Unit 3 Day 10 – Transformations of Logarithmic Functions

Warm-Up 31. Find the inverse of:

2. Your parents put $50 into a savings account when you were born to save up money for college. The savings account gains interest at a rate of 2% annually. Write an explicit function to model this situation.

3. Evaluate:

23

xy

5.310log

Warm Up 3 Describe the transformation using words!!

Essential Question(Last one!)

How can we use equations to determine the transformation of exponential and logarithmic functions?

EQ’s are due on Friday!!! Questions and ANSWERS!

Definitions

• Domain– The x values!

• Range– The y values!

Asymptote:

is a line that a graph approaches, but does not intersect

AsymptoteExponential functions will always have a horizontal

asymptote (y = #)Parent function: Has horizontal asymptote of y = 0

This asymptote changes when the graph is moved up and down.

xy 10

Asymptote

Logarithmic functions will always have a vertical asymptote (x = #)

Parent function: Has vertical asymptote of x = 0

This asymptote changes when you move the graph left and right.

)log(xy

• X – intercept– Where you cross the x – axis!

• Y – intercept– Where you cross the y – axis!

• Exponential Function– A model to model exponential growth or decay– In the form

• Logarithmic Function– The inverse of an exponential Function– In the form :

xbay

xy log

Look at # 1 and # 2

Transformations of Logarithmic Functions

Parent Function y = logbxShift up y = logbx + k

Shift down y = logbx - k

Shift left y = logb(x + h)

Shift right y = logb(x - h)

Combination Shift y = logb(x ± h) ± k

Reflect over the x-axis y = -logbx

Stretch vertically y = a logbx

Stretch horizontally y = logbax

Translations of logarithmic functions are very similar to those for other functions. Describe each translation for parent function y = log x.

1. y = log (x + 2)2. y = log (x) – 3 3. y = 5 log x4. y = -log x5. y = log (x – 4) + 5

1. Left 22. Down 33. Vertical stretch by 54. Reflect over x-axis5. Right 4, up 5

Identify the asymptote

1. y = log (x + 2)2. y = log (x) – 3 3. y = 5 log x4. y = -log x5. y = log (x – 4) + 5

1. Left 2 so x = -22. x = 03. X=04. X=05. Right 4, so x = 4

Translations of exponential functions are very similar to those for other functions. Describe each translation for parent function

1.

2.

3.

4.

5.

xy 10xy )10(23)10( xy

1)10( xy

4)10( 6 xy

5)10(21

xy

1. Vertical stretch of 2

2. Left 3

3. Down 1

4. Right 6 and down 4

5. Reflect over x-axis, vertical compression of ½, up 5

Translations of exponential functions are very similar to those for other functions. Describe each translation for parent function

1.

2.

3.

4.

5.

xy 10xy )10(23)10( xy

1)10( xy

4)10( 6 xy

5)10(21

xy

1. Y = 0

2. Y = 0

3. Down 1 so y = -1

4. down 4 so y = -4

5.up 5 so y = 5

Graph the following function on the graph at right. Describe each transformation, give the domain and range, and identify any asymptotes.

y = -2log (x + 2) – 4 • Domain:

• Range:

• Asymptote:

• Description of transformations:

Guided Practice

Homework

Independent Practice with Logarithmic Functions