Transformations

I. Parent Functions

𝑦=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑦=𝑥

𝑦= 𝑥 2

𝑓 𝑥 = 1 𝑥

𝑦= 3 𝑥

𝑦= 𝑥 3

𝑓 𝑥 = 𝑥

𝑦= 𝑥

𝑓 𝑥 =log⁡(𝑥)

Transformations address how the changes in a function’s equation, result in changes to its graph.

II. Vertical Transformations

𝑓 𝑥

𝑓 𝑥 +𝑘 𝑓 𝑥 𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑢𝑝 𝑘 𝑢𝑛𝑖𝑡𝑠.

𝑘 𝑓 𝑥 +𝑘

𝑓 𝑥 −𝑘 𝑓 𝑥 𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑑𝑜𝑤𝑛 𝑘 𝑢𝑛𝑖𝑡𝑠.

𝑘 𝑓 𝑥 −𝑘

𝑓 𝑥 = 𝑥

𝑓 𝑥 = 𝑥 +3

𝑓 𝑥 = 𝑥 −4 𝑓 𝑥 = 𝑥

𝑓 𝑥 = 𝑥

𝑓 𝑥 = 𝑥 +2 𝑓 𝑥 = 𝑥

𝑓 𝑥 = 𝑥

𝑓 𝑥 = 𝑥 −4 𝑓 𝑥 = 𝑥

III. Horizontal Transformations

𝑓 𝑥

𝑓 𝑥+ℎ 𝑓 𝑥 𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑙𝑒𝑓𝑡 ℎ 𝑢𝑛𝑖𝑡𝑠.

ℎ 𝑓 𝑥+ℎ

𝑓 𝑥−ℎ 𝑓 𝑥 𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑟𝑖𝑔ℎ𝑡 ℎ 𝑢𝑛𝑖𝑡𝑠.

ℎ 𝑓 𝑥−ℎ

𝑓 𝑥 = 3 𝑥

𝑓 𝑥 = 3 𝑥−3 𝑓 𝑥 = 3 𝑥

𝑓 𝑥 = 1 𝑥

4 𝑓 𝑥 = 1 𝑥+4 𝑓 𝑥 = 1 𝑥

IV. Reflection

𝑓 𝑥

−𝑓 𝑥 𝑓 𝑥 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑜𝑣𝑒𝑟 𝑡ℎ𝑒 𝑥−𝑎𝑥𝑖𝑠.

−𝑓 𝑥 𝑓 𝑥

𝑓 −𝑥 𝑓 𝑥 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑜𝑣𝑒𝑟 𝑡ℎ𝑒 𝑦−𝑎𝑥𝑖𝑠.

𝑓 𝑥

𝑓 𝑥 = 𝑥 2

𝑓 𝑥 = −𝑥 2 𝑓 𝑥 = 𝑥 2

𝑓 𝑥 =log⁡(𝑥)

𝑓 𝑥 =log⁡(−𝑥) 𝑓 𝑥 =log⁡(𝑥)