Graphing logarithmic functions with transformations

Graphing logarithmic functions with transformations

 $y=\log_bx$  parent function

• points:  $\left(\frac{1}{b},\ -1\right),\ \left(1,\ 0\right),\ \left(b,1\right)$  

• asymptote:  x = 0

• The graph to the right shows the parent function  $y=\log_2\left(x\right)$  

Transformations

• Follow your order of operations

• 1st: parenthesis - follow order of operations inside parenthesis

• 2nd: multiplication and division

• 3rd: addition and subtraction

• Asymptote only moves with shift left/right

Graph

 $y=-\log_2\left(x+4\right)$  

• Parent function points:  $\left(\frac{1}{2},\ -1\right),\left(1,0\right),\ \left(2,1\right)$  

• 1st transformation:  shift left 4 x -4 

• 2nd transformation:  reflect over x axis y(-1)

• New rule:  (x - 4, -1y) don't forget to move asymptote!

• (-3.5, 1), (-3, 0), (-2, -1) asymptote:  x = -4

 $y=-\log_2\left(x+4\right)$  

• (-3.5, 1), (-3,0), (-2, -1)

• asymptote:  x = -4

Multiple Choice

 $y\ =\ \log_2\left(-x\right)+3$  has the following parent points:  $\left(\frac{1}{2},-1\right),\ \left(1,0\right),\ \left(2,1\right)$  What would happen after the first transformation?

• Reflect over the x axis:  y(-1)

• Reflect over the y axis:  x(-1)

• Shift up 3:  x + 3

• Shift up 3:  y + 3

Multiple Choice

 $y\ =\ \log_2\left(-x\right)+3$  has the following parent points:  $\left(\frac{1}{2},-1\right),\ \left(1,0\right),\ \left(2,1\right)$  What would happen after the second transformation?

• Reflect over the x axis:  y(-1)

• Reflect of the y axis:  x(-1)

• Shift up 3:  x + 3

• Shift up 3:  y + 3

Multiple Choice

 $y=\log_2\left(-x\right)+3$  has parent points:  $\left(\frac{1}{2},-1\right),\ \left(1,0\right),\ \left(2,\ 1\right)$  The transformations gave you: (-1x, y +3).  What would be the transformed points?

•  $\left(-\frac{1}{2},2\right),\left(-1,3\right),\ \left(-2,4\right)$  

•  $\left(2.5,-1\right),\left(2,0\right),\ \left(0,1\right)$  

Multiple Choice

 $y=-2\log_3\left(x+1\right)$  has parent points:   $\left(\frac{1}{3},\ -1\right),\ \left(1,\ 0\right),\ \left(3,1\right)$  What is the 1st transformation?

• reflect over x axis and vertical stretch:  -2(x)

• reflect over x axis and vertical stretch: -2(y)

• shift left 1:  x - 1

• shift left 1:  y - 1

Multiple Choice

 $y=-2\log_3\left(x+1\right)$  has parent points:   $\left(\frac{1}{3},\ -1\right),\ \left(1,\ 0\right),\ \left(3,1\right)$  What is the 2nd transformation?

• reflect over x axis and vertical stretch:  -2(x)

• reflect over x axis and vertical stretch: -2(y)

• shift left 1:  x - 1

• shift left 1:  y - 1

Multiple Choice

 $y=-2\log_3\left(x+1\right)$  has parent points:   $\left(\frac{1}{3},\ -1\right),\ \left(1,\ 0\right),\ \left(3,1\right)$  Transformations gave the rule (x - 1, -2y).  What are the points of the new function?

•  $\left(-\frac{2}{3},2\right),\left(0,0\right),\left(2,-2\right)$  
  

•  $\left(-\frac{2}{3},-2\right),\left(-2,-1\right),\ \left(-6,0\right)$  
  

 $y=-2\log_3\left(x+1\right)$  

•  $\left(-\frac{2}{3},2\right),\left(0,0\right),\left(2,-2\right)$  

• asymptote:  x = -1

Homework in Khan Academy

• 1. Graphs of logarithmic functions due tomorrow 11:59 pm

• Test on Monday/Tuesday