Answer:
Part 1) [tex]y=-x^{2}[/tex] ---> Translated up by 1 units
Part 2) [tex]y=x^{2}+1[/tex] ---> Reflected across the x-axis
Part 3) [tex]y=-(x+1)^{2}-1[/tex] ---> Translated left by 1 unit
Part 4) [tex]y=-(x-1)^{2}-1[/tex] ----> Translated right by 1 unit
Part 5) [tex]y=-x^{2}-1[/tex] ----> Reflected across the y-axis
Part 6) [tex]y=-x^{2}-2[/tex] ----> Translated down by 1 unit
Step-by-step explanation:
we know that
The parent function is
[tex]y=-x^{2}-1[/tex] ----> this is a vertical parabola open downward with vertex at (0,-1)
Calculate each case
Part 1) Translated up by 1 unit
The rule of the translation is
(x,y) -----> (x,y+1)
so
(0,-1) ----> (0,-1+1)
(0,1) ----> (0,0) ----> the new vertex
The new function is equal to
[tex]y=-x^{2}[/tex]
Part 2) Reflected across the x-axis
The rule of the reflection is
(x,y) -----> (x,-y)
so
(0,-1) ----> (0,1) ----> the new vertex
The new function is equal to
[tex]y=x^{2}+1[/tex]
Part 3) Translated left by 1 unit
The rule of the translation is
(x,y) -----> (x-1,y)
so
(0,-1) ----> (0-1,-1)
(0,1) ----> (-1,-1) ----> the new vertex
The new function is equal to
[tex]y=-(x+1)^{2}-1[/tex]
Part 4) Translated right by 1 unit
The rule of the translation is
(x,y) -----> (x+1,y)
so
(0,-1) ----> (0+1,-1)
(0,1) ----> (1,-1) ----> the new vertex
The new function is equal to
[tex]y=-(x-1)^{2}-1[/tex]
Part 5) Reflected across the y-axis
The rule of the reflection is
(x,y) -----> (-x,y)
so
(0,-1) ----> (0,-1) ----> the new vertex
The new function is equal to
[tex]y=-x^{2}-1[/tex]
Part 6) Translated down by 1 unit
The rule of the translation is
(x,y) -----> (x,y-1)
so
(0,-1) ----> (0,-1-1)
(0,1) ----> (0,-2) ----> the new vertex
The new function is equal to
[tex]y=-x^{2}-2[/tex]
