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)?

shift left: x -> x+4

stretch: y -> 3y
shift up: y ->y+5

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

To find the function rule for g(x), we need to understand each transformation step by step.

Step 1: Shift 4 units left:
This means that the graph is shifted horizontally by 4 units to the left. To achieve this, we replace the "x" in the original function with "(x + 4)". So the new function becomes f(x) = (x + 4)^2.

Step 2: Vertically stretch by a factor of 3:
A vertical stretch multiplies the y-values of the function by a constant factor. In this case, the factor is 3. To perform the vertical stretch, we multiply the entire function by 3. The new function becomes f(x) = 3(x + 4)^2.

Step 3: Shift 5 units up:
This means that the graph is shifted vertically by 5 units upwards. To achieve this, we add 5 to the function. The new function becomes g(x) = 3(x + 4)^2 + 5.

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