If ƒ(x) = x + a, where a is a positive integer, write the equation of the new function if ƒ(x) is translated (1) up a units. (2) down a units. (3) up 4 units. (4) down 1 units.

The graphs of the functions can be transformed, obtaining related functions.We consider two kinds of transformations.
 Vertical and horizontal displacements: It is assumed that c is a positive
 constant, ie c> 0 and y = f (x) then
 y = f (x) + c moves the graph c units up.
 y = f (x) - c moves the graph c units down.
 y = f (x + c) shifts the graph c units to the right.
 y = f (x - c) shifts the graph c units to the left

 (1) UP a units.
 ƒ (x) = x + a
 new function
 g (x) = x + a + a
 g (x) = x + 2a

 (2) DOWN a units.
 ƒ (x) = x + a
 new function
 g (x) = x + a - a
 g (x) = x

 (3) UP 4 units.
 ƒ (x) = x + a
 new function
 g (x) = x + a + 4

 (4) DOWN 1 units
 ƒ (x) = x + a
 new function
 g (x) = x + a - 1

When rates are expressed as a quantity of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates. if you have a multiple-unit rate such as 120 students for every 3 buses, and want to find the single-unit rate, write a ratio equal to the multiple-unit rate with 1 as the second term.