Graphing Logarithmic functions

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Presentation transcript:

Graphing Logarithmic functions Transformations of the Parent Function Graph

𝑦= 𝑙𝑜𝑔 2 𝑥 is the inverse of 𝑦= 2 𝑥

Logarithmic parent graph Always passes through (1,0) and (b, 1) Vertical asymptote x = 0 Domain: x > 0 Range: All real numbers

Transformations of the parent log 𝑦=𝑎• 𝑙𝑜𝑔 𝑏 𝑐 𝑥 −ℎ +𝑘 a<0 reflects over x-axis multiply y-values by a c<0 reflects over y-axis divide x-values by c 𝑥 −ℎ translates right + k translates up 𝑥+ℎ translates left - k translates down

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote:

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph (1,0) (___, 1)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph (1,0) (2, 1)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = (1,0) (2, 1)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = Add h = to x (1,0) (2, 1)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = Add h = to x Add k = to y (1,0) (2, 1)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (2, 1)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (2, 1)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (2, 1)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (2, 1)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (-1, -3) (2, 1)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (-1, -3) (2, 1) (2, -2)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (-1, -3) (2, 1) (2, -2) (0, -2)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (-1, -3) (2, 1) (2, -2) (0, -2) (0, -5)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x ? - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (-1, -3) (2, 1) (2, -2) (0, -2) (0, -5)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x > - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (-1, -3) (2, 1) (2, -2) (0, -2) (0, -5)

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x > - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (-1, -3) (2, 1) (2, -2) (0, -2) (0, -5) Reflects over x-axis, dilation

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x > - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (-1, -3) (2, 1) (2, -2) (0, -2) (0, -5) Reflects over x-axis, dilation, translates left 2 units

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x > - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (-1, -3) (2, 1) (2, -2) (0, -2) (0, -5) Reflects over x-axis, dilation, translates left 2 units, translates down 3 units

Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2 Domain: x > - 2 Range: all real numbers Pts on parent graph Mult. y by a = -2 Add h = to x Add k = -3 to y (1,0) (1, 0) (-1,0) (-1, -3) (2, 1) (2, -2) (0, -2) (0, -5) Reflects over x-axis, dilation, translates left 2 units, translates down 3 units

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote:

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph (1,0) (___ , 1)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph (1,0) (1/3 , 1)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = (1,0) (1/3 , 1)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = Add h = to x (1,0) (1/3 , 1)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = Add h = to x Add k = to y (1,0) (1/3 , 1)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1/3 , 1)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1/3 , 1)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (1/3 , 1)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (1/3 , 1)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (4, 2) (1/3 , 1)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (4, 2) (1/3 , 1) (1/3, 3)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (4, 2) (1/3 , 1) (1/3, 3) (3 1/3, 3)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (4, 2) (1/3 , 1) (1/3, 3) (3 1/3, 3) (3 1/3, 5)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x ? 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (4, 2) (1/3 , 1) (1/3, 3) (3 1/3, 3) (3 1/3, 5)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x > 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (4, 2) (1/3 , 1) (1/3, 3) (3 1/3, 3) (3 1/3, 5)

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x > 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (4, 2) (1/3 , 1) (1/3, 3) (3 1/3, 3) (3 1/3, 5) Dilation

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x > 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (4, 2) (1/3 , 1) (1/3, 3) (3 1/3, 3) (3 1/3, 5) Dilation, translates right 3 units

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x > 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (4, 2) (1/3 , 1) (1/3, 3) (3 1/3, 3) (3 1/3, 5) Dilation, translates right 3 units, translates up 2 units

Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3 Domain: x > 3 Range: all real numbers Pts on parent graph Mult. y by a = 3 Add h = to x Add k = 2 to y (1,0) (1, 0) (4,0) (4, 2) (1/3 , 1) (1/3, 3) (3 1/3, 3) (3 1/3, 5) Dilation, translates right 3 units, translates up 2 units

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote:

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph (1,0) (___ , 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph (1,0) (3 , 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = (1,0) (3 , 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = Add h = to x (1,0) (3 , 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = Add h = to x Add k = to y (1,0) (3 , 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (3 , 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (3 , 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (3 , 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (3 , 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (1 1/2, 3) (3 , 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (1 1/2, 3) (3 , 1) (1 1/2, 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (1 1/2, 3) (3 , 1) (1 1/2, 1) (2 1/2, 1)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (1 1/2, 3) (3 , 1) (1 1/2, 1) (2 1/2, 1) (2 1/2, 4)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x ? 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (1 1/2, 3) (3 , 1) (1 1/2, 1) (2 1/2, 1) (2 1/2, 4)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x > 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (1 1/2, 3) (3 , 1) (1 1/2, 1) (2 1/2, 1) (2 1/2, 4)

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x > 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (1 1/2, 3) (3 , 1) (1 1/2, 1) (2 1/2, 1) (2 1/2, 4) Dilation

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x > 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (1 1/2, 3) (3 , 1) (1 1/2, 1) (2 1/2, 1) (2 1/2, 4) Dilation, translates right 1 unit

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x > 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (1 1/2, 3) (3 , 1) (1 1/2, 1) (2 1/2, 1) (2 1/2, 4) Dilation, translates right 1 unit, translates up 3 units

Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1) 𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3 Asymptote: x= 1 Domain: x > 1 Range: all real numbers Pts on parent graph Divide x by c = 2 Add h = 1 to x Add k = 3 to y (1,0) (1/2, 0) (1 1/2,0) (1 1/2, 3) (3 , 1) (1 1/2, 1) (2 1/2, 1) (2 1/2, 4) Dilation, translates right 1 unit, translates up 3 units

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote:

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph (1,0) (___ , 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph (1,0) (2 , 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = (1,0) (2 , 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = Add h = to x (1,0) (2 , 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = Add h = to x Add k = to y (1,0) (2 , 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (2 , 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (2 , 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (2 , 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (2 , 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (3, -3) (2 , 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (3, -3) (2 , 1) (-2, 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (3, -3) (2 , 1) (-2, 1)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (3, -3) (2 , 1) (-2, 1) (2, -2)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x ? 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (3, -3) (2 , 1) (-2, 1) (2, -2)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x < 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (3, -3) (2 , 1) (-2, 1) (2, -2)

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x < 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (3, -3) (2 , 1) (-2, 1) (2, -2) Reflects over the y-axis

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x < 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (3, -3) (2 , 1) (-2, 1) (2, -2) Reflects over the y-axis, translates right 4 units

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x < 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (3, -3) (2 , 1) (-2, 1) (2, -2) Reflects over the y-axis, translates right 4 units, translates down 3 units

Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4) 𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3 Asymptote: x= 4 Domain: x < 4 Range: all real numbers Pts on parent graph Divide x by c = -1 Add h = 4 to x Add k = -3 to y (1,0) (-1, 0) (3,0) (3, -3) (2 , 1) (-2, 1) (2, -2) Reflects over the y-axis, translates right 4 units, translates down 3 units

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote:

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph (1,0) (___ , 1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph (1,0) (4 , 1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = (1,0) (4 , 1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = Divide x by c = (1,0) (4 , 1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = Divide x by c = Add k = to y (1,0) (4 , 1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (4 , 1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (4 , 1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (4 , 1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (4 , 1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (1/2, -1) (4 , 1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (1/2, -1) (4 , 1) (4, -1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (1/2, -1) (4 , 1) (4, -1) (2 , -1)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (1/2, -1) (4 , 1) (4, -1) (2 , -1) (2, -2)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x ? 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (1/2, -1) (4 , 1) (4, -1) (2 , -1) (2, -2)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x > 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (1/2, -1) (4 , 1) (4, -1) (2 , -1) (2, -2)

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x > 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (1/2, -1) (4 , 1) (4, -1) (2 , -1) (2, -2) Reflects over the x-axis

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x > 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (1/2, -1) (4 , 1) (4, -1) (2 , -1) (2, -2) Reflects over the x-axis, dilation

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x > 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (1/2, -1) (4 , 1) (4, -1) (2 , -1) (2, -2) Reflects over the x-axis, dilation, translates down 1 unit

Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1 𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1 Asymptote: x= 0 Domain: x > 0 Range: all real numbers Pts on parent graph Mult. y by a = -1 Divide x by c = 2 Add k = to y (1,0) (1, 0) (1/2, 0) (1/2, -1) (4 , 1) (4, -1) (2 , -1) (2, -2) Reflects over the x-axis, dilation, translates down 1 unit

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote:

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph (1,0) ( ___, 1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph (1,0) (1/2 , 1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph Mult. y by a = (1,0) (1/2 , 1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph Mult. y by a = Add h= to x (1,0) (1/2 , 1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1/2 , 1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1/2 , 1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1, 0) (1/2 , 1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1, 0) (-3, 0) (1/2 , 1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1, 0) (-3, 0) (1/2 , 1) (1/2, -1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1, 0) (-3, 0) (1/2 , 1) (1/2, -1) (-3 1/2 , -1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x ? - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1, 0) (-3, 0) (1/2 , 1) (1/2, -1) (-3 1/2 , -1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x > - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1, 0) (-3, 0) (1/2 , 1) (1/2, -1) (-3 1/2 , -1)

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x > - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1, 0) (-3, 0) (1/2 , 1) (1/2, -1) (-3 1/2 , -1) Reflects over the x-axis

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x > - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1, 0) (-3, 0) (1/2 , 1) (1/2, -1) (-3 1/2 , -1) Reflects over the x-axis, translates left 4 units

Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0 𝑎=−1 𝑏= 1 2 𝑐=1 ℎ=−4 𝑘=0 Asymptote: x= -4 Domain: x > - 4 Range: all real numbers Pts on parent graph Mult. y by a = -1 Add h= -4 to x (1,0) (1, 0) (-3, 0) (1/2 , 1) (1/2, -1) (-3 1/2 , -1) Reflects over the x-axis, translates left 4 units