Question

1. find the image of c(2,3), after the given transformation.

(a) reflected over the y - axis.
(b) reflected over y = x.
(c) rotated 90° about the origin.
(d) rotated 270° about the origin.
(e) translated down 3, left 4.
(f) translated up 4.

1. reflect a\b\c\ across x = 3

match the vocabulary term with its definition.
a. the resulting figure after a transformation

Explanation:

1. For question 11:

• Explanation:
• Step 1: Reflection over y - axis rule
• The rule for reflecting a point \((x,y)\) over the \(y\) - axis is \((x,y)\to(-x,y)\). For the point \(C(2,3)\), when reflected over the \(y\) - axis, \(x = 2\) becomes \(-2\) and \(y = 3\) remains the same. So the new point is \((-2,3)\).
• Step 2: No further transformation in this case
• Since there are no other transformations specified after the reflection over the \(y\) - axis.
• Answer: \((-2,3)\)

1. For question 10 (assuming we are just dealing with the transformation of a general point or figure):

• Explanation:
• Step 1: 90 - degree counter - clockwise rotation rule about the origin
• The rule for rotating a point \((x,y)\) 90 - degrees counter - clockwise about the origin is \((x,y)\to(-y,x)\).
• Step 2: Translation down 5 units rule
• After rotation, if we have a point \((x_1,y_1)\) and we translate it down 5 units, the rule is \((x_1,y_1)\to(x_1,y_1 - 5)\). Let's assume the original point is \((x,y)\), after 90 - degree counter - clockwise rotation it becomes \((-y,x)\), and after translation down 5 units, it becomes \((-y,x - 5)\).
• Answer: If the original point is \((x,y)\), the final point is \((-y,x - 5)\)

For the vocabulary - matching part:

• Brief Explanations:
• (a) “reflected over the \(y\) - axis” means that for each point \((x,y)\) in the pre - image, the new point is \((-x,y)\).
• (b) “reflected over \(y=x\)” means that for each point \((x,y)\) in the pre - image, the new point is \((y,x)\).
• (c) “rotated 90° about the origin” for a point \((x,y)\) gives the new point \((-y,x)\) (counter - clockwise).
• (d) “rotated 270° about the origin” is equivalent to a 90° clockwise rotation. For a point \((x,y)\), the new point is \((y,-x)\) (since 270° counter - clockwise rotation about the origin is the same as 90° clockwise rotation).
• (e) “translated down 3, left 4” means for a point \((x,y)\), the new point is \((x - 4,y - 3)\).
• (f) “translated up 4” means for a point \((x,y)\), the new point is \((x,y + 4)\).
• Answer:
• (a) Matches with the description of reflecting a point over the \(y\) - axis.
• (b) Matches with the description of reflecting a point over the line \(y = x\).
• (c) Matches with the 90° counter - clockwise rotation about the origin.
• (d) Matches with the 270° counter - clockwise (or 90° clockwise) rotation about the origin.
• (e) Matches with the translation of a point down 3 units and left 4 units.
• (f) Matches with the translation of a point up 4 units.

Answer:

1. For question 11:

• Explanation:
• Step 1: Reflection over y - axis rule
• The rule for reflecting a point \((x,y)\) over the \(y\) - axis is \((x,y)\to(-x,y)\). For the point \(C(2,3)\), when reflected over the \(y\) - axis, \(x = 2\) becomes \(-2\) and \(y = 3\) remains the same. So the new point is \((-2,3)\).
• Step 2: No further transformation in this case
• Since there are no other transformations specified after the reflection over the \(y\) - axis.
• Answer: \((-2,3)\)

1. For question 10 (assuming we are just dealing with the transformation of a general point or figure):

• Explanation:
• Step 1: 90 - degree counter - clockwise rotation rule about the origin
• The rule for rotating a point \((x,y)\) 90 - degrees counter - clockwise about the origin is \((x,y)\to(-y,x)\).
• Step 2: Translation down 5 units rule
• After rotation, if we have a point \((x_1,y_1)\) and we translate it down 5 units, the rule is \((x_1,y_1)\to(x_1,y_1 - 5)\). Let's assume the original point is \((x,y)\), after 90 - degree counter - clockwise rotation it becomes \((-y,x)\), and after translation down 5 units, it becomes \((-y,x - 5)\).
• Answer: If the original point is \((x,y)\), the final point is \((-y,x - 5)\)

For the vocabulary - matching part:

• Brief Explanations:
• (a) “reflected over the \(y\) - axis” means that for each point \((x,y)\) in the pre - image, the new point is \((-x,y)\).
• (b) “reflected over \(y=x\)” means that for each point \((x,y)\) in the pre - image, the new point is \((y,x)\).
• (c) “rotated 90° about the origin” for a point \((x,y)\) gives the new point \((-y,x)\) (counter - clockwise).
• (d) “rotated 270° about the origin” is equivalent to a 90° clockwise rotation. For a point \((x,y)\), the new point is \((y,-x)\) (since 270° counter - clockwise rotation about the origin is the same as 90° clockwise rotation).
• (e) “translated down 3, left 4” means for a point \((x,y)\), the new point is \((x - 4,y - 3)\).
• (f) “translated up 4” means for a point \((x,y)\), the new point is \((x,y + 4)\).
• Answer:
• (a) Matches with the description of reflecting a point over the \(y\) - axis.
• (b) Matches with the description of reflecting a point over the line \(y = x\).
• (c) Matches with the 90° counter - clockwise rotation about the origin.
• (d) Matches with the 270° counter - clockwise (or 90° clockwise) rotation about the origin.
• (e) Matches with the translation of a point down 3 units and left 4 units.
• (f) Matches with the translation of a point up 4 units.