Question

explore translations
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. arguments why is figure e not a translation of figure c?

Explanation:

Problem 1: Translate Figure A using the rule \((x, y) \to (x + 1, y - 4)\)

Step 1: Identify Coordinates of Figure A

First, find the coordinates of the vertices of Figure A. From the graph, Figure A (the diamond) has vertices at \((1, 3)\), \((2, 2)\), \((3, 3)\), and \((2, 4)\) (approximate, based on grid). Wait, more accurately, looking at the grid: the diamond A is centered at (2,3)? Wait, no, let's check the grid. The grid has x from -8 to 8 and y from -8 to 8. Figure A is at x=2, y=3? Wait, the diamond: let's list the four vertices. Let's see, the diamond A: top vertex (2,4), bottom (2,2), left (1,3), right (3,3). So vertices are \((1, 3)\), \((2, 4)\), \((3, 3)\), \((2, 2)\).

Step 2: Apply Translation Rule

For each vertex \((x, y)\), apply \((x + 1, y - 4)\):

• For \((1, 3)\): \(x + 1 = 2\), \(y - 4 = -1\) → \((2, -1)\)
• For \((2, 4)\): \(x + 1 = 3\), \(y - 4 = 0\) → \((3, 0)\)
• For \((3, 3)\): \(x + 1 = 4\), \(y - 4 = -1\) → \((4, -1)\)
• For \((2, 2)\): \(x + 1 = 3\), \(y - 4 = -2\) → \((3, -2)\)

Step 3: Draw the Translated Figure

Plot the new vertices \((2, -1)\), \((3, 0)\), \((4, -1)\), \((3, -2)\) on the grid and connect them to form the translated image of Figure A.

Step 1: Identify Coordinates of B and D

Find vertices of Figure B (the lower triangle) and Figure D (the upper triangle).

Figure B: Let's say its bottom vertex is at \((-2, -9)\)? Wait, no, looking at the grid: Figure B is at x=-2, y=-8 (bottom vertex), and two top vertices at \((-3, -7)\) and \((-1, -7)\) (approximate). Figure D is at x=-7, y=-4 (bottom vertex), and top vertices at \((-8, -3)\) and \((-6, -3)\).

Wait, better: count the horizontal and vertical shifts. From B to D:

• Let's take a corresponding vertex. Let's take the bottom vertex of B: let's say B's bottom vertex is at \((-2, -9)\)? No, the grid: Figure B is at the bottom, around x=-2, y=-8. Figure D is at x=-7, y=-4. So horizontal shift: from x=-2 to x=-7: that's \( -7 - (-2) = -5\) (so left 5 units), vertical shift: from y=-8 to y=-4: \( -4 - (-8) = 4\) (up 4 units). So the translation rule is \((x, y) \to (x - 5, y + 4)\).

Check with another vertex: B's top left vertex: say \((-3, -7)\), D's top left vertex: \((-8, -3)\). \(x\): \(-8 - (-3) = -5\), \(y\): \(-3 - (-7) = 4\). Correct. So the translation is \((x, y) \to (x - 5, y + 4)\).

Step 1: Recall Translation Definition

A translation is a rigid transformation that slides a figure without changing its shape, size, or orientation. It only shifts the figure horizontally and/or vertically.

Step 2: Compare Figure C and E

Figure C is a vertical rectangle (tall and narrow), Figure E is a horizontal rectangle (short and wide). The orientation (length and width) is different. In a translation, the shape (including the ratio of length to width, orientation) should remain the same. Since Figure E has a different orientation (rotated or resized in aspect ratio) compared to Figure C, it is not a translation.

Answer:

The translated figure (image of A) has vertices at \((2, -1)\), \((3, 0)\), \((4, -1)\), \((3, -2)\) (draw these points and connect them).

Problem 2: How was Figure B translated to get Figure D?