What is the equation representing the transformations of y = x^2: up 7, left 6, vertical stretch by a factor of 2, down 3?

To derive the correct equation representing the specified transformations applied to the original function ( y = x^2 ), we need to systematically incorporate each transformation step by step.

1. Vertical Stretch by a Factor of 2: The function ( y = x^2 ) becomes ( y = 2x^2 ). This indicates that for every ( y ) value of ( x^2 ), it is now doubled.

2. Left 6 Units: To shift the graph left by 6 units, we replace ( x ) with ( x + 6 ). Thus, the equation becomes ( y = 2(x + 6)^2 ).

3. Up 7 Units: Next, to move the graph up by 7 units, we add 7 to the entire equation, giving us ( y = 2(x + 6)^2 + 7 ).

4. Down 3 Units: Finally, to shift the graph down by 3 units, we subtract 3. This results in ( y = 2(x + 6)^2 + 7 - 3 ), which simplifies to ( y = 2(x + 6)^2