Section 3.5 – Transformation of Functions

Symmetry Symmetric with respect to an axis: You can fold a graph along an axis and the graph will fall on top of itself. Each part of the graph is covered by its image. Symmetric with respect to the origin: Rotating a graph 180⁰ about the origin results in the original graph. You can also fold along the x-axis AND the y-axis and the graph will fall on top of itself.

Symmetric with respect to 𝑥−axis   If the graph is symmetric with respect to the x-axis, the x-value stays the same!

Symmetric with respect to 𝑦−axis   If the graph is symmetric with respect to the y-axis, the y-value stays the same!

Symmetric With Respect to the Origin   If the graph is symmetric with respect to the origin, NOTHING stays the same!

Even and Odd Functions     Functions CAN’T be BOTH even and odd! They may be neither!

Even and Odd Functions Determine whether the function is even, odd, or neither even nor odd.   NOT EVEN   NOT ODD NEITHER

Even and Odd Functions Determine whether the function is even, odd, or neither even nor odd.   NOT EVEN   ODD

Horizontal Translations 𝑓 𝑥 = 𝑥 2 𝑓 𝑥 = (𝑥+5) 2 𝑓 𝑥 = (𝑥 −3) 2 𝑓 𝑥 = 𝑥 2 𝑓 𝑥 = (𝑥+5) 2 𝑓 𝑥 = (𝑥 −3) 2

Horizontal Translations 𝑓 𝑥 = 𝑥 3 𝑓 𝑥 = (𝑥+2) 3 𝑓 𝑥 = (𝑥 −4) 3 𝑓 𝑥 = 𝑥 3 𝑓 𝑥 = (𝑥+2) 3 𝑓 𝑥 = (𝑥 −4) 3

Horizontal Translations 𝑓 𝑥 = 𝑥 𝒇 𝒙 = 𝒙−𝟏 𝒇 𝒙 = 𝒙+𝟑 𝒇 𝒙 = 𝒙 𝒇 𝒙 = 𝒙−𝟏 𝒇 𝒙 = 𝒙+𝟑

Horizontal Translations  

VERTICAL Translations 𝑓 𝑥 = 3 𝑥 𝑓 𝑥 = 3 𝑥 +3 𝑓 𝑥 = 3 𝑥 −5 𝑓 𝑥 = 3 𝑥 𝑓 𝑥 = 3 𝑥 +3 𝑓 𝑥 = 3 𝑥 −5

VERTICAL Translations 𝑓 𝑥 = 1 𝑥 𝑓 𝑥 = 1 𝑥 −3 𝑓 𝑥 = 1 𝑥 +1 𝑓 𝑥 = 1 𝑥 𝑓 𝑥 = 1 𝑥 −3 𝑓 𝑥 = 1 𝑥 +1

VERTICAL Translations  

REFLECTIONS ACROSS THE 𝑥−AXIS 𝑓 𝑥 = 1 𝑥 2 𝑓 𝑥 =− 1 𝑥 2 𝑓 𝑥 = 1 𝑥 2 𝑓 𝑥 =− 1 𝑥 2

REFLECTIONS ACROSS THE 𝑥−AXIS 𝑓 𝑥 = 3 𝑥 𝑓 𝑥 =− 3 𝑥 𝑓 𝑥 = 3 𝑥 𝑓 𝑥 =− 3 𝑥

REFLECTIONS ACROSS THE 𝑥−AXIS  

VERTICAL STRETCHING OR SHRINKING 𝑓 𝑥 = 𝑥 𝑓 𝑥 =6 𝑥 𝑓 𝑥 = 1 3 𝑥 𝑓 𝑥 = 𝑥 𝑓 𝑥 =6 𝑥 𝑓 𝑥 = 1 3 𝑥

VERTICAL STRETCHING OR SHRINKING 𝑓 𝑥 = 𝑥 𝑓 𝑥 = 1 2 𝑥 𝑓 𝑥 =4 𝑥 𝑓 𝑥 = 𝑥 𝑓 𝑥 = 1 2 𝑥 𝑓 𝑥 =4 𝑥

VERTICAL STRETCHING OR SHRINKING  

Transformations Describe the transformations associated with the function and then graph the function. 𝑓 𝑥 = (𝑥+1) 2 −5 Basic Function: 𝑓 𝑥 = 𝑥 2 Shift 1 unit to the left Shift down 5 units

Transformations Describe the transformations associated with the function and then graph the function. 𝑓 𝑥 =− 3 𝑥−4 +1 Basic Function: 𝑓 𝑥 = 3 𝑥 Shift 4 units to the right Reflect over 𝑥−axis Shift up 1 unit

Transformations