Function shifts

Vertical Shifts

Vertical function shifts are highly intuitive. They occur from either a coefficient placed in front of the variable(s) in the function or because of a constant term added to the function equation.

Sample Function: y = 2(x+1)² − 5

Here the function is shifted down by 5 units, and is multiplied by a factor of 2. The directions of these changes are intuitive, as they are the same size and direction as the numbers in the equation.

Horizontal Shifts

Horizontal shifts can be a bit more counterintuitive. For example, in the sample equation above the x variable has a + 1 next to it. Following the same rule as for vertical shifts, it would make sense that the x-intercept would then move one unit in the positive direction on the x-axis (to the right).

However, it’s actually the opposite. (X+1) means a horizontal shift one unit to the left. This is also true for leading coefficients. (2X) does not expand a graph in the horizontal direction, it shrinks it.