how do I solve this quadratic transformation question?

Question

Describe a function  in terms of  if the graph of  is obtained by vertically stretching  by a factor of 6, then shifting the graph of  to the right 4 units and upward 7 units.

Answer 1

We want to start with the parent function for any quadratic, f(x) = x².

The transformation of a quadratic function is g(x) = a(x - h)² + k.

1.) The graph of g(x) is translating 4 units to the right and upward 7 units. That means h = 4 and k = 7. I think of the mnemonic device: "opposite h, same k" anytime when I describe a transformation of a function.

(x - 4)² + 7

2.) The graph of g(x) is vertically stretched by a factor of 6. That means a = 6. The parabola or U-shaped curve is very steep upward.

6x²

Once the transformation is done, the graph of g(x) will look like this.

g(x) = 6(x - 4)² + 7