Final answer:
The equation of the new function after applying the described transformations to the parent function y = 2 is not represented by any of the available options (A-D). The correct equation after a vertical stretch and translation is f(x) = 9.
Step-by-step explanation:
The student is asking for the equation that results from applying several transformations to the parent function y = 2. To achieve this, we must apply a vertical stretch by a factor of 2, horizontal translation left by 4 units, and a vertical translation up by 5 units.
The vertical stretch by a factor of 2 will multiply the function by 2. Since y = 2 is a horizontal line, stretching vertically will not change its shape or position, but the y-value should be multiplied by 2, giving us y = 4.
Next, we horizontally translate the function left by 4 units. In algebra, moving a function left by d units is represented by f(x + d). Since our original function is y = 4 after stretching, this translation will create f(x) = 4 at every point that was f(x + 4) = 4 which is 4 units left of where it was. This transformation does not impact the equation, as the function is still a horizontal line.
Finally, the vertical translation up by 5 units means adding 5 to the y-value of the function. Applying this to the function y = 4 yields the new function f(x) = 4 + 5 which simplifies to f(x) = 9.
Since the only transformations that affect the equation of a horizontal line are vertical translations, and we applied a vertical stretch that did not alter the line's position followed by a vertical translation, the correct equation of the new function is simply f(x) = 9. This means that none of the options A-D provided are correct. So, the student may need to review the question or the options provided.