Function Notation Transformations

𝑓 𝑥 “F of x" 𝑓 𝑥 is a substitute for “y.” 𝑓 𝑥 represents the output when using x as the input. The function described by ƒ(x) = 5x + 3 is the same as the function described by y = 5x + 3.

Evaluating Functions Just plugging it in 𝑓 𝑥 =5𝑥+3 𝑓 3 = Find the following outputs for each function. 𝑓 𝑥 =5𝑥+3 Just plugging it in 𝑓 3 = 5 3 +3=15+3=18 𝑓 0 = 5 0 +3=0+3=3 𝑓 −5 = 5 −5 +3=−25+3=−22

Transforming 𝑓(𝑥) 𝑓 𝑥 +𝑘 Vertical Shift: Up(+) or down(-) k units 𝑓 𝑥+𝑘 Horizontal Shift: Left(+) or right(-) k units 𝑘𝑓 𝑥 Stretch −𝑓 𝑥 Flip

Example Function x y -2 4 -1 1 2 x y -2 9 -1 6 5 1 2 𝑓 𝑥 = 𝑥 2 𝑓 𝑥 +5= 𝑥 2 +5 (Standard Function) (Vertical Shift) x y -2 4 -1 1 2 x y -2 9 -1 6 5 1 2

Example Function x y -7 4 -6 1 -5 -4 -3 x y -2 20 -1 5 1 2 𝑓 𝑥+5 = (𝑥+5) 2 5𝑓 𝑥 = 5𝑥 2 (Horizontal Shift) (Stretch) x y -7 4 -6 1 -5 -4 -3 x y -2 20 -1 5 1 2

Example Function 2 x y -2 2 -1 1 x y -2 -4 -1 1 2 4 𝑓 𝑥 =|𝑥| −2𝑓 𝑥 = −2|𝑥| Standard Function Stretch and Flip x y -2 2 -1 1 x y -2 -4 -1 1 2 4

Example Function 2 x y 1 2 3 4 5 x y -2 -1 -3 1 2 𝑓 𝑥−3 = |𝑥−3| 𝑓 𝑥 −3= 𝑥 −3 Horizontal Shift Vertical Shift x y 1 2 3 4 5 x y -2 -1 -3 1 2

Multiple Shifts x y 1 2 1.4 3 1.7 4 x y -3 -5 -2 -4 -1 -3.6 -3.3 1 𝑥+3 −5 𝑓 𝑥 = 𝑥 𝑓 𝑥+3 −5= Standard Function Horizontal and Vertical Shifts x y 1 2 1.4 3 1.7 4 x y -3 -5 -2 -4 -1 -3.6 -3.3 1

Multiple Shifts x y 2 1 -3 -5.1 3 -6.7 4 -8 −5𝑓 𝑥 +2= −5 𝑥 +2 −5 𝑥 +2 Stretch and Vertical Shift x y 2 1 -3 -5.1 3 -6.7 4 -8