Question

find the equation of line y=x+1 under each of the following transformations
(a) (x,Y)-(x+3,Y+5)
(b) reflections across the x-axis
(c) reflection across the y-axis
(d) half turn about origin

Answer 1

I would replace the old x and the old y with
(x-3) and (y-5) respectively to get
y-5 = x-3 + 1
y = x + 3

Since you are dealing with straight lines, a good way to understand this would be to take any two points on the original line, and then applying each of the transformations to those points
e.g. (1,2) and (3,4) are on the old line

a)
under the translation
(1,2) ---> (4,7)
(3,4) ---> (6,9)

Notice the slope is still (9-7)/(6-4) = 1
new line: y = x + b
point (4,7) on it
7 = 4 + b
b = 3
new line : y = x + 3 , just like I had

b) using the reflection in x-axis
(1,2) ---> (1,-2)
(3,4) ---> (3,-4)
slope = -2/2 = -1
new equation:
y = -x + b
(1,-2) on it
-2 = 1+b
b = -3
new equation: y = -x - 3

c) do it the same way

d) don't know what you mean by "half-turn"
- a rotation of 180° ?
if so, then (x,y) ---> (-x,-y)