Bellringer August 23rd Graph the following equations 𝑦= 𝑥 𝑦= 𝑥 2 𝑦= 𝑥

Graphing techniques: Transformations MAFS.912.F-BF.1.1

Transformations on Functions consist of being ________, ___________________, or reflected. Shifted stretched /compressed

Assuming c is a positive constant Adding/Subtracting a constant ___________ the function corresponds to a __________ shift that goes ____________________. To Graph: Shift graph 𝑓 𝑥 +𝑐 c units _________ 𝑓 𝑥 +𝑐 c units _________ outside vertical With the sign up down

Assuming c is a positive constant Adding/Subtracting a constant ___________ the function corresponds to a __________ shift that goes ____________________. To Graph: Shift graph 𝑓 𝑥+𝑐 c units _________ 𝑓 𝑥+𝑐 c units _________ inside horizontal opposite the sign left right

Examples: Given the Parent function 𝑓 𝑥 = 𝑥 2 describe the shifts for the following functions 𝑔 𝑥 = 𝑥 2 +6 ____________ ℎ 𝑥 = (𝑥−3) 2 ____________ 𝐹 𝑥 = (𝑥+4) 2 −2 ____________ Up 6 Right 3 Left 4 down 2

Horizontal and Vertical Shifts and Changes in the Domain and Range As a function is transformed the ___________ and ____________ may also change. Consider the following: When 𝑓 𝑥 = 𝑥 the domain: _______ range: _______ However, for 𝑓 𝑥+1 = 𝑥+1 the Domain: _______ range: _______ Domain Range [0, ∞) [0, ∞) [−1, ∞) [0, ∞)

Reflections The graph of ______ is obtained by reflecting the graph of 𝑓(𝑥) about the _____. −𝑓(𝑥) 𝑥−𝑎𝑥𝑖𝑠 𝑓(−𝑥) y−𝑎𝑥𝑖𝑠

Examples: Graph the following reflections and shifts 𝑓 𝑥 = 𝑥 ________________ 𝑓 𝑥+2 = 𝑥+2 ________________ 𝑓 −𝑥+2 = −𝑥+2 ________________ 𝑓 −𝑥+2 +1= 2−𝑥 +1 ________________ Start with the square root function Shift left 2 units Reflect over the y axis Shift up 1 unit

Stretching and compressing the graph Horizontal: The graph of 𝑓(𝑐𝑥) is found by: Horizontally _____________ the graph of 𝑓(𝑥) if __________ Vertical: The graph of 𝑐𝑓(𝑥) is found by: Vertically _____________ the graph of 𝑓(𝑥) if __________ Note: c is any positive real number. stretching 0<𝑐<1 compressing 𝑐>1 stretching 𝑐>1 compressing 0<𝑐<1

Sketching the graph of a function using multiple Transformations Graph the following : 𝑓 𝑥 = 𝑥 2 𝑓 𝑥−3 =___________ 2𝑓 𝑥−3 =___________ −2𝑓 𝑥−3 =__________ Right 3 units Vert stretch by 2 Reflect over x-axis

Homework Due August 24th TB p 274 2-24 even,28, 32, 34, 36, 50, 54, 58, 62, 66, 70, 74 (23 problems total USE GRAPH PAPER)