Part 4/6:
When we write ( y_3 = f(x + a) ), we are indicating a shift of ( a ) units to the left. The reasoning lies in the requirement for the argument ( (x + a) ) to equal zero; hence, to find the corresponding ( x ) value for the zero point, we must solve ( x + a = 0 ) leading to ( x = -a ).
Shifting Right
On the other hand, ( y_4 = f(x - a) ) results in a shift to the right by ( a ) units. Here, the equation ( (x - a) ) must equal zero, hence solving this results in ( x = a ).
The distinction between shifts is critical: a plus inside the function's argument shifts the graph left, while a minus shifts it right.
Graphical Representation and Examples
To illustrate these transformations, let’s consider the basic function ( f(x) = x^2 ) — a simple parabola.