Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:A reflection flips a figure so each point lands the same distance on the other side of a line.
Common stuck point:The procedure for reflection is the easy part; the trap is changing both coordinates for an axis reflection without checking the axis. Asking "Can I identify the mirror line and equal distances from it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Can I identify the mirror line and equal distances from it?
Worked Examples
Example 1
easy
Reflect the point P(3,−4) over the x-axis. What are the coordinates of the image?Reflect P(3,−4) over the x-axis
Answer
P′=(3,4)
First step
1
Step 1: Reflecting over the x-axis: the rule is (x,y)→(x,−y).
Full solution
2
Step 2: Apply to P(3,−4): P′=(3,−(−4))=(3,4).
3
Step 3: The point flips from below to above the x-axis.
Reflecting over the x-axis keeps the x-coordinate the same and negates the y-coordinate. The x-axis acts as a mirror: points below the x-axis map to the corresponding points above it, and vice versa.
Example 2
medium
Reflect the point Q(−2,5) over the line y=x. Find the image.Reflect Q(−2,5) over y=x
Example 3
medium
Reflect (2,7) across the horizontal line y=4.
Example 4
medium
After reflecting P(3,4) across the x-axis to P′, find the length of PP′.Find the length PP' after reflecting P(3,4) across the x-axis
Example 5
hard
Reflect y=2x+1 across the y-axis. Write the new equation.
Example 6
hard
Reflect (3,7) across the line y=x+2.Reflect (3,7) across y=x+2
Example 7
challenge
If a figure has both vertical and horizontal lines of symmetry, what point symmetry must it also have?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
Reflect (5,3) over the y-axis. What is the image?Reflect (5,3) over the y-axis
Example 2
hard
Reflect the point R(4,1) over the line y=2. Find the image coordinates and explain your method.
Example 3
easy
Reflect the point (3,2) across the y-axis.Reflect (3,2) across the y-axis
Example 4
easy
Reflect (5,4) across the x-axis.Reflect (5,4) across the x-axis
Example 5
easy
Does a reflection change a figure's size or shape?
Example 6
easy
Does a reflection reverse orientation (clockwise/counterclockwise order)?
Example 7
easy
A point lies exactly on the mirror line. Where does its reflection go?
Example 8
easy
Reflect (2,7) across the x-axis.
Example 9
easy
Reflecting a figure twice across the same line returns it to where?
Example 10
easy
The mirror image of your right hand looks like which hand?
Example 11
medium
Reflect (4,1) across the line y=x.Reflect (4,1) across y=x
Example 12
medium
Reflect the point (2,5) across the vertical line x=4.
Example 13
medium
Why is the mirror line the perpendicular bisector of the segment joining a point to its image?
Example 14
medium
Triangle vertices (1,2), (3,2), (1,5) are reflected across the y-axis. Find the images.Reflect triangle ABC across the y-axis
Example 15
medium
A figure is reflected across the x-axis, giving a congruent image. Can a single rotation produce the same image from the original?
Example 16
medium
Reflect (−2,3) across the line y=x.Reflect (−2,3) across y=x
Example 17
medium
A shape has a line of symmetry. What does reflecting it across that line do?
Example 18
medium
Reflect (6,1) across the horizontal line y=3.
Example 19
challenge
A ball bounces off a flat wall. Explain why the reflection law (angle in = angle out) is a geometric reflection across the wall's perpendicular.
Example 20
challenge
Reflect (3,5) across the y-axis, then across the x-axis. What single transformation results, and what's the image?Reflect (3,5) across y-axis, then x-axis
Example 21
challenge
A laser at (0,4) must hit a target at (6,2) by bouncing off the mirror line y=0 (the x-axis). Find the bounce point using reflection.Find the bounce point on y=0
Example 22
challenge
Explain why reflecting twice across two parallel lines produces a translation, not a reflection.
Example 23
easy
Reflect (7,−2) across the x-axis.Reflect (7,−2) across the x-axis
Example 24
easy
Reflect (0,4) across the y-axis.
Example 25
easy
Reflect (5,5) across the line y=x.
Example 26
easy
True or false: reflection changes a clockwise letter to a counterclockwise one.
Example 27
medium
Reflect (−1,5) across the vertical line x=3.
Example 28
medium
A triangle has vertices A(1,1),B(4,1),C(4,5). Reflect across the x-axis.Reflect triangle ABC across the x-axis
Example 29
medium
Reflect (−4,3) across the y-axis, then across the x-axis. Final image?Reflect (−4,3) across y-axis then x-axis
Example 30
medium
An equilateral triangle has how many lines of symmetry?
Example 31
hard
Reflect the line y=2x+1 across the x-axis. Write the new equation.
Example 32
hard
After reflecting (a,b) across the line y=x, then across the x-axis, what is the image?
Example 33
hard
Triangle ABC has vertices A(2,3),B(4,1),C(5,6). Find the image of A under reflection over y=x.Find A' after reflecting triangle ABC over y=x
Example 34
hard
If a figure is reflected over line ℓ1 then over line ℓ2 (parallel to ℓ1, distance d apart), what single transformation results?
Example 35
hard
A point on the mirror line y=x is its own image under reflection. Give a non-origin example.
Example 36
challenge
A laser at (2,5) must hit a target at (8,1) by reflecting off the x-axis. Find the bounce point.Find the bounce point on y=0
Example 37
challenge
The reflection of (x0,y0) across the line y=mx has a known formula. Apply it: reflect (1,0) across y=x (so m=1).Reflect (1,0) across y=x