10.5: Composite Transformations
Objectives
The lesson objectives for composite transformations are:
- What is a composite transformation?
- The order of a composite transformation
- The notation for a composite transformation
What Is a Composite Transformation?
Concept Content
Up until now you have worked with single transformations. That is you worked with problems involving translations, rotations, reflections, and dilations. In each of the problems you have worked with in the previous four lessons, there was only one transformation taking place. For example, you would have a preimage and rotate it about the origin \begin{align*}90^\circ\end{align*}CCW.
In lesson Composite Transformations, you will study composite transformations. Composite transformations result when two or more transformations are combined to form a new figure from the preimage. The result is a composition of transformations. In this first concept of lesson Composite Transformations, you will learn to recognize and draw composite transformations.
Guidance
Look at the following diagram. It involves two translations. Identify the two translations of triangle \begin{align*}ABC\end{align*} where the vertices of \begin{align*}\Delta ABC\end{align*} are \begin{align*}A(-1, 0)\end{align*}, \begin{align*}B(4, 0)\end{align*}, and \begin{align*}C(2, 6)\end{align*}. Pay particular attention to the notation of the vertices as the triangle moves.
\begin{align*}\Delta ABC\end{align*} moves over 6 to the left and down 5 to produce \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}. Then \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*} moves over 14 to the right and up 3 to produce \begin{align*}\Delta A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}\end{align*}. These translations are represented by the blue arrows in the diagram.
All together \begin{align*}\Delta ABC\end{align*} moves over 8 to the right and down 2 to produce \begin{align*}\Delta A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}\end{align*}. The total translations for this movement are seen by the green arrow in the diagram above.
Examples
Example A
Describe the transformations in the diagram below. The transformations involve a reflection and a rotation.
The transformations involve a reflection and a rotation. First line \begin{align*}AB\end{align*} is rotated about the origin by \begin{align*}90^\circ\end{align*}CCW.
Then the line \begin{align*}A^\prime B^\prime\end{align*} is reflected about the \begin{align*}y\end{align*}-axis to produce line \begin{align*}A^{\prime \prime}B^{\prime \prime}\end{align*}.
Example B
Describe the transformations in the diagram below.
The flag in diagram S is rotated about the origin \begin{align*}180^\circ\end{align*} to produce flag T. You know this because if you look at one point you notice that both \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinate points is multiplied by –1, which is consistent with a \begin{align*}180^\circ\end{align*} rotation about the origin. Flag T is then reflected about the line \begin{align*}x = -8\end{align*} to produce Flag U.
Example C
Triangle \begin{align*}ABC\end{align*} where the vertices of \begin{align*}\Delta ABC\end{align*} are \begin{align*}A(-1, -3)\end{align*}, \begin{align*}B(-4, -1)\end{align*}, and \begin{align*}C(-6, -4)\end{align*} undergoes a composition of transformations described as:
a) a translation 10 units to the right, then
b) a reflection in the \begin{align*}x\end{align*}-axis.
Draw the diagram to represent this composition of transformations. What are the vertices of the triangle after both transformations are applied.
Triangle \begin{align*}A^\prime B^{\prime \prime} C^{\prime \prime}\end{align*} is the final triangle after all transformations are applied. It has vertices of \begin{align*}A^{\prime \prime}(9, 3)\end{align*}, \begin{align*}B^{\prime \prime}(6, 1)\end{align*}, and \begin{align*}C^{\prime \prime}(4, 4)\end{align*}.
Vocabulary
- Composite Transformations
- Composite transformations result when two or more transformations are combined to form a new figure from the preimage. The result is a composition of transformations.
Guided Practice
1. Describe the transformations in the diagram below. The transformations involve a rotation and a reflection.
2. Triangle \begin{align*}XYZ\end{align*} has coordinates \begin{align*}X (1, 2)\end{align*}, \begin{align*}Y (-3, 6)\end{align*} and \begin{align*}Z (4, 5)\end{align*}.The triangle undergoes a translation of 2 units to the right and 1 unit down to form triangle \begin{align*}X^\prime Y^\prime Z^\prime \end{align*}. Triangle \begin{align*}X^\prime Y^\prime Z^\prime\end{align*} is then reflected about the \begin{align*}y\end{align*}-axis to form triangle \begin{align*}X^{\prime \prime} Y^{\prime \prime} Z^{\prime \prime}\end{align*}. Draw the diagram of this composite transformation and determine the vertices for triangle \begin{align*}X^{\prime \prime} Y^{\prime \prime} Z^{\prime \prime}\end{align*}.
3. The coordinates of the vertices of \begin{align*}\Delta JAK\end{align*} are \begin{align*}J(1, 6)\end{align*}, \begin{align*}A(2, 9)\end{align*}, and \begin{align*}K(7, 10)\end{align*}.
a) Draw and label \begin{align*}\Delta JAK\end{align*}.
b) \begin{align*}\Delta JAK\end{align*} is reflected over the line \begin{align*}y=x\end{align*}. Graph and state the coordinates of \begin{align*}\Delta J^\prime A^\prime K^\prime\end{align*}.
c) \begin{align*}\Delta J^\prime A^\prime K^\prime\end{align*} is then reflected about the \begin{align*}x\end{align*}-axis. Graph and state the coordinates of \begin{align*}\Delta J^{\prime \prime} A^{\prime \prime} K^{\prime \prime}\end{align*}.
d) \begin{align*}\Delta J^{\prime \prime} A^{\prime \prime} K^{\prime \prime}\end{align*}undergoes a translation of 5 units to the left and 3 units up. Graph and state the coordinates of \begin{align*}\Delta J^{\prime \prime \prime} A^{\prime \prime \prime} K^{\prime \prime \prime}\end{align*}.
Answers
1. The transformations involve a reflection and a rotation. First line \begin{align*}AB\end{align*} is reflected about the \begin{align*}x\end{align*}-axis to produce line \begin{align*}A^\prime B^\prime\end{align*}.
Then the line \begin{align*}A^\prime B^\prime\end{align*} is rotated about the origin by \begin{align*}90^\circ\end{align*}CCW to produce line \begin{align*}A^{\prime \prime} B^{\prime \prime}\end{align*}.
2.
3.
Summary
In this concept you began your study of composite transformations. Composite transformations involve two or more transformations combined to form a new figure from the preimage. In this lesson, you worked with diagrams where preimages undergo a translation, a reflection, or a rotation to the transformed image \begin{align*}T^\prime\end{align*} and then a second transformation is performed on image \begin{align*}T^\prime\end{align*} to form image \begin{align*}T^{\prime \prime}\end{align*}. You also had a quick look at a diagram with three transformations in that image \begin{align*}T^{\prime \prime}\end{align*} was transformed into image \begin{align*}T^{\prime \prime \prime}\end{align*}.
Problem Set
Find the measure of the transformed image given the following information:
- A point \begin{align*}X\end{align*} has coordinates (–1, –8). The point is reflected in the \begin{align*}y\end{align*}-axis to form \begin{align*}X^\prime\end{align*}. \begin{align*}X^\prime\end{align*} is translated over 4 to the right and up 6 to form \begin{align*}X^{\prime \prime}\end{align*}. What are the coordinates of \begin{align*}X^\prime\end{align*} and \begin{align*}X^{\prime \prime}\end{align*}?
- A point \begin{align*}A\end{align*} has coordinates (2, –3). The point is translated over 3 to the left and up 5 to form \begin{align*}A^\prime\end{align*}. \begin{align*}A^\prime\end{align*} is reflected in the \begin{align*}x\end{align*}-axis to form \begin{align*}A^{\prime \prime}\end{align*}. What are the coordinates of \begin{align*}A^\prime\end{align*} and \begin{align*}A^{\prime \prime}\end{align*}?
- A point \begin{align*}P\end{align*} has coordinates (5, –6). The point is reflected in the line \begin{align*}y = -x\end{align*} to form \begin{align*}P^\prime\end{align*}. \begin{align*}P^\prime\end{align*} is rotated about the origin \begin{align*}90^\circ\end{align*}CCW to form \begin{align*}P^{\prime \prime}\end{align*}. What are the coordinates of \begin{align*}P^\prime\end{align*} and \begin{align*}P^{\prime \prime}\end{align*}?
- Line \begin{align*}JT\end{align*} has coordinates \begin{align*}J(-2, -5)\end{align*} and \begin{align*}T(2, 3)\end{align*}. The line is rotated about the origin \begin{align*}180^\circ\end{align*} to form \begin{align*}J^\prime T^\prime \end{align*}. \begin{align*}J^\prime T^\prime \end{align*} is translated over 6 to the right and down 3 to form \begin{align*}J^{\prime \prime} T^{\prime \prime} \end{align*}. What are the coordinates of \begin{align*}J^\prime T^\prime \end{align*} and \begin{align*}J^{\prime \prime} T^{\prime \prime} \end{align*}?
- Line \begin{align*}SK\end{align*} has coordinates \begin{align*}S(-1, -8)\end{align*} and \begin{align*}K(1, 2)\end{align*}. The line is translated over 4 to the right and up 4 to form \begin{align*}S^\prime K^\prime\end{align*}. \begin{align*}S^\prime K^\prime\end{align*} is rotated about the origin \begin{align*}90^\circ\end{align*}CCW to form \begin{align*}S^{\prime \prime} K^{\prime \prime}\end{align*}. What are the coordinates of \begin{align*}S^\prime K^\prime\end{align*} and \begin{align*}S^{\prime \prime} K^{\prime \prime}\end{align*}?
Describe the following composite transformations:
The Order of a Composite Transformation
Concept Content
In this second concept of lesson Composite Transformations, you will learn about the order of a composite transformation. The order is very important. Imagine if you rotate, then dilate, and then translate a rectangle of vertices \begin{align*}A(1, 1), B(1, 3), C(5, 3)\end{align*}, and \begin{align*}D(5, 1)\end{align*}. You would end up with a diagram similar to that found below:
If you take the same preimage and rotate, translate it, and finally dilate it, you could end up with the following diagram:
Therefore the order is important when performing a composite transformation. Remember that the composite transformation involves a series of one or more transformations in which each transformation after the first is performed on the preceding image that was transformed. Only the first transformation will be performed on the initial preimage.
Guidance
Quadrilateral \begin{align*}WXYZ\end{align*} has coordinates \begin{align*}W(-5, -5), Z(-2, 0), Y(2, 3)\end{align*} and \begin{align*}X(-1, 3)\end{align*}. Draw the quadrilateral on the Cartesian plane. The quadrilateral has a dilation centered at the origin of scale factor \begin{align*}\frac{1}{3}\end{align*} and is then translated 4 units to the right and 5 units down. Show the resulting image.
Examples
Example A
Line \begin{align*}\overline{AB}\end{align*} drawn from (–4, 2) to (3, 2) has undergone a reflection in the \begin{align*}x\end{align*}-axis. It then undergoes a translation up one unit and over 3 units to the right to produce \begin{align*}A^{\prime \prime} B^{\prime \prime}\end{align*}. Draw a diagram to represent this composite transformation and indicate the vertices for each transformation.
Example B
For the composite transformation in Example A, suppose the preimage \begin{align*}AB\end{align*} undergoes a translation up one unit and over 3 units to the right and then undergoes a reflection in the \begin{align*}x\end{align*}-axis. Does the order matter?
For this example \begin{align*}A^{\prime \prime} B^{\prime \prime}\end{align*} is not the same as \begin{align*}A^{\prime \prime} B^{\prime \prime}\end{align*} for the previous example (example A). Therefore order does matter.
Example C
Triangle \begin{align*}BCD\end{align*} is rotated \begin{align*}90^\circ\end{align*}CCW about the origin. The resulting figure is then translated over 3 to the right and down 7. Does order matter?
Order: Rotation then Translation
Order: Translation then Rotation
The blue triangle represents the final image after the composite transformation. In this example, order does matter as the blue triangles do not have the same coordinates.
Vocabulary
- Composite Transformations
- Composite transformations result when two or more transformations are combined to form a new figure from the preimage. The result is a composition of transformations.
Guided Practice
1. Line \begin{align*}\overline{ST}\end{align*} drawn from (–3, 4) to (–4, 8) has undergone a rotation about the origin at \begin{align*}90^\circ \end{align*}CW and then a reflection in the \begin{align*}x\end{align*}-axis. Draw a diagram with labeled vertices to represent this composite transformation.
2. Line \begin{align*}\overline{ST}\end{align*} drawn from (–3, 4) to (–3, 8) has undergone a reflection in the \begin{align*}x\end{align*}-axis and then a rotation about the origin at \begin{align*}90^\circ\end{align*}CW. Draw a diagram with labeled vertices to represent this composite transformation. Is the graph the same as the diagram in #1?
3. The triangle with vertices \begin{align*}J(-5, -2), K (-1, 4)\end{align*} and \begin{align*}L (1, -3)\end{align*} has undergone a transformation of up 4 and over to the right 4 and then a reflection in the \begin{align*}x\end{align*}-axis. Draw and label the composite transformation. Does order matter?
Answers
1.
2.
If you compare the graph above to that found in Question 1, you see that the final transformation image \begin{align*}S^{\prime \prime}T^{\prime \prime}\end{align*} has different coordinates than the image \begin{align*}S^{\prime \prime}T^{\prime \prime}\end{align*} in question 2. Therefore order does matter.
3.
Order: Translation then Reflection
Order: Reflection then Transformation
In this problem, order did matter. The final image after the composite transformation changed when the order changed.
Summary
In this concept of Chapter Is it a Slide, a Flip, or a Turn?, lesson Composite Transformations, you learned that in the majority of cases, the order in which you complete composite transformations is important. Composite transformations occur in any combination of reflections, translations, and rotations. When graphing the composite image, it is necessary for you to perform the transformations in the order they are written.
Problem Set
- Draw the composite image for the figure \begin{align*}MATH\end{align*} on the grid below that undergoes a reflection in the \begin{align*}y\end{align*}-axis and then a reflection in the \begin{align*}x\end{align*}-axis.
- Draw a composite image for Image A on the grid below that has undergone a reflection in the \begin{align*}y\end{align*}-axis and then a transformation of 3 units down and 4 units to the left.
- Draw a composite image for the figure below on the grid below that undergoes a translation of 5 units to the right and 3 units up. Then this image is rotated across the origin \begin{align*}90^\circ\end{align*}CW.
- Draw a composite image for the Image A on the grid below that undergoes a rotation of \begin{align*}180^\circ\end{align*} about the point \begin{align*}B\end{align*} and then a translation of 3 units to the left and 3 units up.
- Draw a composite image for the figure on the grid below that undergoes a reflection in the line \begin{align*}y = -x\end{align*} and then a rotation about the origin at \begin{align*}270^\circ\end{align*}CW.
For each of the following diagrams the images undergo a reflecting in the \begin{align*}y\end{align*}-axis, then a rotation of \begin{align*}90^\circ\end{align*}CW and finally a translation of 4 units right and 7 units up. Draw and label the composite images.
The Notation for a Composite Transformation
Concept Content
As you have learned in the previous two concepts, a composite transformation depends on order. In this last concept, you will learn how to describe a composite transformation using a rule. Notation for composite transformations combines the transformations using the symbol \begin{align*}\circ\end{align*}.
In Lesson Translations of Geometric Shapes, you learned that the notation for translations used the symbol \begin{align*}T\end{align*}. Therefore a translation with notation \begin{align*}T_{3, 5}(x, y) = (x + 3, y + 5)\end{align*} or a translation of 3 units to the right and 5 units up. In Lesson Reflections of Geometric Shapes, you learned that the notation for reflections could be in the form \begin{align*}P(x,y) \rightarrow P^{\prime}(-x, y)\end{align*} or \begin{align*}r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*} with the most common notation of \begin{align*}r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*}. In lesson Rotations of Geometric Shapes, you learned that the notation for rotations uses the form \begin{align*}R_{90^\circ}(x,y)=(-y,x), R_{180^\circ}(x,y)=(-x,-y),\end{align*} or \begin{align*}R_{270^\circ}(x,y)=(y,-x)\end{align*} depending on the rotation angle with respect to the center point, the origin.
In this final concept of lesson Composite Transformations, you will learn to combine these notations using the symbol \begin{align*}\circ\end{align*} in order to write the notation for the composite transformations.
Guidance
The figure below shows a composite transformation of a trapezoid. Write the mapping rule for the composite transformation.
The transformation from Image A to Image B is a reflection in the \begin{align*}y\end{align*}-axis. The notation for this is \begin{align*}r_{y-axis}\end{align*}. The transformation for image B to form image C is a rotation about the origin of \begin{align*}90^\circ\end{align*}CW or \begin{align*}270^\circ\end{align*}CCW. The notation for this transformation is \begin{align*}R_{270^\circ}\end{align*}. Therefore the notation to describe the transformation of Image A to Image C is:
\begin{align*}r_{y-axis}\ \circ \ R_{270^\circ}\end{align*}
Example A
Graph the line \begin{align*}XY\end{align*} given that \begin{align*}X(2, -2)\end{align*} and \begin{align*}Y(3, -4)\end{align*}. Also graph the composite image that satisfies the following rule
\begin{align*}R_{90^\circ} \ \circ \ r_{y-axis}\end{align*}
The first translation is a \begin{align*}90^{\circ}\end{align*}CCW turn about the origin to produce \begin{align*}X^{\prime}Y^{\prime}\end{align*}. The second translation is a reflection about the \begin{align*}y\end{align*}-axis to produce \begin{align*}X^{\prime \prime}Y^{\prime \prime}\end{align*}. State the coordinates of \begin{align*}X^{\prime \prime}Y^{\prime \prime}\end{align*}.
Example B
Image A with vertices \begin{align*}A(3, 5), B(4, 2)\end{align*} and \begin{align*}C(1, 1)\end{align*} undergoes a composite transformation with mapping rule \begin{align*}r_{y-axis} \ \circ \ r_{x-axis}\end{align*}. Draw the preimage and the composite image and show the vertices of the composite image.
Example C
Image D with vertices \begin{align*}D(-3, 7), E(-1, 3), F(-7, 5)\end{align*} and \begin{align*}G(-5, 1)\end{align*} undergoes a composite transformation with mapping rule \begin{align*}r_{x-axis} \ \circ \ T_{3,4}\end{align*}. Draw the preimage and the composite image and show the vertices of the composite image.
Vocabulary
- Composite Transformations
- Composite transformations result when two or more transformations are combined to form a new figure from the preimage. The result is a composition of transformations.
Guided Practice
1. Graph the line \begin{align*}XY\end{align*} given that \begin{align*}X(2, -2)\end{align*} and \begin{align*}Y(3, -4)\end{align*}. Also graph the composite image that satisfies the following rule
\begin{align*}r_{y-axis} \ \circ \ R_{90^\circ}\end{align*}
The first translation is a reflection about the \begin{align*}y\end{align*}-axis to produce \begin{align*}X^{\prime}Y^{\prime}\end{align*}. The second translation is a \begin{align*}90^\circ\end{align*}CCW turn about the origin to produce \begin{align*}X^{\prime \prime}Y^{\prime \prime}\end{align*}. State the coordinates of \begin{align*}X^{\prime \prime}Y^{\prime \prime}\end{align*}.
2. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure \begin{align*}ABCD\end{align*} to \begin{align*}A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}D^{\prime \prime}\end{align*}.
3. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure \begin{align*}ABC\end{align*} to \begin{align*}A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}\end{align*}.
Answers
1.
2. There are two transformations shown in the diagram. The first transformation is a reflection about the line \begin{align*}X = 2\end{align*} to produce \begin{align*}A^{\prime}B^{\prime}C^{\prime}D^{\prime}\end{align*}. The second transformation is a \begin{align*}90^{\circ}\end{align*}CW (or \begin{align*}270^{\circ}\end{align*}CCW) rotation about the point (2, 0) to produce the figure \begin{align*}A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}D^{\prime \prime}\end{align*}. Notation for this composite transformation is:
\begin{align*}r_{x=2} \ \circ \ R_{270^{\circ}}\end{align*}
3. There are two transformations shown in the diagram. The first transformation is a translation of 1 unit to the left and 5 units down to produce \begin{align*}A^{\prime}B^{\prime}C^{\prime}\end{align*}. The second reflection in the \begin{align*}y\end{align*}-axis to produce the figure \begin{align*}A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}\end{align*}. Notation for this composite transformation is:
\begin{align*}T_{-1,-5} \ \circ \ r_{y-axis}\end{align*}
Summary
In this last concept of lesson Composite Transformations you have learned to describe composite transformations in notation form. Notations can allow you to quickly determine what is happening to the preimage to form the final composite image. The notations for composite transformations combine the notations for reflections, rotations, and translations using the symbol \begin{align*}\circ\end{align*}. Remember to work left to right as you complete the transformations. For example a notation of \begin{align*}T_{3,-4} \ \circ \ R_{90^{\circ}}\end{align*} represents a translation of 3 units to the right and 4 units down then a rotation about the origin of \begin{align*}90^{\circ}\end{align*}CCW. Always remember that order is important.
Problem Set
Complete the following table:
Starting Point | \begin{align*}T_{3,-4} \ \circ \ R_{90^{\circ}}\end{align*} | \begin{align*}r_{x-axis} \ \circ \ r_{y-axis}\end{align*} | \begin{align*}T_{1,6} \ \circ \ r_{x-axis}\end{align*} | \begin{align*}r_{y-axis} \ \circ \ R_{180^{\circ}}\end{align*} |
---|---|---|---|---|
1. (1, 4) | ||||
2. (4, 2) | ||||
3. (2, 0) | ||||
4. (–1, 2) | ||||
5. (–2, –3) |
Write the notation that represents the composite transformation of the preimage A to the composite images in the diagrams below.
Summary
In this fifth lesson of Chapter Is it a Slide, a Flip, or a Turn? you have been introduced to composite transformations. Composite transformations result when two or more transformations are combined to form a new figure from the preimage. The result is a composition of transformations. Composite transformations involve the combination of reflections, rotations and translations of a point, a line, or a figure. In the first concept, you learned how to draw and recognize composite transformations. In concept Reflections of Geometric Shapes you learned that order is important. If you were asked to draw a figure, translate it and then rotate it, this is not the same as taking the same figure and rotating it then translating it. The key is to work left to right.
In the last concept of lesson Composite Transformations, you learned to determine the notation for a composite transformation or to draw a figure from a composite transformation. Remember the notation uses the symbol \begin{align*}\circ\end{align*} to indicate the composite transformation. For example a notation of \begin{align*}T_{-3,5} \ \circ \ r_{y-axis}\end{align*} represents a translation of 3 units to the left and 5 units up then a reflection in the \begin{align*}y\end{align*}-axis. You need to remember that order is important.
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