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.
Show less Show more expand_more| Student Learning Objectives: |
|---|
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| | 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. The coordinate plane is used to visualize two-dimensional quantities such as a location on a map. This lesson will explore how to locate points on a coordinate plane and examine 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 left side and rotate them by clicking on the right side.
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. A coordinate plane is a grid formed by two number lines that meet at 0. The horizontal number line is the x-axis, and the vertical number line is 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 on a map.
| Position | |
|---|---|
| Dominika's House | D=(2,1) |
| Jordan's House | J=(2,4) |
| Emily's House | E=(-1,4) |
a
Which coordinate plane correctly displays the positions of the three houses?
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
We can graph an ordered pair 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 Dominika's and her friends' houses are given as ordered pairs. Dominika's house is at the point D=( 2, 1), so the x-coordinate is 2 and the y-coordinate is 1. To graph it on the coordinate plane, we should move 2 units to the right of the origin and then 1 unit up.
Now let's add Jordan and Emily's houses to the coordinate plane. Starting from the origin, Jordan’s house is 2 units to the right and 4 units up. Similarly, Emily’s house is 1 unit to the left and 4 units up.
Our graph is the same as Option III.
b
We start by adding the coordinates of the basketball court to the graph we made in Part A. Since it is located at the point C=( -1, 1), we move 1 unit to the left and 1 up starting from the origin.
Now we can draw the path that Dominika is taking on the way from her house to the court. D → J → E → F → D She start at her house, then she takes her friends by going to E and J. Later, they go to the court C, and after the game, she returns home to D.
This graph matches Option I.
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Example
A Robot's Range
A local restaurant is located at (2,1) on a coordinate plane. Each unit on the grid represents one town block. The restaurant recently invested in a delivery robot. The robot can travel up to 5 blocks along the streets from the restaurant. Which graph shows the robot's delivery range?
{"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 the robot can only stay on the straight roads, its position can only move up, down, left, and right on the graph.
Solution
On the coordinate plane, each unit represents one block. Since the robot only moves along straight roads, it can move only up, down, left, or right. For example, starting at the restaurant, point (2,1), the robot can move up to 5 units to the right.
The delivery robot can also turn during its path as long as it stays on the roads. For example, it could go one block up and then 4 blocks right.
Let's follow the same reasoning to find the other farthest places that the robot can travel to and plot each as ordered pairs on the coordinate plane.
Now that all the points are plotted, we can connect them with lines to see that the overall shape is a square. Since these points mark the farthest places the robot can reach, its range includes everything inside the square.
Showing Our Work
Finding All the Places the Delivery Robot Can ReachWe can find the robot's delivery range by identifying all locations that are 5 blocks from the restaurant along the streets. Moving left or down represents a negative direction on the coordinate plane, so these movements are shown with negative numbers.
| Route from the Restaurant | 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) |
After plotting all destinations, we get the same shape as in the main solution.
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Discussion
Quadrants
In a coordinate plane, the intersection of the x- and y-axes 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 of a Point
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Example
Matching Points and Quadrants
Match 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
We can find the quadrant to which a point belongs based on the signs of the point's coordinates. For example, both coordinates of (-2, -8) are negative, so the point lies in Quadrant III. Let's use this idea to find the quadrants of the other points.
| Point | Signs of Coordinates | Quadrant |
|---|---|---|
| ( -2, -8) | ( -, -) | Quadrant III |
| ( 4, -2) | ( +, -) | Quadrant IV |
| ( -11, 3) | ( -, +) | Quadrant II |
| ( 4, 3) | ( +, +) | Quadrant I |
Now that we found the quadrant of each point, let's plot the points on a coordinate plane to check our work. 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.
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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 axes 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 To reflect a point across the x-axis, keep the same x-coordinate and change the y-coordinate to its opposite. Point A is reflected across the x-axis by changing its y-coordinate 8 to its opposite, -8.
A=(5, 8) ⇒ A’=(5, -8)
To reflect a point across the y-axis, change the x-coordinate to its opposite and keep the same y-coordinate. Point B is reflected across the y-axis by changing the x-coordinate 10 to its opposite, -10
B=( 10,-5) ⇒ B’=( -10,-5)
To reflect a point across both the x- and y-axes, change both coordinates to their opposites. 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 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
Reflect this point across the horizontal central line of the court.
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b
What is the mirror image of (-4,-3) across the vertical central line of the court?
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Hint
a
The x-coordinate of the reflection stays the same and the y-coordinate is changed to its opposite.
b
Reflect the point across the y-axis. The y-coordinate stays the same and the x-coordinate is changed to its opposite.
Solution
a
The horizontal central line of the court is the x-axis. To reflect (-4,-3) across the x-axis, we keep the x-coordinate the same and change the y-coordinate to its opposite.
ccc Point&&Reflection Acrossx-axis [0.5em] (-4, -3)&&(-4, 3) Let's graph this point on the coordinate plane. Start at the origin and move 4 units to the left and 3 units up.
b
The vertical central line of the court is the y-axis. To reflect the point (-4,-3) across the y-axis, we should keep the y-coordinate and change the x-coordinate to its opposite.
ccc Point&&Reflection Acrossy-axis [0.5em] ( -4,-3)&&( 4,-3) Let's add this point to our graph.
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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. Let's explore this concept.
| Description | |
|---|---|
| x-axis | Tells us if something is in front of or behind the origin |
| y-axis | Tells us if something is to the left or right of the origin |
| z-axis | Tells us 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 1.1
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Start by looking at the given graph.
Let's draw a straight line upwards from point P to the x-axis to find its x-coordinate. Then, we will draw another straight line from point P to the y-axis, going horizontally to determine its y-coordinate.
The vertical line intersects the x-axis at 1 and the horizontal line intersects the y-axis at -4. This indicates that the x-coordinate of point P is 1 and the y-coordinate is -4. Now we can express this as an ordered pair. P( 1, -4)
We can use a similar process to determine the coordinates of point Q. Let's start by drawing a vertical line from point Q to the x-axis. We will then draw a horizontal line from point Q to the y-axis.
In this case, the vertical line intersects the x-axis at -7 and the y-axis at 8. This means that the x-coordinate of point Q is -7 and the y-coordinate is 8. Let's write it as an ordered pair! Q( -7, 8)
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Solution
Consider the given set of ordered pairs. M(2,4), N(-5,0), O(-3,-5), P(7,1) Which of the graphs correctly displays these ordered pairs on the coordinate plane?
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
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Let's graph each ordered pair on the coordinate plane one by one to determine the correct option. Remember that we graph an ordered pair as a point on a coordinate plane by moving horizontally from the origin the number of units specified by its x-coordinate and vertically the number of units specified by its y-coordinate.
Now, let's take a look at the first given ordered pair. M( 2, 4) Point M has an x-coordinate of 2 and a y-coordinate of 4. We need to move 2 units to the right of the origin and 4 units up to graph this ordered pair.
We can use the same approach to plot the other ordered pairs. Let's start by identifying the x- and y-coordinates for each point. Once we have these coordinates, we can plot them on the graph.
| Ordered Pair | x-coordinate | y-coordinate |
|---|---|---|
| M( 2, 4) | 2 | 4 |
| N( -5, 0) | -5 | 0 |
| O( -3, -5) | -3 | -5 |
| P( 7, 1) | 7 | 1 |
We can add now these points to the graph.
This graph corresponds to option A.
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Solution
Match each point with its corresponding quadrant.
{"type":"pair","form":{"alts":[[{"id":1,"text":"(-5,14)"},{"id":3,"text":"(14,-11)"},{"id":2,"text":"(-7,-22)"},{"id":0,"text":"(21,19)"}],[{"id":0,"text":"Quadrant II"},{"id":3,"text":"Quadrant IV"},{"id":2,"text":"Quadrant III"},{"id":1,"text":"Quadrant I"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[1,3,2,0],[0,3,2,1]]}
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We can check the signs of the x- and y-coordinates to find out where a point is located.
With this information in mind, let's examine the coordinates of the point ( -5, 14). ( -5, 14) ⇓ ( -, +) This point has a negative x-coordinate and a positive y-coordinate. This means that the point ( -5, 14) is located in Quadrant II. We can use a similar reasoning to find the corresponding quadrant of the remaining points.
| Point | Coordinates' Signs | Quadrant |
|---|---|---|
| ( -5, 14) | ( -, +) | Quadrant II |
| ( 14, -11) | ( +, -) | Quadrant IV |
| ( -7, -22) | ( -, -) | Quadrant III |
| ( 21, 19) | ( +, +) | Quadrant I |
We have found the quadrant where each of the given points is located. Let's plot them on a coordinate plane to confirm our matches. For each point, we will move horizontally from the origin according to its x-coordinate and vertically based on its y-coordinate.
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Solution
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{"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":null,"formTextAfter":null,"answer":{"text1":"-4","text2":"15"}}
{"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":null,"formTextAfter":null,"answer":{"text1":"-2","text2":"9"}}
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We are asked to reflect the point ( -5, -1) across the x-axis. This means keeping its x-coordinate the same and changing the y-coordinate to its opposite. The x-coordinate is -5. The opposite of the y-coordinate -1 is 1. Point&Reflection Acrossx-axis ( -5, -1)&( -5, 1) We can plot both points on a coordinate plane to visualize that they are a mirror of each other across the x-axis.
Let's reflect the point ( 4, 15) across the y-axis. In this case, the y-coordinate remains the same and the x-coordinate changes to its opposite. The y-coordinate is 15. The opposite of the x-coordinate 4 is -4. Point&Reflection Acrossy-axis ( 4, 15)&( -4, 15) Again, it is useful to plot these points on a coordinate plane. Let's do it!
Let's reflect the point ( 2, -9) across both the x- and y-axes. To do that, we need to change the x- and y-coordinates both to their opposites. The x-coordinate is 2, so we change it to -2. The y-coordinate is -9, so we change it to 9 to get its reflection. Point&Reflection Across Both Axes ( 2, -9)&( -2, 9) Let's graph both points on a coordinate plane.
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{"type":"choice","form":{"alts":["D is a reflection of C over the x-axis.","D is a reflection of C over the y-axis.","D is a reflection of C over both the x- and y-axes.","No relationship."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
{"type":"choice","form":{"alts":["F is a reflection of E over the x-axis.","F is a reflection of E over the y-axis.","F is a reflection of E over both the x- and y-axes.","No relationship."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":2}
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Let's review how to reflect a point across each axis.
| Axis of Reflection | Procedure |
|---|---|
| x-axis | Change the y-coordinate to its opposite and keep the x-coordinate the same. |
| y-axis | Change the x-coordinate to its opposite and keep the y-coordinate the same. |
| x- and y-axes | Change the x- and y-coordinates to their opposites. |
Now let's take a look at the points A and B. A( -8, 2), B( -8, -2) They both have an x-coordinate of -8, but their y-coordinates are opposites of each other. This means that B is a reflection of A across the x-axis.
Let's begin by looking at the points C and D.
C( 4, 12), D( -4, 12)
In this case, C and D both have a y-coordinate of 12, but their x-coordinates are opposites. This fits the rule for a reflection across the y-axis, so D is a reflection of C across the y-axis.
Consider the points E and F.
E( -11, 3), F( 11, -3)
Notice that the x-coordinate of E is the opposite of the x-coordinate of F and the y-coordinate of E is also the opposite of the y-coordinate of F. This means that F is the reflection of E across the x- and y-axes, following the rule for reflecting a point across both axes.
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Solution
|
A point that lies on the y-axis has a y-coordinate of 0. |
{"type":"choice","form":{"alts":["always","sometimes","never"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
|
Points in Quadrant II have negative x-coordinates. |
{"type":"choice","form":{"alts":["always","sometimes","never"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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Consider the given statement.
A point that lies on the y-axis has a y-coordinate of 0.
Let's examine some points on the y-axis to find out if the given statement is always, sometimes, or never true.
Point A is 5 units up and 0 units left or right from the origin, so its x-coordinate is 0 and its y-coordinate is 5. A( 0, 5) Points B and D also have an x-coordinate of 0, but their y-coordinates are not 0. B( 0, 2) D( 0, -3) Does this mean that the y-coordinate of a point on the y-axis is never 0? Not necessarily. Point C, which lies on the origin of the coordinate plane, is on the y-axis and has coordinates ( 0, 0). C( 0, 0) We found examples of points on the y-axis whose y-coordinate is not 0 and an example of a point on the y-axis whose y-coordinate is 0. Therefore, the given statement is sometimes true.
Let's look at the given statement.
Points in Quadrant II have negative x-coordinates.
Recall that quadrants of a coordinate plane are numbered counterclockwise, from Quadrant I in the top right to Quadrant IV in the bottom right.
Quadrant II is the top left part of the coordinate plane. Any point in this area is to the left of the y-axis. Since all points to the left of the y-axis have an x-coordinate less than 0, any point in Quadrant II will have a negative x-coordinate. Therefore, this statement is always true.
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The Coordinate Plane
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