The graph of

g(x) is the graph of f (x)= x^2
shifted 4 units left, vertically stretched by a factor of 3, then shifted 5 units up. What is the function rule for
g(x)?

f(x) = x^2

shift left: (x+4)^2
vert stretch: 3(x+4)^2
up: 3(x+4)^2 + 5

Why did the function go to the party? It wanted to have a good "shift" and a vertical "stretch" with its friends!

The function rule for g(x) is g(x) = 3(x + 4)^2 + 5.

To determine the function rule for g(x), we can apply each transformation to the function f(x) = x^2 step-by-step.

1. Shifted 4 units left:
This means that every x-coordinate in the graph of f(x) will decrease by 4. Therefore, we subtract 4 from the x-coordinate: f(x + 4) = (x + 4)^2.

2. Vertically stretched by a factor of 3:
This means that the y-coordinate of each point on the graph will be multiplied by 3. So, we multiply the function by 3: 3f(x + 4) = 3(x + 4)^2.

3. Shifted 5 units up:
This means that every y-coordinate in the graph will increase by 5. Therefore, we add 5 to the function: g(x) = 3(x + 4)^2 + 5.

Thus, the function rule for g(x) is g(x) = 3(x + 4)^2 + 5.

To find the function rule for g(x), we need to apply each transformation to the original function f(x) = x^2.

1. Shifting 4 units left:
To shift the graph 4 units to the left, we replace x with (x + 4) in the original function.

f(x) = x^2 -> g(x) = (x + 4)^2

2. Vertically stretching by a factor of 3:
To vertically stretch the graph by a factor of 3, we multiply the function by 3.

g(x) = 3(x + 4)^2

3. Shifting 5 units up:
To shift the graph 5 units up, we add 5 to the function.

g(x) = 3(x + 4)^2 + 5

Therefore, the function rule for g(x) is g(x) = 3(x + 4)^2 + 5.