Lesson #1: Exponential Functions and Their
Inverses Day 2
Unit 5:
Logarithmic Functions
Lesson #1: Exponential Functions and Their Inverses
Day 2
Exponential Functions & Their Inverses


Exponential Functions are in the form
.
The inverse of an exponential is a reflection in the line
. The and coordinates are swapped.
Inverse of the
exponential function
:
Solve for :
log
Algebra II with Trigonometry
Unit 8
1
Lesson #1: Exponential Functions and Their
Inverses Day 2
Properties of a Logarithmic Function •The domain is all
positive real numbers
0 or 0, ∞ .
•The range is all real
numbers, ∞, ∞ .
•The vertical asymptote
is located at
0
(the -axis).
•The ‐intercept is (1,0).
Example 1: Graph Algebra II with Trigonometry
Unit 8
of 2
.
2
Lesson #1: Exponential Functions and Their
Inverses Day 2
Transformations of Logarithmic Functions
Using HSRV
Logarithmic Functions and Reflections
•Recall, when reflecting over the ‐axis, the ‐value is negated.
•The vertical asymptote remains at 0.
Algebra II with Trigonometry
Unit 8
3
Lesson #1: Exponential Functions and Their
Inverses Day 2
Logarithmic Functions and Reflections
•Recall, when reflecting over the ‐axis, the ‐value is negated.
•The vertical asymptote remains at 0.
Logarithmic Functions and Translations
•If h is positive, the graph shifts to the right
units.
(Remember: if h “looks negative” h is positive)
•If h is negative, the graph shifts to the left
units.
(Remember: if h “looks positive” h is negative)
•The line
•The domain is
•The range is
Algebra II with Trigonometry
Unit 8
is the vertical asymptote.
.
∞, ∞ or all real numbers.
4
Lesson #1: Exponential Functions and Their
Inverses Day 2
Logarithmic Functions and Translations
•If
is positive, the graph shifts up
•If
is negative, the graph shifts down
•If
•If 0
units.
units.
1, the graph moves up to the right,.
1the graph moves down to the right.
Logarithmic Functions: Stretch & Shrink
1, multiply each ‐coordinate of by , vertically stretching the graph of by the factor of .
•If •If 0
1, multiply each ‐coordinate of by , vertically shrinking the graph of by the factor of .
•If 1, divide each ‐coordinate of by , horizontally shrinking the graph of by the factor of .
•If 0
1, divide each ‐coordinate of by , horizontally stretching the graph of by the factor of .
Algebra II with Trigonometry
Unit 8
5
Lesson #1: Exponential Functions and Their
Inverses Day 2
Transformations of Functions
Recall:CombinationsofTransformations:
A function involving more than one transformation can be graphed by performing transformations in the following order:
1. Horizontal Shifting
2. Stretching or Shrinking
3. Reflecting
4. Vertical Shifting
Ex 1: Describe the graph of 2log
1.
Transformation: H none
S vertical stretch by factor of 2 (multiply s by 2)
R reflect over the – axis V up 1
⟹ Domain:
Range:
or
∞,
∞, ∞
Asymptote: Algebra II with Trigonometry
Unit 8
6
Lesson #1: Exponential Functions and Their
Inverses Day 2
Ex 2: Describe the graph of log 2
3
4.
Transformation: H left 3
S horizontal shrink by factor of 2 (divide ′s by 2)
R none
V down 4
Domain:
Range:
⟹ or
,∞
∞, ∞
Asymptote: Ex 3: Use the graph of log
and transformations to sketch the graph of log
1
4. Also, find the domain and vertical asymptote of . H left 1 unit
S none
R none
V up 4 units
Domain: ,∞
VA @
Algebra II with Trigonometry
Unit 8
7
Lesson #1: Exponential Functions and Their
Inverses Day 2
log
Ex 4: Use the graph of and transformations to 2log
sketch the graph of domain & vertical asymptote of 4
1. Also, find the . H left 4 units
S vertical stretch by factor (multiply ‐values by ) R none
V down 1 unit
Domain: ,∞
VA @
Ex 5: Use the graph of log
sketch the graph of log
domain & vertical asymptote of and transformations to 5
4. Also, find the . H left 5 units
S horizontal stretch by factor (divide ‐values by ) R reflection over ‐axis
(negate ‐values)
V down 4 units
Domain: ,∞
VA @
Algebra II with Trigonometry
Unit 8
8
Lesson #1: Exponential Functions and Their
Inverses Day 2
Ex 6: Determine the domain and vertical asymptote of the graph of log
2
16
1.
Recall: Domain of a logarithmic function is all real numbers greater than zero.
∞,
Domain:
VA @
Ex 7: Determine the domain and vertical asymptote of the graph of log 3
5.
3
Domain:
,∞
VA @
Algebra II with Trigonometry
Unit 8
9