Question

true or false! a) the translation rule (x,y)→(x - 3,y + 8) means the figure will be translated to the left 3 units and up 8 units. b) the translation rule (x,y)→(x + 7,y - 9) means the figure will be translated to the left 7 units and up 9 units. c) the translation rule (x,y)→(x - 1,y + 1) means the figure will be translated to the left 1 unit and up 1 unit. d) the translation rule (x,y)→(x + 4,y - 7) means the figure will be translated to the right 4 units and down 7 units. e) the translation rule (x,y)→(x - 2,y + 7) means the figure will be translated to the right 7 units and down 2 units. f) the rule for a reflection over the x - axis is (x,y)→(-x,y) g) the rule for a reflection over the y - axis is (x,y)→(-x,y) h) the rule for a reflection over the line y = x is (x,y)→(-y,x) i) the rule for a reflection over the line y = -x is (x,y)→(-y,-x) j) the rule for a rotation 90° clockwise around the origin is (x,y)→(y,-x) k) the rule for a rotation 270° clockwise around the origin is (x,y)→(-y,x) multiple choice: the vertices of △abc are a(-4,2), b(6,6), c(2,7). a translation maps point a to the point a(6,-3). if b and c are mapped by the same translation, what are the coordinates of b and c? a. b(-4,11) and c(-8,-12) b. b(8,5) and c(4,6) c. b(12,3) and c(8,4) d. b(16,1) and c(12,2) if △abc is reflected over the line y = x with coordinates a(-3,2), b(4,-1), c(2,-3), what are the coordinates of △abc? a. a(2,1), b(4,1), c(2,3) b. a(-1,2), b(-1,4), c(-3,2) c. a(-2,1), b(-4,1), c(-2,3) d. a(-2,-1), b(-4,-1), c(-2,-3)

Explanation:

Step1: Determine translation rules

In a translation $(x,y)\to(x + a,y + b)$, $a$ represents horizontal shift (positive is right, negative is left) and $b$ represents vertical shift (positive is up, negative is down).
For $(x,y)\to(x - 3,y + 8)$, it's left 3 and up 8, so a) is True.
For $(x,y)\to(x + 7,y - 9)$, it's right 7 and down 9, so b) is False.
For $(x,y)\to(x - 1,y + 1)$, it's left 1 and up 1, so c) is True.
For $(x,y)\to(x + 4,y - 7)$, it's right 4 and down 7, so d) is True.
For $(x,y)\to(x - 2,y + 7)$, it's left 2 and up 7, so e) is False.

Step2: Determine reflection rules

Reflection over x - axis: $(x,y)\to(x,-y)$, so f) is False.
Reflection over y - axis: $(x,y)\to(-x,y)$, so g) is True.
Reflection over $y = x$: $(x,y)\to(y,x)$, so h) is False.
Reflection over $y=-x$: $(x,y)\to(-y,-x)$, so i) is False.

Step3: Solve multiple - choice for translation

Find translation rule from $A(-4,2)$ to $A'(6,-3)$.
$x$ - translation: $6-(-4)=10$, $y$ - translation: $-3 - 2=-5$.
For $B(6,6)$, $B'$ has $x$ - coordinate $6 + 10=16$ and $y$ - coordinate $6-5 = 1$.
For $C(2,7)$, $C'$ has $x$ - coordinate $2 + 10=12$ and $y$ - coordinate $7-5 = 2$. So the answer for the translation multiple - choice is D.

Step4: Solve multiple - choice for reflection

Reflection over $y = x$ has the rule $(x,y)\to(y,x)$.
If $A(-4,2)$, $A'$ is $(2,-4)$.
If $B(4,-1)$, $B'$ is $(-1,4)$.
If $C(2,-3)$, $C'$ is $(-3,2)$. So the answer for the reflection multiple - choice is B.

Answer:

a) True
b) False
c) True
d) True
e) False
f) False
g) True
h) False
i) False
Multiple Choice (translation): D. B'(16,1) and C'(12,2)
Multiple Choice (reflection): B. A'(-1,2), B'(-1,4), C'(-3,2)