Transformations In The Coordinate Plane
by Mrs. Raquel A. Pesce
There are cerain types of transformations such as reflections, translations, and dilations. With these types of transformations there exist special rules for determining the coordinates, of the images of points of a figure, that are reflected, translated, or dilated in the coordinate plane.
In this lesson module we will cover the rules for these types of transformations.
Are transformations used in real life situations?
Reflection of a point in the Origin
The image of P(x,y) after a reflection in the origin is P'(-x,-y). This is the same thing as a rotation of 180 degrees.
Reflections in the Coordinate Axis
|The reflection of P(x,y) in the x-axis is P'(x,-y).|
| The reflection of P(x,y) in the y-axis is P'(-x,y).|
To reflect a line segment in a coordinate axis, flip it over the reflecting line by reflecting each endpoint in that line.
Example 1: What is the image of the point A(2,3) after a reflection in the origin?
To reflect a point from the origin follow the formula P(x,y) to P'(-x,-y).
The point (2,3) becomes (-2,-3).
Example 2: What is the image of point (3,-4) after:
a) reflection in the x-axis
b) reflection in the y-axis
| a) Reflecting (3,-4) in the x-axis equals (3,4).|
|b) Reflecting (3,-4) in the y-axis equals (-3,-4).|
Example 3: If the endpoints of AB are A(5,9) and B(8,10), what are the endpoints of AB after a reflection of AB in the x-axis?
Reflect each point in the x-axis.
A(5,9) becomes A'(5,-9).
B(8,10) becomes B'(8,-10).
Sliding a point P(x,y) horizontally h units and then vertically k units places the image at P'(x + h, y + k).
Example 4: The coordinates of the vertices of triangle ABC are A(2,-3), B(0,4), and C(-1,5). If the image of point A under a translation is point A'(0,0), find the images of points B and C under this translation.
|Step ||Instruction ||Example |
|1 || First we have to find the translation of A(2, -3). ||A(2,-3) becomes A'(0,0). |
|2 ||Remember our formula for a translation of P'(x + h, y + k). This gives us......|| A'(2 + h, -3, + k) = A'(0,0) |
2 + h = 0 and -3 + k = 0
h = -2 and k = 3
|3 ||Apply this translation to points B and C.|| |
B(0,4) becomes B'(0 + (-2), 4 + 3) = B'(-2,7)
C(-1,5) becomes C'(-1 + (-2), 5 + 3) = C'(-3,8)
The image of P(x,y) under a dilation from the origin is P'(cx,cy), where c is the constant of dilation, so c does not equal 0.
Example 5: After a dilation from the origin, the image of A(2,3) is A'(4,6). What are the coordinates of the point that are the image of B(1,5) after the same dilation?
We have to first find the constant of dilation.
The constant of dilation is 2 since A(2,3) becomes A'(2 x 2, 2 x 3) = A'(4,6).
Use the same dilation to find point B.
B(1,5) becomes B'(2,10).
Example 6: Graph triangle ABC with coordinates A(0,3), B(4,-1), and C(6,4)). On the same set of axes, graph triangle A'B'C', the reflection of triangle ABC in the origin.
Follow the formula P(x,y) becomes P'(-x,-y) under a reflection from the origin.
A(0,3) becomes A'(0,-3)
B(4,-1) becomes B'(-4,1)
C(6,4) becomes C'(-6,-4).
|Next, using graph paper, graph triangle ABC with coordinates A(-2,-1), B(1,4), and C(5,-1).|
a) On the same set of axes, graph triangle A'B'C', the image of triangle ABC after a dilation of scale factor of 2.
b) What is the ratio of the area of triangle A'B'C' to the area of triangle ABC?
|Finally, using graph paper, graph triangle DEF with coordinates D(-1,-3), E(6,2) and F(8,-5). On the same set of axes, graph:|
a) Triangle D'E'F', the image of triangle DEF after a reflection in the x-axis.
b) Triangle D''E''F'', the image of triangle D'E'F' after a reflection in the y-axis.