Question

1. travel write a rule for the translation that maps the descent of the hot - air balloon. which transformation is suggested by each of the following? 13. mountain range and its image on a lake 14. straight line path of a band marching down a street 15. wings of a butterfly given points f(3,5), g(-1,4), and h(5,0), draw △fgh and its reflection across each of the following lines. 16. the x - axis 17. the y - axis 18. find the vertices of one of the triangles on the graph. then use arrow notation to write a rule for translating the other three triangles. a transformation maps a onto b and c onto d. 19. name the image of a. 20. name the pre - image of b. 21. name the image of c. 22. name the pre - image of d.

Explanation:

Step1: Recall translation rule for vertical movement

For a vertical - downward translation, if a point $(x,y)$ is translated $k$ units down, the rule is $(x,y)\to(x,y - k)$. In the case of the hot - air balloon's descent, assuming it moves down $k$ units, the translation rule is $(x,y)\to(x,y - k)$.

Step2: Identify transformation types

• For the mountain range and its image on a lake, it is a reflection (a mirror - like transformation across a horizontal line, usually the water surface).
• For the straight - line path of a band marching down a street, it is a translation (the band moves along a straight path without rotation or reflection).
• For the wings of a butterfly, it is a reflection (the wings are symmetric about a central line).

Step3: Recall reflection rules

• Reflection across the $x$ - axis: The rule for reflecting a point $(x,y)$ across the $x$ - axis is $(x,y)\to(x, - y)$. For $F(3,5)$, its reflection is $F'(3, - 5)$; for $G(-1,4)$, its reflection is $G'(-1, - 4)$; for $H(5,0)$, its reflection is $H'(5,0)$.
• Reflection across the $y$ - axis: The rule for reflecting a point $(x,y)$ across the $y$ - axis is $(x,y)\to(-x,y)$. For $F(3,5)$, its reflection is $F''(-3,5)$; for $G(-1,4)$, its reflection is $G''(1,4)$; for $H(5,0)$, its reflection is $H''(-5,0)$.

Step4: For triangle translation

Suppose one triangle has vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)$. If we find the vector between the corresponding vertices of the first triangle and the other three triangles, say the translation vector is $(a,b)$, the rule in arrow notation is $(x,y)\to(x + a,y + b)$.

Step5: For transformation mapping

If a transformation maps $A$ onto $B$ and $C$ onto $D$, the image of $A$ is $B$, the pre - image of $B$ is $A$, the image of $C$ is $D$, and the pre - image of $D$ is $C$.

Answer:

1. $(x,y)\to(x,y - k)$ (where $k$ is the number of units of descent)
2. Reflection
3. Translation
4. Reflection
5. $F'(3, - 5),G'(-1, - 4),H'(5,0)$
6. $F''(-3,5),G''(1,4),H''(-5,0)$
7. First, find the vertices of a triangle. Then, find the translation vector $(a,b)$ between the first triangle and the other three triangles and write the rule $(x,y)\to(x + a,y + b)$
8. $B$
9. $A$
10. $D$
11. $C$