2.5 Horizontal and Vertical Shifts

Required Videos

Here are the required notes for your convenience.


Supplemental Videos

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)