Question

group practice worksheet:
transformations in the coordinate
plane

part a: quick review
fill in the blanks for the transformation rules:

1. translation: (x, y) → (x + , y + )
2. reflection over x-axis: (x, y) → (x, __)
3. reflection over y-axis: (x, y) → (__, y)
4. rotation 90° clockwise: (x, y) → (, )
5. rotation 90° counterclockwise: (x, y) → (, )

part b: practice with points

1. translate a(-2, 5) right 3, down 4 ______
2. reflect b(4, -1) across the y-axis ______
3. rotate c(3, 6) 90° clockwise ______
4. rotate d(-5, -2) 90° counterclockwise ______
5. plot and label a(4, 2), b(-3, 5), c(-6, -4).

Explanation:

Part A: Quick Review

1. Translation:

Step1: Recall translation rule

For a translation, moving right/left affects the \(x\)-coordinate (right is \(+\), left is \(-\)) and moving up/down affects the \(y\)-coordinate (up is \(+\), down is \(-\)). The general rule is \((x,y)\to(x + h,y + k)\), where \(h\) is the horizontal shift and \(k\) is the vertical shift.
So the blanks are \(h\) and \(k\) (or in the context of a general translation, we can say the first blank is the horizontal shift amount and the second is the vertical shift amount. If we consider a specific translation, but since it's a general rule, we use \(h\) and \(k\)).
\((x,y)\to(x + \boldsymbol{h},y + \boldsymbol{k})\)

1. Reflection over \(x\)-axis:

Step1: Recall reflection over \(x\)-axis rule

When reflecting a point \((x,y)\) over the \(x\)-axis, the \(x\)-coordinate remains the same and the \(y\)-coordinate changes sign. So the rule is \((x,y)\to(x,-y)\).
\((x,y)\to(x,\boldsymbol{-y})\)

1. Reflection over \(y\)-axis:

Step1: Recall reflection over \(y\)-axis rule

When reflecting a point \((x,y)\) over the \(y\)-axis, the \(y\)-coordinate remains the same and the \(x\)-coordinate changes sign. So the rule is \((x,y)\to(-x,y)\).
\((x,y)\to(\boldsymbol{-x},y)\)

1. Rotation \(90^\circ\) clockwise:

Step1: Recall \(90^\circ\) clockwise rotation rule

The rule for rotating a point \((x,y)\) \(90^\circ\) clockwise about the origin is \((x,y)\to(y,-x)\).
\((x,y)\to(\boldsymbol{y},\boldsymbol{-x})\)

1. Rotation \(90^\circ\) counterclockwise:

Step1: Recall \(90^\circ\) counterclockwise rotation rule

The rule for rotating a point \((x,y)\) \(90^\circ\) counterclockwise about the origin is \((x,y)\to(-y,x)\).
\((x,y)\to(\boldsymbol{-y},\boldsymbol{x})\)

Part B: Practice with Points

1. Translate \(A(-2,5)\) right 3, down 4:

Step1: Apply translation rule

For a translation right 3 (so \(h = 3\)) and down 4 (so \(k=- 4\)), we use the rule \((x,y)\to(x + 3,y-4)\).
Substitute \(x=-2\) and \(y = 5\):
\(x+3=-2 + 3=1\)
\(y - 4=5-4 = 1\)
So the new point is \((1,1)\).

1. Reflect \(B(4,-1)\) across the \(y\)-axis:

Step1: Apply reflection over \(y\)-axis rule

The rule for reflection over \(y\)-axis is \((x,y)\to(-x,y)\).
Substitute \(x = 4\) and \(y=-1\):
\(-x=-4\)
\(y=-1\)
So the new point is \((-4,-1)\).

1. Rotate \(C(3,6)\) \(90^\circ\) clockwise:

Step1: Apply \(90^\circ\) clockwise rotation rule

The rule is \((x,y)\to(y,-x)\).
Substitute \(x = 3\) and \(y = 6\):
\(y=6\)
\(-x=-3\)
So the new point is \((6,-3)\).

1. Rotate \(D(-5,-2)\) \(90^\circ\) counterclockwise:

Step1: Apply \(90^\circ\) counterclockwise rotation rule

The rule is \((x,y)\to(-y,x)\).
Substitute \(x=-5\) and \(y=-2\):
\(-y=-(-2)=2\)
\(x=-5\)
So the new point is \((2,-5)\).

1. Plot and label \(A(4,2)\), \(B(-3,5)\), \(C(-6,-4)\):

To plot these points:

• For \(A(4,2)\): Move 4 units to the right on the \(x\)-axis and 2 units up on the \(y\)-axis.
• For \(B(-3,5)\): Move 3 units to the left on the \(x\)-axis and 5 units up on the \(y\)-axis.
• For \(C(-6,-4)\): Move 6 units to the left on the \(x\)-axis and 4 units down on the \(y\)-axis. Then label each point with their respective letters.

Part A Answers:

1. \(h\), \(k\) (or specific shift amounts, but in general form \(h\) and \(k\))
2. \(-y\)
3. \(-x\)
4. \(y\), \(-x\)
5. \(-y\), \(x\)

Part B Answers:

1. \((1,1)\)
2. \((-4,-1)\)
3. \((6,-3)\)
4. \((2,-5)\)
5. Points plotted as described (visual plot not shown here, but coordinates are \(A(4,2)\), \(B(-3,5)\), \(C(-6,-4)\))

Answer:

Part A: Quick Review

1. Translation:

Step1: Recall translation rule

For a translation, moving right/left affects the \(x\)-coordinate (right is \(+\), left is \(-\)) and moving up/down affects the \(y\)-coordinate (up is \(+\), down is \(-\)). The general rule is \((x,y)\to(x + h,y + k)\), where \(h\) is the horizontal shift and \(k\) is the vertical shift.
So the blanks are \(h\) and \(k\) (or in the context of a general translation, we can say the first blank is the horizontal shift amount and the second is the vertical shift amount. If we consider a specific translation, but since it's a general rule, we use \(h\) and \(k\)).
\((x,y)\to(x + \boldsymbol{h},y + \boldsymbol{k})\)

1. Reflection over \(x\)-axis:

Step1: Recall reflection over \(x\)-axis rule

When reflecting a point \((x,y)\) over the \(x\)-axis, the \(x\)-coordinate remains the same and the \(y\)-coordinate changes sign. So the rule is \((x,y)\to(x,-y)\).
\((x,y)\to(x,\boldsymbol{-y})\)

1. Reflection over \(y\)-axis:

Step1: Recall reflection over \(y\)-axis rule

When reflecting a point \((x,y)\) over the \(y\)-axis, the \(y\)-coordinate remains the same and the \(x\)-coordinate changes sign. So the rule is \((x,y)\to(-x,y)\).
\((x,y)\to(\boldsymbol{-x},y)\)

1. Rotation \(90^\circ\) clockwise:

Step1: Recall \(90^\circ\) clockwise rotation rule

The rule for rotating a point \((x,y)\) \(90^\circ\) clockwise about the origin is \((x,y)\to(y,-x)\).
\((x,y)\to(\boldsymbol{y},\boldsymbol{-x})\)

1. Rotation \(90^\circ\) counterclockwise:

Step1: Recall \(90^\circ\) counterclockwise rotation rule

The rule for rotating a point \((x,y)\) \(90^\circ\) counterclockwise about the origin is \((x,y)\to(-y,x)\).
\((x,y)\to(\boldsymbol{-y},\boldsymbol{x})\)

Part B: Practice with Points

1. Translate \(A(-2,5)\) right 3, down 4:

Step1: Apply translation rule

For a translation right 3 (so \(h = 3\)) and down 4 (so \(k=- 4\)), we use the rule \((x,y)\to(x + 3,y-4)\).
Substitute \(x=-2\) and \(y = 5\):
\(x+3=-2 + 3=1\)
\(y - 4=5-4 = 1\)
So the new point is \((1,1)\).

1. Reflect \(B(4,-1)\) across the \(y\)-axis:

Step1: Apply reflection over \(y\)-axis rule

The rule for reflection over \(y\)-axis is \((x,y)\to(-x,y)\).
Substitute \(x = 4\) and \(y=-1\):
\(-x=-4\)
\(y=-1\)
So the new point is \((-4,-1)\).

1. Rotate \(C(3,6)\) \(90^\circ\) clockwise:

Step1: Apply \(90^\circ\) clockwise rotation rule

The rule is \((x,y)\to(y,-x)\).
Substitute \(x = 3\) and \(y = 6\):
\(y=6\)
\(-x=-3\)
So the new point is \((6,-3)\).

1. Rotate \(D(-5,-2)\) \(90^\circ\) counterclockwise:

Step1: Apply \(90^\circ\) counterclockwise rotation rule

The rule is \((x,y)\to(-y,x)\).
Substitute \(x=-5\) and \(y=-2\):
\(-y=-(-2)=2\)
\(x=-5\)
So the new point is \((2,-5)\).

1. Plot and label \(A(4,2)\), \(B(-3,5)\), \(C(-6,-4)\):

To plot these points:

• For \(A(4,2)\): Move 4 units to the right on the \(x\)-axis and 2 units up on the \(y\)-axis.
• For \(B(-3,5)\): Move 3 units to the left on the \(x\)-axis and 5 units up on the \(y\)-axis.
• For \(C(-6,-4)\): Move 6 units to the left on the \(x\)-axis and 4 units down on the \(y\)-axis. Then label each point with their respective letters.

Part A Answers:

1. \(h\), \(k\) (or specific shift amounts, but in general form \(h\) and \(k\))
2. \(-y\)
3. \(-x\)
4. \(y\), \(-x\)
5. \(-y\), \(x\)

Part B Answers:

1. \((1,1)\)
2. \((-4,-1)\)
3. \((6,-3)\)
4. \((2,-5)\)
5. Points plotted as described (visual plot not shown here, but coordinates are \(A(4,2)\), \(B(-3,5)\), \(C(-6,-4)\))