Geometrical transformations in the plane

Consider the figure below. Each coordinate systems depicts a geometrical transformation of the point $A$ . The resulting point is denoted by $A^\prime$ .

Translations: (a) and (b)

Shifting upwards by $\Delta y$ . Of course, shifting downwards is possible as well.
Shifting to the right by $\Delta x$ . Again, we can also shift to the left.

Stretching: (c ) and (d)

Stretching by a factor of $c$ in $y$ -direction.
Stretching by a factor of $c$ in $x$ -direction.

Reflections: (e)- (h)

Reflecting about $x$ -axis
Reflecting about $y$ -axis
Reflecting about origin
Reflecting about the diagonal

Exercise 1

Assume the following coordinates for point $A$ :

A(1|2)

Determine the coordinates of the transformed point $A^\prime$ , if we apply the following transformation:

Shift upwards by $0.5$
Shift downwards by $2$
Shift left by $3$
Shift right by $0.8$
Stretch in $x$ -direction by a factor of $3$
Stretch in $y$ -direction by a factor of $0.2$
Reflect about the $x$ -axis
Reflect about the $y$ -axis
Reflect about the origin
Reflect about the diagonal

Solution

Shift upwards by $0.5$ : $A^\prime(1|2.5)$
Shift downwards by $2$ : $A^\prime(1|0)$
Shift left by $3$ : $A^\prime(-2|2)$
Shift right by $0.8$ : $A^\prime(1.8|2)$
Stretch in $x$ -direction by a factor of $3$ : $A^\prime(3|2)$
Stretch in $y$ -direction by a factor of $0.2$ : $A^\prime(1|0.4)$
Reflect about the $x$ -axis: $A^\prime(1|-2)$
Reflect about the $y$ -axis: $A^\prime(-1|2)$
Reflect about the origin: $A^\prime(-1|-2)$
Reflect about the diagonal: $A^\prime(2|1)$

Exercise 2

Express a reflection about the $x$ -axis by stretching a stretching transformation.
Express a reflection about the $y$ -axis by a stretching transformation.
Express the reflection about the origin with the help of other reflections.

Solution

Stretching in $y$ -direction by a factor of $-1$ .
Stretching in $x$ -direction by a factor of $-1$ .
Reflection about the $x$ -axis, followed by a reflection about the $y$ -axis (or first reflecting about the $y$ -axis, then about the $x$ -axis).

Theorem 1

In general, the coordinates of a point $A(x|y)$ change as follows:

Shift in $y$ -direction by $\Delta y$ : $A^\prime(x|y+\Delta y)$ . The shift is downwards for $\Delta y<0$ and upwards for $\Delta y>0$ .
Shift in $x$ -direction by $\Delta x$ : $A^\prime(x+\Delta x|y)$ . The shift is to the left for $\Delta x<0$ and to the right for $\Delta x>0$ .
Stretch in $x$ -direction by a factor of $c$ : $A^\prime(cx|y)$
Stretch in $y$ -direction by a factor of $c$ : $A^\prime(x|cy)$
Reflection about the $x$ -axis: $A^\prime(x|-y)$
Reflection about the $y$ -axis: $A^\prime(-x|y)$
Reflection about the origin: $A^\prime(-x|-y)$
Reflection about the diagonal: $A^\prime(y|x)$

The transformation of a curve is done by transforming every point on this curve.

Exercise 3

Copy the curve $f$ below into a coordinate system, apply the transformations below to it, and sketch the resulting curve $f^\prime$ .

Shift $f$ to the right by $2$ , then stretch the result by a factor of $0.5$ in $y$ -direction.
Reflect $f$ about the $x$ -axis, shift the result upwards by $1$ , and then stretch it by a factor of $1.5$ in $x$ -direction.

Solution