Question

explore transformations
use the graph to answer problems 1–4.

1. translate figure a using the rule ((x, y) \to (x + 1, y - 4)). draw the image of figure a. practice reading the rule aloud.
2. how was figure b translated to get figure d? write your answer using mapping notation.
3. (constructed response) why is figure e not a translation of figure c?
4. figure f has vertices with coordinates at ((-6, 1)), ((-9, 1)), ((-8, 2)), ((-9, 3)), ((-6, 3)), and ((-7, 2)). if the figure is translated using the rule ((x, y) \to (x + 2, y + 5)), what are the coordinates of the new vertices?
5. math on the spot draw the translation of (\triangle xyz) 4 units right and 2 units down.

Explanation:

Step1: Identify the translation rule

The translation rule given is \((x, y) \to (x + 2, y + 5)\). This means we add 2 to the \(x\)-coordinate and 5 to the \(y\)-coordinate of each vertex of Figure \(F\).

Step2: Apply the rule to each vertex

• For the vertex \((-6, 1)\):
• New \(x\)-coordinate: \(-6 + 2 = -4\)
• New \(y\)-coordinate: \(1 + 5 = 6\)
• So the new vertex is \((-4, 6)\)
• For the vertex \((-9, 1)\):
• New \(x\)-coordinate: \(-9 + 2 = -7\)
• New \(y\)-coordinate: \(1 + 5 = 6\)
• So the new vertex is \((-7, 6)\)
• For the vertex \((-8, 2)\):
• New \(x\)-coordinate: \(-8 + 2 = -6\)
• New \(y\)-coordinate: \(2 + 5 = 7\)
• So the new vertex is \((-6, 7)\)
• For the vertex \((-9, 3)\):
• New \(x\)-coordinate: \(-9 + 2 = -7\)
• New \(y\)-coordinate: \(3 + 5 = 8\)
• So the new vertex is \((-7, 8)\)
• For the vertex \((-6, 3)\):
• New \(x\)-coordinate: \(-6 + 2 = -4\)
• New \(y\)-coordinate: \(3 + 5 = 8\)
• So the new vertex is \((-4, 8)\)
• For the vertex \((-7, 2)\):
• New \(x\)-coordinate: \(-7 + 2 = -5\)
• New \(y\)-coordinate: \(2 + 5 = 7\)
• So the new vertex is \((-5, 7)\)

Answer:

The new coordinates of the vertices of Figure \(F\) after translation are \((-4, 6)\), \((-7, 6)\), \((-6, 7)\), \((-7, 8)\), \((-4, 8)\), and \((-5, 7)\).