What is the equation of the graph obtained when the parent graph y equals x3 is translated 4 units left and 7 units down?

Answer 1

y = x3

After translating 4 units left and 7 units down, this function would become:

y = (x+4)3-7

In single-variable mathematical functions, vertical translations are always achieved by simply adding or subtracting a number constant from the original function. Addition causes upward shift, subtraction causes downward shift.

Horizontal translations for simple functions are achieved by adding or subtracting number constants within the argument of the function in question. Addition causes leftward shift, subtraction causes rightward shift.

Vertical translations are easy to comprehend, horizontal translations are not always so easy. Some more examples:

y = sin(x) shifted pi units to the left is y = sin(x+(pi)). [this is also equal to y = cos(x)]

y = ln(x) shifted 5 units right is y = ln(x-5)

Take note: for more complex functions or polynomials not in vertex form, it is not always so simple. For example, a function like:

y = ln(sin(x)-cos(x))3

cannot have horizontal shifts so easily applied to it. Vertical shifting is still the same for any function regardless of complexity, so this function could still be vertically shifted by adding or subtracting number constants. I don't know off the top of my head how exactly to precisely shift this function horizontally, but I would recommend messing around on a graphing calculator to really gain a good knowledge of how changing certain parts of functions affects their appearance.