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282 Chapter 6 Integers and the Coordinate Plane
You can reflect a point in the x -axis, in the y -axis, or in both axes. The red points are mirror images of each other in the x -axis because the x -coordinates are the same and the y -coordinates are opposites. So, the red points are 3 units from the x -axis in opposite directions. The red points represent a reflection in the x-axis. The blue points are mirror images of each other in the y -axis because the y -coordinates are the same and the x -coordinates are opposites. So, the blue points are 4 units from the y -axis in opposite directions. The blue points represent a refl ection in the y-axis.
EXAMPLE 11 Reflecting Points in One Axis
a. Reflect ( − 2, 4) in the x -axis. Plot (−2, 4). To refl ect (−2, 4) in the x -axis, use the same x -coordinate, −2, and take the opposite of the y -coordinate. The opposite of 4 is −4.
So, the reflection of (−2, 4) in the x -axis is (−2, −4). b. Reflect ( − 3, − 1) in the y -axis. Plot (−3, −1). To refl ect (−3, −1) in the y -axis, use the same y -coordinate, −1, and take the opposite of the x -coordinate. The opposite of −3 is 3.
So, the reflection of (−3, −1) in the y -axis is (3, −1).
Lesson Tutorials
Reflecting Points in the
Coordinate Plane
Extension
Refl ecting a Point in the Coordinate Plane ● To refl ect a point in the x -axis, use the same x -coordinate and take the opposite of the y -coordinate. ● To refl ect a point in the y -axis, use the same y -coordinate and take the opposite of the x -coordinate.
Ź 4 Ź 3 Ź 2 1 2 3 4
Ź 3
Ź 2
x
y 3 2 1 O
(Ź4, 1) (4, 1)
(1, 3)
(1, Ź3)
x
y 3
4
2 1
Ź 3 Ź 4
Ź 2
Ź 4 Ź 3 Ź 2 O 1 2 3 4
(Ź2, Ź4)
(Ź2, 4)
x
y 3
4
2 1
Ź 3 Ź 4
Ź 2
Ź 4 Ź 3 Ź 2 O 1 2 3 4 (Ź3, Ź1) (3, Ź1)
COMMON
CORE
Coordinate Plane In this extension, you will ● (^) understand reflections of points in the coordinate plane. Learning Standard 6.NS.6b
Extension 6.5 Reflecting Points in the Coordinate Plane 283
EXAMPLE 22 Reflecting a Point in Both Axes
Refl ect (2, 1) in the x -axis followed by the y -axis.
Step1: First, plot (2, 1).
Step 2: Next, reflect (2, 1) in the x -axis. Use the same x -coordinate, 2, and take the opposite of the y -coordinate. The opposite of 1 is −1. The point (2, 1) reflected in the x -axis is (2, −1).
Step 3: Finally, reflect (2, −1) in the y -axis. Use the same y -coordinate, −1, and take the opposite of the x -coordinate. The opposite of 2 is −2. The point (2, −1) reflected in the y -axis is (−2, −1).
So, (2, 1) reflected in the x -axis followed by the y -axis is (−2, −1).
Refl ect the point in (a) the x -axis and (b) the y -axis.
1. (3, 2) 2. (−4, 4) 3. (−5, − 6 ) 4. (4, − 7 )
5. (0, − 1 ) 6. (−8, 0) 7. (2.5, 4.5) 8. ( − 5 —^1
2
Refl ect the point in the x -axis followed by the y -axis.
9. (4, 5) 10. (−1, 7) 11. (−2, − 2 ) 12. (6.5, −10.5) 13. REASONING A point is reflected in the x -axis. The reflected point is (3, − 9 ). What is the original point? What is the distance between the points? 14. REASONING A point is reflected in the y -axis. The reflected point is (5.75, 0). What is the original point? What is the distance between the points? 15. a. STRUCTURE In Exercises 9−12, reflect the point in the y -axis followed by the x -axis. Do you get the same results? Explain.
b. LOGIC Make a conjecture about how to use the coordinates of a point to find its reflection in both axes.
16. GEOMETRY The vertices of a triangle are (−1, 3), (−5, 3), and (−5, 7). How would you reflect the triangle in the x -axis? in the y -axis? Give the coordinates of the reflected triangle for each case.
When reflecting a second time, be sure to use the reflected point and not the original point.
Common Error Ź^4 Ź^3 Ź^21234
Ź 3
Ź 2
x
y 3 2 1 O
(2, 1)
(2, Ź1)
Ź 4 Ź 3 Ź 2 1 2 3 4
Ź 3
Ź 2
x
y 3 2 1 O
(2, 1)
(Ź2, Ź1) (2, Ź1)
