2.5 Horizontal and Vertical Shifts

Two graphs may look exactly alike in shape, but differ in their positions within the xy-plane.

Adding or subtracting values from a function will shift the graph of the function on the coordinate plane. 

If f is a function and c is a positive constant, then the graph of

  • y = f(x) + c is the graph of y = f(x) shifted up c units
  • y = f(x) - c is the graph of y = f(x) shifted down c units
  • y = f(x + c) is the graph of y = f(x) shifted left c units
  • y = f(x - c) is the graph of y = f(x) shifted right c units

Vertical translations: adding or subtracting after the function (outside the parenthesis)

y = f(x) +3 shifts f(x) up 3 spaces on the y axis

y = f(x) - 2 shifts f(x) down 2 spaces on the y axis

Horizontal translations: adding or subtracting with the function (inside the parenthesis)

y = f(x+4) shifts the graph of f(x) left 4 spaces on the x axis

y = f(x-3) shifts the graph of f(x) right 3 spaces on the x axis

Vertical & Horizontal translation                                                        

y = f(x-5) +2 shifts the graph of f(x) up 2 spaces on the y-axis and right 5 on the x-axis.

If f(5) = 2, f(2) = 7, and f(-3) = -14

Find the coordinates for h(x) = f(x) + 5 and g(x) = f(x – 2)

 f(x)(5,-2)  (2,7)(-3,-14)
f(x)+5   (5,3)(2,12)(-3,-9)
f(x-2)(7,-2)(4,7)(-1,-14)