7. The Coordinate Plane
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Chapter 1
7.
The Coordinate Plane
If one is an aspiring athlete like Dominika, the coordinate plane can be used to improve performance in sports. The coordinate plane is a two-dimensional grid created by the intersection of a vertical and a horizontal number line. It is divided into four areas known as quadrants. Points on this plane are represented by ordered pairs, where the first number indicates the distance along the horizontal axis and the second number indicates the distance along the vertical axis. These ordered pairs can mark the locations of homes, friends' homes, and sports facilities. The concept of reflection can also be explored, which involves flipping a point across an axis to produce its mirror image. This concept can help identify advantageous positions on a sports field or court. The coordinate plane is not merely a mathematical abstraction; it is a practical tool that can be applied in various real-life situations such as sports strategy, navigation, and design.
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Lesson Settings & Tools
| | 13 Theory slides |
| | 13 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
The Coordinate Plane
Slide of 13
A number line helps visualize one-dimensional quantities like temperature and time. However, there are situations where two dimensions are needed, such as when latitude and longitude when locating places on a map. The coordinate plane is used in these situations! This lesson will teach how to locate points on a coordinate plane and explore some of its uses.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Explore
The Battleship Game Board
Get ready for an adventure! Discover the location of enemy ships hidden on the game board by using specific points. Each point on the board is represented by the combination of a letter and a number. Place the boats by dragging them from the stern, or the back, and rotate them by clicking on the bow, or the front.
External credits: Freepik
- How does using the letter and number combination to locate enemy ships in the game relate to finding a location on a map in real life?
- What are other situations where two numbers, two letters, or a combination of both help to indicate a location?
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Discussion
Coordinate System
A coordinate system is a reference framework used to describe the positions of objects like points, lines, and surfaces in a space. A fixed point called the origin is used as a reference in the coordinate system. The most common types of coordinate systems are one-, two-, or three-dimensional.
Coordinate Plane
A coordinate plane, also known as a Cartesian or a rectangular coordinate system, is a two-dimensional coordinate system. It is a grid that results from intersecting a vertical number line with a horizontal number line at their zero points. The horizontal number line is usually named the x-axis and the vertical number line is usually the y-axis.
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Discussion
Coordinate
A coordinate is the position of an object in a coordinate system relative to the corresponding axis. Coordinates are often seen together with other coordinates, which can describe the position of an object in a coordinate system with one, two, three, or even more dimensions.
In Two Dimensions
In a two-dimensional coordinate system, points are usually expressed as a coordinate pair — also called an ordered pair — denoted by (x,y). The first coordinate states the position along the x-axis and the second coordinate states the position along the y-axis.
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Pop Quiz
Random Points on a Coordinate System
Identify the coordinates of the given point. Write the coordinates as an ordered pair (x,y), where x represents the x-coordinate and y the y-coordinate.
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Example
The Path From Home to the Basketball Court
Dominika plays basketball with her friends every afternoon. She warms up by running and picks up two friends, Jordan and Emily, on the way to the court. Dominika uses ordered pairs to represent the positions of her and her friends' houses.
| Position | |
|---|---|
| Dominika's House | D=(2,1) |
| Jordan's House | J=(2,4) |
| Emily's House | E=(-1,4) |
a Dominika wants to display this information on a coordinate plane. Which graph correctly displays the positions of the three houses?
{"type":"choice","form":{"alts":["I","II","III","IV"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
b The basketball court is at the point C=(-1,1), and Dominika's whole path from her house to the court and back can be represented by joining points D, J, E, C, and D. Which graph shows Dominika's whole path?
{"type":"choice","form":{"alts":["I","II","III","IV"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
Hint
a An ordered pair can be graphed on a coordinate plane by moving horizontally from the origin the number of units specified in its x-coordinate and then vertically the number of units specified in its y-coordinate.
b Plot the coordinates of the basketball court. Connect the points using line segments in the given order.
Solution
a The positions of the girls' houses are given as ordered pairs. Recall that an ordered pair can be graphed on a coordinate plane by moving horizontally from the origin the number of units specified in its x-coordinate and vertically the number of units specified in its y-coordinate.
With this information in mind, consider the position of Dominika's house. Dominika's House D=( 2, 1) The x-coordinate is 2 and the y-coordinate is 1. Move 2 units to the right of the origin and then 1 unit up to graph Dominika's house on the coordinate plane.
Now, consider the coordinates of Jordan's and Emily's houses. Jordan's House & Emily's House J=( 2, 4)&E=( -1, 4) On the coordinate plane, move 2 units horizontally to the right of the origin and then 4 units up to graph the position of Jordan's house. To plot Emily's house, start at the origin again. Move 1 unit to the left and 4 units up to graph Emily's house.
Notice that this corresponds to graph III.
b Begin by considering the coordinates of the position of the basketball court.
Basketball Court C=( -1, 1) Add this point to the graph from the previous part. Move 1 unit horizontally to the left of the origin and then 1 unit up.
The points can now be joined according to the given order D, J, E, F, and D. Draw a line starting from point D to point J. Next, from point J, go to point E, and so on until point C is connected to point D.
This graph matches the one given in option I.
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Example
Dominika's Running Routine
Dominika's house is represented by the point (2,1) on a coordinate plane. Every morning, she runs to a place five blocks away from her house, then returns home using the same route.
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
{"type":"choice","form":{"alts":["Square","Triangle","Star","Circle"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
Hint
Moving one unit on the x-axis or y-axis on the coordinate plane is counted as moving one block. Since Dominika can only stay on the straight roads, her position can only move up, down, left, and right on the graph.
Solution
Moving one unit on the x-axis or y-axis on the coordinate plane is the same as Dominika moving one block. Dominika only runs on the straight roads, so her progress on the graph can only move in four directions — up, down, left, and right.
Keeping this in mind, one route she can take is moving five blocks directly to the right. Move 5 units to the right starting her house to show this possible destination. Remember that Dominika's house is located at (2,1), not the origin!
Another spot Dominika might go is one block up and four blocks to the right of her starting point. It does not matter if she moves up on her first, second, third, fourth, or final move. Her destination will always be the same if she moves one block up out of the five total moves.
Follow the same reasoning to find the other places that Dominika can go. Plot each as ordered pairs on the coordinate plane.
Now that all the points are plotted on the graph, connect them with lines to see the overall shape.
The shape is a rotated square.
Showing Our Work
Finding All the Places Dominika Can GoThe places Dominika can go can be determined by identifying all the possible routes she can take. Remember, she can only go five blocks away from her home. Moving left or down indicates a negative direction on the corresponding axis, represented by negative numbers.
| Route from Home | Calculations | Destination |
|---|---|---|
| 5 blocks to the right | (2+ 5,1) | (7,1) |
| 4 blocks to the right and 1 block up | (2+ 4,1+ 1) | (6,2) |
| 3 blocks to the right and 2 blocks up | (2+ 3,1+ 2) | (5,3) |
| 2 blocks to the right and 3 blocks up | (2+ 2,1+ 3) | (4,4) |
| 1 block to the right and 4 blocks up | (2+ 1,1+ 4) | (3,5) |
| 5 blocks up | (2,1+ 5) | (2,6) |
| 1 block to the left and 4 blocks up | (2- 1,1+ 4) | (1,5) |
| 2 blocks to the left and 3 blocks up | (2- 2,1+ 3) | (0,4) |
| 3 blocks to the left and 2 blocks up | (2- 3,1+ 2) | (-1,3) |
| 4 blocks to the left and 1 block up | (2- 4,1+ 1) | (-2,2) |
| 5 blocks to the left | (2- 5,1) | (-3,1) |
| 4 blocks to the left and 1 block down | (2- 4,1- 1) | (-2,0) |
| 3 blocks to the left and 2 blocks down | (2- 3,1- 2) | (-1,-1) |
| 2 blocks to the left and 3 blocks down | (2- 2,1- 3) | (0,-2) |
| 1 block to the left and 4 blocks down | (2- 1,1- 4) | (1,-3) |
| 5 blocks down | (2,1- 5) | (2,-4) |
| 1 block to the right and 4 blocks down | (2+ 1,1- 4) | (3,-3) |
| 2 blocks to the right and 3 blocks down | (2+ 2,1- 3) | (4,-2) |
| 3 blocks to the right and 2 blocks down | (2+ 3,1- 2) | (5,-1) |
| 4 blocks to the right and 1 block down | (2+ 4,1- 1) | (6,0) |
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Discussion
Quadrants
In a coordinate plane, the intersection of the x-axis and the y-axis produces four regions called quadrants. The quadrants are numbered counterclockwise from the top right quadrant as Quadrant I to Quadrant IV in the bottom right.
The signs of the coordinates of a point can be determined based on which quadrant the point lies in.
Conversely, the quadrant of a point can be found by looking at the signs of its coordinates.
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Pop Quiz
Finding the Quadrant in Which a Point Lies
A coordinate system has four quadrants that provide information about the signs of the x- and y-coordinates of a point. In the following applet, identify in which quadrant the given point lies.
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Example
Designing the Team's Logo
Dominika is drafting a logo for her basketball team to wear on their jerseys. She wants the logo to have a cool geometric shape, so she sets some points that she thinks will look good. Help her design the logo by matching each point with its corresponding quadrant on the coordinate plane.
{"type":"pair","form":{"alts":[[{"id":0,"text":"Quadrant I"},{"id":1,"text":"Quadrant II"},{"id":2,"text":"Quadrant III"},{"id":3,"text":"Quadrant IV"}],[{"id":1,"text":"(4,3)"},{"id":0,"text":"(-11,3)"},{"id":2,"text":"(-2,-8)"},{"id":3,"text":"(4,-2)"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3],[1,0,2,3]]}
Hint
Look at the signs of the coordinates of each point to identify their quadrants.
Solution
The quadrant in which a point lies can be determined by looking at the signs of its coordinates. For instance, a point in Quadrant I has a positive x-coordinate and a positive y-coordinate. Consider the given points and see if any of them fit this description.
(-2,-8) (4,-2) (-11,3) ( 4, 3)
The point ( 4, 3) is located in Quadrant I because both of its coordinates are positive. In contrast, a point in Quadrant II has a negative x-coordinate and a positive y-coordinate. Look at the remaining points to see which one fits this description. (-2,-8) (4,-2) ( -11, 3) (4,3)
The point ( -11, 3) is in Quadrant II because its x-coordinate is negative but its y-coordinate is positive. Follow a similar reasoning to find the corresponding quadrants of the remaining points.
| Point | Signs of Coordinates | Quadrant |
|---|---|---|
| ( -2, -8) | ( -, -) | Quadrant III |
| ( 4, -2) | ( +, -) | Quadrant IV |
| ( -11, 3) | ( -, +) | Quadrant II |
| ( 4, 3) | ( +, +) | Quadrant I |
Now that the quadrant of each point has been identified, plot the points on a coordinate plane. Move horizontally from the origin the number of units indicated by the x-coordinate and vertically the number of units specified by the y-coordinate of each point.
External credits: Image by Freepik
The logo is a hit, and the team gains more and more fans as a result!
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Discussion
Reflecting a Point Across an Axis on the Coordinate Plane
Reflecting a point means making a mirror image of that point by flipping it across a certain line or axis. The following rules show how to find reflected points across a specific axis.
| Point | Axis of Reflection | Procedure | Reflected Point |
|---|---|---|---|
| (x_1,y_1) | x | Change the y-coordinate to its opposite. | (x_1,- y_1) |
| (x_1,y_1) | y | Change the x-coordinate to its opposite. | (- x_1,y_1) |
| (x_1,y_1) | x and y | Change the x- and y-coordinates to their opposites. | (- x_1,- y_1) |
Next, the process for reflecting points across the x-axis, the y-axis, or both, on a coordinate plane will be shown. Consider the points A, B, and C.
| Point | Axis of Reflection | Reflected Point |
|---|---|---|
| A = (5,8) | x | ? |
| B = (10,-5) | y | ? |
| C = (-4,-7) | x and y | ? |
There are two steps to follow to reflect a point.
1
Find the Reflected Point
expand_more Keep the same x-coordinate of a point and change the y-coordinate to its opposite to reflect it across the x-axis. With this information, point A can be reflected across the x-axis by changing its y-coordinate 8 to its opposite, -8.
A=(5, 8) ⇒ A’=(5, -8)
In contrast, change the x-coordinate of a point to its opposite and keep its y-coordinate to reflect it across the y-axis. Point B can be reflected across the y-axis by changing the x-coordinate 10 to its opposite, -10
B=( 10,-5) ⇒ B’=( -10,-5)
Finally, change the x-coordinate and the y-coordinate of a point to their opposites to reflect it across both the x-axis and y-axis. Therefore, reflect point C across both axes by changing its x-coordinate from -4 to 4 and its y-coordinate from -15 to 15.
C=( -4, -7) ⇒ C’=( 4, 7)
2
Plot the Point and Its Reflection on the Coordinate Plane
expand_more Once each initial point and its reflection across a given axis are found, plot both points on the coordinate plane.
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Example
Improving Basketball Skills With Math
Dominika's team is only one game away from winning the basketball tournament! Dominika is a shooting guard and wants to use her math skills to improve her game. She starts by figuring out the point on the court from where she usually scores using a coordinate plane. This point is (-4,-3).
a Dominika knows that if she mirrors that point across the horizontal central line of the court, she can score from there, too. Find this mirror point.
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b When Dominika is defending, she needs to be quick on her feet to get the rebounds. To do that, she needs to know the point that is the mirror image of (-4,-3) across the vertical central line of the court. What is this point?
{"type":"text","form":{"type":"point2d","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false,"decimal":false,"function":false},"text":" "},"formTextBefore":"(x,y)=","formTextAfter":null,"answer":{"text1":"4","text2":"-3"}}
Hint
a Reflect the point across the x-axis. The x-coordinate remains the same and the y-coordinate is changed to its opposite.
b Reflect the point across the y-axis. The y-coordinate remains the same and the x-coordinate is changed to its opposite.
Solution
a Reflect the point across the x-axis to find the mirror point of (-4,-3). Keep the x-coordinate the same and change the y-coordinate to its opposite. The x-coordinate is -4. The opposite of the y-coordinate -3 is 3.
Point&Reflection Acrossx-axis (-4, -3)&(-4, 3) Graph this point on the coordinate plane. Start at the origin and move 4 units to the left and 3 units up.
Dominika is interested in the mirror point of (-4, -3) because it is a great shooting position. This point is as far away from the basket as the original point but is oriented in a different direction. This can give her an advantage over her opponents and help her score more points during game!
b Reflect the point (-4,-3) across the y-axis to find the defense point. Keep the y-coordinate and change the x-coordinate to its opposite. In this case, the opposite of -4 is 4.
Point&Reflection Acrossy-axis ( -4,-3)&( 4,-3) Add this point to the graph.
Now that Dominika understands her attack and defense positions better, she can perform much better on the court. Her team is doing great and they keep scoring point after point until before they know it, they win first place in the tournament. They are the champions!
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Closure
Introducing the Three-Dimensional Coordinate System
This lesson showed how to locate points with respect to the x- and y-axis on a coordinate plane. A third axis, usually called the z-axis, can be used to provide more information about the position of objects. Think of it like this.
| Description | |
|---|---|
| x-axis | Tells if something is in front of or behind the origin. |
| y-axis | Tells if something is to the left or right of the origin |
| z-axis | Tells if something is above or below the origin. |
These three axes create a 3D coordinate system that is similar to the coordinate plane but with the addition of an extra dimension. This additional dimension means that points are represented as ordered triples (x,y,z) instead of as ordered pairs, as in the coordinate plane.
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The Coordinate Plane
Exercise 2.1
Diego gathered some information for his homework about Jefferson Junior High School. He surveyed some students and asked how many hours they spent studying each week. He wrote the information in a table and made a graph showing his results.
| Hours of Study per Week | Number of Students |
|---|---|
| 1 | 3 |
| 2 | 5 |
| 3 | 10 |
| 4 | 15 |
| 5 | 12 |
| 6 | 8 |
| 7 | 4 |
| 8 | 2 |
However, his dog got into his backpack pulled everything out of it. Diego must choose which of four graphs matches his data. Help him find the correct graph.
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
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Let's make a graph for the data Diego recorded to determine the correct option. We will first write an ordered pair for each row of data from the table. The x-coordinate is the hours of study per week and the y-coordinate is the number of students.
| Hours of Study per Week, x | Number of Students, y | (x,y) |
|---|---|---|
| 1 | 3 | ( 1, 3) |
| 2 | 5 | ( 2, 5) |
| 3 | 10 | ( 3, 10) |
| 4 | 15 | ( 4, 15) |
| 5 | 12 | ( 5, 12) |
| 6 | 8 | ( 6, 8) |
| 7 | 4 | ( 7, 4) |
| 8 | 2 | ( 8, 2) |
Our graph will be in the first quadrant because the coordinates of each ordered pair are non-negative. The x-coordinate indicates how far to move on the x-axis and the y-coordinate indicates how far to move on the y-axis. Let's plot our data!
This graph corresponds to option B.
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Solution
Mark wants to determine the shape formed by this set of ordered pairs. (-3, 1), (-2, 4), (1, 3), (3, 0), (2, -3), (-1, -4) Help him identify the shape that these pairs create when they are joined in the given order.
{"type":"choice","form":{"alts":["Triangle","Square","Circle","Hexagon"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":3}
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Let's plot the set of ordered pairs as points on a coordinate plane. Once we have plotted them, we can connect them to see what shape they create. Remember, in an ordered pair, the first number represents the x-coordinate and the second number represents the y-coordinate. Ordered Pair (x-coordinate,y-coordinate) The x-coordinate tells us how far to move along the x-axis and the y-coordinate tells us how far to move along the y-axis. We can move 3 units to the left from the origin and then 1 unit up to get to the point (-3, 1).
We can follow a similar process to graph the rest of the coordinate pairs.
Let's connect the points to identify the shape they create.
The shape that these points form is a six-sided figure called a hexagon.
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Solution
Look at the following ordered pairs. (3,-4), (-5,6), (7,-2), (2,-8) Identify the ordered pair that does not belong with the other three.
{"type":"choice","form":{"alts":["(3,-4)","(-5,6)","(7,-2)","(2,-8)"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
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We can examine the signs of the coordinates for each ordered pair to determine the quadrant where the point is located. First, let's review how to determine a point's quadrant based on the signs of the coordinates.
With this information in mind, let's now consider the first given ordered pair. (3,-4) Since the x-coordinate is positive and the y-coordinate is negative, we can tell that the ordered pair (3, -4) is in Quadrant IV. We can identify the quadrant for the other points by following a similar reasoning.
| Ordered Pair | Signs of the Coordinates | Quadrant |
|---|---|---|
| ( 3, -4) | ( +, -) | Quadrant IV |
| ( -5, 6) | ( -, +) | Quadrant II |
| ( 7, -2) | ( +, -) | Quadrant IV |
| ( 2, -8) | ( +, -) | Quadrant IV |
After identifying the quadrant of each ordered pair, we can see that three of them belong to Quadrant IV, but the ordered pair (-5,6) is in Quadrant II. This means that (-5,6) is the ordered pair that is different from the others.
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Solution
{"type":"choice","form":{"alts":["Office","Cafeteria","Science Lab","Gym"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
{"type":"choice","form":{"alts":["Science Lab","Art Room","Library","Office"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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We want to find the place located at the reflection of point ( 5, -4) across the y-axis. We can do this by keeping the y-coordinate of -4 and changing the x-coordinate 5 to its opposite -5. Point&Reflection Acrossy-axis ( 5, -4)&( -5, -4) We will move 5 units to the left from the origin and 4 units down to identify the place located at the point ( -5, -4).
This means that the place located at the reflection of (5,-4) across the y-axis is the cafeteria.
Let's begin by determining the coordinates of the gym. We can do this by drawing arrows from this point to each axis.
The arrow touches the x-axis at x= 5 and the horizontal arrow touches the y-axis at y= 3. These numbers represent the gym's coordinates. Gym's Coordinates ( 5, 3) We can now reflect this point across the x-axis. We keep its x-coordinate and switch the sign of its y-coordinate to get its reflection. Gym&Reflection Acrossx-axis ( 5, 3)&( 5, -3) Let's find the reflected point on the coordinate plane.
The science lab is located at the reflection of the gym across the x-axis.
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Heichi is solving a math puzzle to find a hidden shape using a set of points on the coordinate plane. He has already identified the following points. B(1,-1), C(6,-1), E(4,-9) F(0,-5), H(-2,-3) He needs to find the missing ordered pairs which fit the following descriptions.
| Point | Description |
|---|---|
| A | Reflection of F across the x-axis. |
| D | Reflection of H across the y-axis. |
| G | Reflection of E across the y-axis. |
| I | Reflection of C across the y-axis. |
| J | Reflection of B across the y-axis. |
If Heichi connects all the points in alphabetical order, what shape will the points form?
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
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Let's start by graphing the given points on a coordinate plane. Move left or right the number of units specified by the x-coordinate and then move up or down the number of units specified by the y-coordinate to plot each point.
Now that we have the graph of the known points, we can begin by finding the corresponding reflections to identify the missing points. Let's look at the descriptions of the unknown points.
| Point | Description |
|---|---|
| A | Reflection of F across the x-axis. |
| D | Reflection of H across the y-axis. |
| G | Reflection of E across the y-axis. |
| I | Reflection of C across the y-axis. |
| J | Reflection of B across the y-axis. |
Let's find the coordinates of point A, which is the reflection of point F across the x-axis. We keep the x-coordinate and change the y-coordinate to its opposite to reflect a point across the x-axis. For point F( 0, -5), we change its y-coordinate to 5 and keep its x-coordinate 0 the same to get the location of point A. cc Point&Reflection Acrossx-axis F( 0, -5)&A( 0, 5) We can use a similar method to find point D, which is a reflection of point H across the y-axis. We keep the y-coordinate and change the x-coordinate to its opposite to reflect a point across the y-axis. cc Point&Reflection Acrossy-axis H( -2, -3)&D( 2, -3) The rest of the unknown points can be determined following a similar fashion.
| Point | Description | Reflection |
|---|---|---|
| A | Reflection of F across the x-axis. | F( 0, -5) ⇒ A( 0, 5) |
| D | Reflection of H across the y-axis. | H( -2, -3) ⇒ D( 2, -3) |
| G | Reflection of E across the y-axis. | E( 4, -9) ⇒ G( -4, -9) |
| I | Reflection of C across the y-axis. | C( 6, -1) ⇒ I( -6, -1) |
| J | Reflection of B across the y-axis. | B( 1, -1) ⇒ J( -1, -1) |
Finally, let's add the unknown points and connect them in alphabetical order to identify the shape they form.
The points form a star, corresponding to option B.
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The Coordinate Plane
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