Answer:
C
Step-by-step explanation:
given a function f(x) then
f(x) + c ← denotes a vertical translation of f(x)
• If c > 0 then shift vertically up
• If c < 0 then shift vertically down
Here the vertex of f(x) is at the origin and the vertex of g(x) is (0, 3 ), that is 3 units vertically up
Hence g(x) = x² + 3 → C
The graph that shows the same shape is:
C. [tex]g(x)=x^2+3[/tex]
We are given the parent function f(x) as:
[tex]f(x)=x^2[/tex]
Now, after looking at the graph we see that the graph is a translation of a function f(x) 3 units upward.
since, the vertex of graph is at (0,3).
Now, we will check which graph passes through the point (0,3)
A)
[tex]g(x)=x^2-3[/tex]
at x=0 we have:
[tex]g(x)=-3\neq 3[/tex]
Hence, option: A is incorrect.
B)
[tex]g(x)=(x+3)^2[/tex]
when x=0 we have:
[tex]g(x)=(3)^2=9\neq 3[/tex]
Hence, option: B is incorrect.
D)
[tex]g(x)=(x-3)^2[/tex]
when x=0 we have:
[tex]g(x)=(-3)^2=9\neq 3[/tex]
Hence, option: D is incorrect.
Hence, we are left with option: C
[tex]g(x)=x^2+3[/tex]
It passes through (0,3).
Also, we know that the translation g(x) of a parent function f(x) k units upward is given by:
[tex]g(x)=f(x)+k[/tex]
Here k=3
Hence,
[tex]g(x)=x^2+3[/tex]
Option: C is correct.
