I have two questions, please help.

1. Determine the quadratic function f whose vertex is (2,-1) and passes through (1,1).
f(x)=?
type in form f(x)=ax^2+bx+c
**pleasae show work**

2. Describe for the graph of g(x)=3square root x+2 can be obtained from the graph of f(x)=3square root x
a. shift graph 2 units right
b. shift graph 2 units down
c. shift graph 2 units up
d. shift graph 2 units left

given the vertex,

y = a(x-2)^2 - 1
given the point,
1 = a(1-2)^2 - 1
1 = a - 1
a = 2
so,
y = 2(x-2)^2 - 1
y = 2x^2 - 8x + 7

g(x) = f(x+2), so
(D)

Sure, I'll help you with both questions. Let's start with the first one.

1. Determine the quadratic function f whose vertex is (2,-1) and passes through (1,1).

To find the quadratic function, we need to use the vertex form of a quadratic equation, which is given by:

f(x) = a(x - h)^2 + k

where (h, k) represents the vertex.

Given that the vertex is (2,-1), we substitute h = 2 and k = -1 into the equation:

f(x) = a(x - 2)^2 - 1

Next, we need to find the value of 'a'. Since the function also passes through the point (1,1), we can substitute x = 1 and f(x) = 1 into the equation:

1 = a(1 - 2)^2 - 1
1 = a(-1)^2 - 1
1 = a - 1

Adding 1 to both sides of the equation, we get:

2 = a

Therefore, the value of 'a' is 2, and substituting it back into the equation:

f(x) = 2(x - 2)^2 - 1

So, the quadratic function f(x) = 2x^2 - 8x + 7.

Moving on to the second question:

2. Describe how the graph of g(x) = 3√x + 2 can be obtained from the graph of f(x) = 3√x.

The function g(x) = 3√x + 2 can be obtained by applying transformations on the graph of f(x) = 3√x.

a. Shift the graph 2 units right:
To shift the graph 2 units to the right, we replace 'x' in f(x) with '(x - 2)'. So, we have f(x - 2):

f(x - 2) = 3√(x - 2)

b. Shift the graph 2 units down:
To shift the graph 2 units down, we subtract 2 from the entire function. So, we have f(x) - 2:

f(x) - 2 = 3√x - 2

c. Shift the graph 2 units up:
To shift the graph 2 units up, we add 2 to the entire function. So, we have f(x) + 2:

f(x) + 2 = 3√x + 2

d. Shift the graph 2 units left:
To shift the graph 2 units to the left, we replace 'x' in f(x) with '(x + 2)'. So, we have f(x + 2):

f(x + 2) = 3√(x + 2)

Therefore, after applying the transformations, the graph of g(x) = 3√x + 2 can be obtained.

I hope this helps! Let me know if you have any other questions.