While doing single transformations may seem pretty simple, multiple transformations can get a little harder to do on plain paper. Focusing on how the points of the figure (called the preimage) map to the transformed figure (called the image) can be more accurate. As you learned in the video, the process of mapping using graph paper is definitely better.
Take a look at the problems in the slideshow below. In each problem, notice how focusing on the points of the original figure can help you with the final transformation.
Problem 1
Problem 2
Problem 3
If trapezoid JKLM were translated 5 units to the left and then down by 5 units, where would trapezoid J'K'L'M' be?
2) Then, imagine that point K were moved 5 units to the left. What would its coordinates be? (4-5, 5) = K'(-1, 5)
3) Now imagine that point K were moved again 5 units down from (-1, 5). What would its coordinates be? (-1, 5-5) = K'(-1, 0)
4) Plot point K.
5) Now, you have to draw the rest of the trapezoid. You can do this by observing the lengths of the sides of the original one.
6) Note that point J is 2 units down from K. Note that point M is 4 units to the right of J. Note that point L is 6 units up from M.
7) Construct points J', K', M' and L' in the same manner. For instance, for J' move 2 units up from K'; for M' move 4 units to the right of J', and so on.
8) Finally, connect the dots to form the new trapezoid J'K'L'M'.
Determine the transformations done on the square 1 to obtain square 2.
1) First, notice that square 1 has a width and length of 2 units, and square 2 has a width and length of 4 units. This is double square 1, which means that square 2 has been dilated or enlarged by a scale factor of 2. We can show the dilation. (figure A)
2) Now, place your pencil on any point on the dotted square. Let's choose the upper left corner (shown by the dot). Use your pencil to count up to the upper left corner of the square 2. (shown by the dot on square 2). This is a translation that is 10 units up.
3) So the transformations to square 2 were: a dilation of scale factor 2 and then a translation up 10 units.
Build a transformation that first reflects the line segment XY across the y-axis. Show the reflected line X'Y'. Then, rotate X'Y' clockwise 90 degrees about the origin to form X"Y". (Hint: Focus on what happens to each point of the line segment during each transformation.)
For the first transformation, the reflection would use the opposite x-coordinate of each point in XY. So X(2, -2) maps to X’(-2, -2) and Y(2, -6) maps to Y’(-2, -6).
For the second transformation, the rotation 90 degrees clockwise about the origin involves turning the same direction as a clock’s hands. The new line X”Y” would be horizontal and lie in the quadrant above X’Y’. So the point X’(-2, -2) maps to X”(-2, 2) and Y’(-2, -6) maps to Y”(-6, 2).
