to$ $(y, - x)$

Predicting rotations

Examine the following Cartesian plane:

Description

A Cartesian plane with a triangle plotted. The scale of the x-axis starts at negative four and goes up by ones to four. The scale of the y-axis starts at negative four and goes up by ones to four. Triangle A B C has the coordinates A (3,-3), B (1,-3), and C (3,0).

Notice that triangle A B C is plotted in quadrant four. If you were to rotate triangle A B C 90° CCW about the origin, in which quadrant would triangle A′B′C′ be plotted?

Record your prediction using a method of your choice.

Press ‘Suggested Answer’ to access the solution.

Suggested Answer

Triangle A B C will be rotated 90° counterclockwise (or 270° clockwise) about the origin (0, 0).
Change each x-coordinate to the y-coordinate and add a negative, and change each y-coordinate to the x-coordinate.	$(x, y) \to$ $(- y, x)$
Therefore, triangle A′B′C′ will be plotted in quadrant one.

Performing rotations

It's time to perform a rotation on a Cartesian plane.

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

You can also record a detailed description of your rotation using a method of your choice.

Activity

Press ‘Suggested Answer’ to reveal the solution.

Suggested Answer

Triangle A B C has been rotated 90° counterclockwise (or 270° clockwise) about the origin, forming triangle A′B′C′.

Description

A Cartesian plane with two triangles plotted. The scale of the x-axis starts at negative four and goes up by twos to four. The scale of the y-axis starts at negative four and goes up by twos to four. Triangle A B C has the coordinates A (3,-3), B (1,-3), and C (3,0). Triangle A-prime B-prime C-prime has the coordinates A-prime (3,3), B-prime (3,1), and C-prime (0,3).

Rotation 90° counterclockwise (or 270° clockwise) about the origin mapping rule:	$(x, y) \to$ $(- y, x)$
Rotating point A	$A (3, - 3) \to A' (3, 3)$
Rotating point B	$B (1, - 3) \to B' (3, 1)$
Rotating point C	$C (3, 0) \to C' (0, 3)$

My Results

Did your rotation look approximately like your prediction?

Record your response using a method of your choice.

Dilations on a Cartesian plane

Dilations (or enlargements and reductions) make a shape larger or smaller by a scale factor, forming a similar shape.

The orientation of the shape remains the same, but the shape's size and position change. The original shape transforms into a dilated shape.

When we describe a dilation, we state the scale factor.

You can think of dilations like enlarging your perspective of an object with a magnifying glass. The orientation of the object remains the same, but the object's size and position change.

Mapping rules for dilations

Dilations can be described as multiplying each x-coordinate and each y-coordinate by $a$ , where $a$ represents the scale factor. A scale factor greater than one enlarges a shape, while a scale factor less than one reduces a shape.

The mapping rule for translations can be written as:

$(x, y) \to (a x, a y)$

To perform a dilation, we apply this mapping rule to all the points of a shape.

Examine the following example of a dilation.

Description

Two triangles are plotted on a Cartesian plane. The scale of the x-axis starts at zero and goes up by twos to ten. The scale of the y-axis starts at zero and goes up by twos to ten. Triangle P Q R has the coordinates P (1,3), Q (3,1) and R (1,1). Triangle P-prime Q-prime R-prime has the coordinates P-prime (3,9), Q-prime (9,3), and R-prime (3,3).

Let’s determine the mapping rule for the translation from our example:

Triangle P Q R has been dilated (or enlarged) by a scale factor of three to form triangle P′Q′R′.
The scale factor is three:	$a = 3$
Use the mapping rule for dilations:	$(x, y) \to$ $(a x, a y)$
Input a from this translation:	$(x, y) \to$ $(3 x, 3 y)$
Each x-coordinate and each y-coordinate are multiplied by 3.
Therefore, the mapping rule for this translation is:	$(x, y) \to$ $(3 x, 3 y)$

Now, let’s use our mapping rule to calculate the coordinates of point P′:

Point P coordinates:	$a = 3$
Use the dilation mapping rule we determined for the example:	$(x, y) \to$ $(a x, a y)$
Input the x and y coordinates of point P:	$(x, y) \to$ $(3 x, 3 y)$
Therefore, the coordinates of point P′ are:	$(x, y) \to$ $(3 x, 3 y)$

Calculate the coordinates of point Q′ and R′ by applying the dilation mapping rule we determined for the example to points Q and R. Record your answers using a method of your choice.

Refer to the example to check your answers. Press ‘Suggested Answers’ to reveal the solution.

Suggested Answers

Solution for point Q′
Point Q coordinates:	$(3, 1)$
Use the dilation mapping rule we determined for the example:	$(x, y) \to$ $(3 x, 3 y)$
Input the x and y coordinates of point Q:	$(3, 1) \to$ $(3 (3), 3 (1))$
Therefore, the coordinates of point Q′ are:	$(9, 3)$

Solution for point R′
Point R coordinates:	$(1, 1)$
Use the dilation mapping rule we determined for the example:	$(x, y) \to$ $(3 x, 3 y)$
Input the x and y coordinates of point R:	$(1, 1) \to$ $(3 (1), 3 (1))$
Therefore, the coordinates of point R′ are:	$(3, 3)$

Compare the coordinates of triangle P Q R and triangle P′Q′R′ then select the correct answer:

Practicing dilations

Review the following mapping rule for dilations:

$(x, y) \to (a x, a y)$

Predicting dilations

Examine the following Cartesian plane:

Description

A Cartesian plane with a triangle plotted. The scale of the x-axis starts at negative eight and goes up by twos to eight. The scale of the y-axis starts at negative eight and goes up by twos to eight. Triangle A B C has the coordinates A (-1,-2), B (3,-3), and C (1,4).

Notice that point A is plotted in quadrant three, point B is plotted in quadrant four, and point C is plotted in quadrant one. If you were to dilate triangle A B C by a scale factor of two, in which quadrants would point A′, point B′, and point C′ be plotted?

Record your prediction using a method of your choice.

Press ‘Suggested Answer’ to access the solution.

Suggested Answer

Triangle A B C will be dilated by a scale factor of two.
Each coordinate of triangle A B C is multiplied by two.	$(x, y) \to$ $(2 x, 2 y)$
Therefore, point A′ will be plotted in quadrant three, point B′ will be plotted in quadrant four, and point C′ will be plotted in quadrant one.

Performing dilations

It's time to perform a dilation on a Cartesian plane.

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

You can also record a detailed description of your dilation using a method of your choice.

Activity

Press ‘Suggested Answer’ to access the solution.

Suggested Answer

Description

A Cartesian plane with two triangles plotted. The scale of the x-axis starts at negative eight and goes up by twos to eight. The scale of the y-axis starts at negative eight and goes up by twos to eight. Triangle A B C has the coordinates A (-1,-2), B (3,-3), and C (1,4). Triangle A-prime B-prime C-prime has the coordinates A-prime (-2,-4), B-prime (6,-6), and C-prime (2,8).

Dilation by a scale factor of two mapping rule:	$(x, y) \to (2 x, 2 y)$
Dilating point A:	$A (- 1, - 2) \to$ $A' (- 2, - 4)$
Dilating point B:	$B (3, - 3) \to$ $B' (6, - 6)$
Dilating point C:	$C (1, 4) \to$ $C' (2, 8)$

My Results

Did your dilation look approximately like your prediction?

Record your response using a method of your choice.

Determining scale factors using coordinates

Determine the scale factor that polygon A B C D E has been dilated by to form polygon A′B′C′D′E′ using the following coordinates.

Record your answer using a method of your choice.

Polygon A B C D E
A(2,8)	B(2,14)	C(6,4)	D(14,8)	E(6,6)

Polygon A′B′C′D′E′
A′(1,4)	B′(1,7)	C′(3,7)	D′(7,4)	E′(3,3)

Press ‘Suggested Answer’ to access the solution.

Suggested Answer

Polygon A B C D E has been dilated by a scale factor of $\frac{1}{2}$ (or 0.5), forming polygon A′B′C′D′E′.

Notice that the original polygon is larger, and the dilated polygon is smaller.

Description

A Cartesian plane with two polygons plotted. The scale of the x-axis starts at negative eight and goes up by twos to eight. The scale of the y-axis starts at negative eight and goes up by twos to eight. Polygon A B C D E has the coordinates A (2,8), B (2,14), C (6,14), D (14,8), E (6,6). Polygon A-prime B-prime C-prime D-prime E-prime has the coordinates A-prime (1,4), B-prime (1,7), C-prime (3,7), D-prime (7,4), E-prime (3,3).

Consolidation

Summarizing transformations

Summarize the four types of transformations that we have explored in this learning activity.

Complete Summary of Transformations in your notebook or using the following fillable and printable document. You can also record your summaries using a method of your choice.

Summary of Transformations
Record a description and/or representation of a translation, a reflection, a rotation, and a dilation.
Translation	Reflection

Rotation	Dilation

Press the ‘Activity’ button to access the Summary of Transformations.

Activity

Create, predict, and perform transformations

Create a four-sided polygon on a Cartesian plane, then predict where each type of transformation (translation, reflection, rotation, and dilation) would move your shape.

After recording your predictions, perform each type of transformation on your shape, moving it around the Cartesian Plane as necessary.

Use a prime marking (′) to indicate your first transformation, a double prime marking (″) to indicate your second transformation, a triple prime marking (‴) to indicate your third transformation, and a quadruple prime marking (⁗) to indicate your fourth transformation.

When you are finished performing and labelling your transformations, determine the mapping rules and record a description of each transformation using mathematical language.

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

Activity

Use the following checklist to make sure that you have completed every step of this activity:

Checklist

Have I:

created a four-sided polygon and plotted it on a Cartesian plane predicted where a translation, reflection, rotation, and dilation will move my shape performed the translation, reflection, rotation, and dilation that I predicted labelled my polygon and four transformations, including their coordinates determined the mapping rules and described each transformation

What are the similarities and differences between your predictions and the actual transformations?

How would you explain performing each type of transformation on a four-sided polygon to someone? What is most important information for them to understand?

Record your responses using a method of your choice.

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.

I feel...

I have an excellent understanding of the concepts I have a very good understanding of the concepts I have some understanding but need to review some more my understanding of the concepts needs a lot more work

Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.

When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.

Connect with a TVO Mathify tutor

Think of TVO Mathify as your own personalized math coach, here to support your learning at home. Press ‘TVO Mathify’ to connect with an Ontario Certified Teacher math tutor of your choice. You will need a TVO Mathify login to access this resource.

Learning goals

Success criteria

Minds On

The four quadrants of Cartesian planes

Identifying and plotting coordinates

Action

Identifying and describing transformations

Translations on a Cartesian plane

Mapping rule for translations

Practicing translations

Predicting translations

Performing translations

My Results

Reflections on a Cartesian plane

Mapping rules for reflections

Practicing reflections

Predicting reflections

Performing reflections

My Results

Rotations on a Cartesian plane

Mapping rule for rotations about the origin

Practicing rotations

Predicting rotations

Performing rotations

My Results

Dilations on a Cartesian plane

Mapping rules for dilations

Practicing dilations

Predicting dilations

Performing dilations

My Results

Determining scale factors using coordinates

Consolidation

Summarizing transformations

Create, predict, and perform transformations

Checklist

Have I:

Reflection

I feel...

Connect with a TVO Mathify tutor