Let’s check
your mastery!
Be the master of
Exponential
Functions
Determine whether the exponential function is a
growth or decay.
f(x) =
𝟑 𝒙
( )
𝟓
f(x) = 7x
f(x) =
𝟖 𝒙
( )
𝟑
Exponential Decay
f(x) = 0.0205x
Exponential Growth
Exponential Decay
Exponential Growth
f(x) = ( 𝟏𝟔)𝒙
Exponential Growth
f(x) = 1.005x
Exponential Growth
Determine whether the exponential function is a growth or
decay.
Exponential
Decay
Exponential Growth
Think of a number between 1
and 10. Multiply it by 9 and
subtract 1. Now close your
eyes.
It’s dark isn’t it!
Happy Friday everyone! 
Lesson 2
Graph of Exponential
Function
continuation
Exponential
Function of the
x
Form f(x) = ab + k
The Vertical Shift
Consider the graphs of exponential functions below
What type of Exponential Function is
being represented by the graph?
Exponential Growth
Whatare
are
characteristics
What
thethe
characteristics
or or
++=6?
features
of of
thethe
graph
f(x) =f(x)
2x +2?
3?
features
graph
2x ?
Domain:
All REAL NUMBERS, (-∞, ∞)
Range:
yy >
3
>6
2
0
y-intercept:
x-intercept:
Asymptote:
f(x) = 2x + 2
f(x) = 2x
f(x) = 2x + 3
(0,7)
(0,4)
(0,3)
(0,1)
f(x) = 2x + 6
No x-intercept:
yy == 620
3
What features
features of
of the
the original
original graph
graph are
are
What
affected when
when we
we translated
translated the
the graph?
graph?
affected
f(x)
=2x
f(x) = abx + k
a = 1, b = 2, k = 0
Range:
y>0
f(x)
=2x+
2
f(x) = abx + k
a = 1, b = 2, k = 2
y>2
f(x)
=2x+
3
f(x) =2x+ 6
f(x) = abx + k
f(x) = abx + k
a = 1, b = 2, k = 3
a = 1, b = 2, k = 6
y>3
y>6
y-intercept:
(0,1)
(0,3)
(0,4)
(0,7)
Asymptote:
y=0
y=2
y=3
y=6
Given an exponential function, can we determine the
characteristics without actually graphing it?
Consider the graphs of exponential functions below
What type of Exponential Function is
being represented by the graph?
f(x) = 2x
What
arethe
thecharacteristics
characteristics
What are
or or
features ofofthe
8?
graph
f(x)f(x)
= 2x=- 4?
features
the
graph
2x ?
Domain:
All REAL NUMBERS, (-∞, ∞)
Range:
4
yy >
>0--8
y-intercept:
-7)
(0,
(0,1)
-3)
x-intercept:
(2,
No 0)
x-intercept
(3,
Asymptote:
y = -0-84
What features of the original graph are
affected when we translated the graph?
f(x) = 2x - 4
f(x) = 2x - 8
f(x) =2x
f(x) =
abx
+ k
a = 1, b = 2, k = 0
Range:
y-intercept:
x-intercept:
Asymptote:
f(x) =2x - 4
+ k
f(x) = abx + k
a = 1, b = 2, k = - 4
a = 1, b = 2, k = -8
f(x) =
y>0
(0,1)
No x-intercept
y=0
f(x) =2x - 8
abx
y>−4
y>- 8
(0,- 3)
(0, - 7)
(2,0)
(3, 0)
y=-4
y = -8
Given an exponential function, can we determine the
characteristics without actually graphing it?
Remember:
Graphing an Exponential Function with a Vertical Shift
An exponential function of the form f(x) = abx+ k is an exponential
function with a vertical shift. The constant k is what causes the vertical
shift to occur. A vertical shift is when the graph of the function is moved
up or down a fixed distance, k. When a vertical shift is applied to an
exponential function, what features of the graph are affected? The features
of the graph that are affected are the intercepts, horizontal asymptote,
and range. The y-intercept will move up or down a fixed amount, k, and
the horizontal asymptote will also move up or down a fixed amount, k.
Moving the horizontal asymptote up or down will then change the range of
the function because the graph cannot touch or go below the horizontal
asymptote.
Exponential Function of the
x-h
Form f(x) = ab
The Horizontal Shift
Consider the graphs of exponential functions below
What type of Exponential Function is
being represented by the graph?
What
What are
are the
the characteristics
characteristics or
or
𝟏𝟏𝒙+𝟐
𝒙−𝟑
features
of
the
graph
f(x)
=
(
f(x) ==((𝟐)𝒙)?
??
features of the graph f(x)
𝟏
f(x) = (𝟐)𝒙−𝟑
𝟐
Domain:
All REAL NUMBERS, (-∞, ∞)
Range:
y>0
𝟏
f(x) = (𝟐)𝒙
y-intercept:
(0,1)
(0,
(0,8)
1/4 )
x-intercept:
No x-intercept:
Asymptote:
𝟏
𝟐
f(x) = ( )𝒙+𝟐
y=0
What features of the original graph are
affected when we translated the graph?
f(x) =
𝟏 𝒙
( )
𝟐
𝒙−𝒉
f(x) = 𝒂𝒃𝒙−𝒉
a = 1, b =
𝟏
( ),
𝟐
Range:
y-intercept:
x-intercept:
Asymptote:
f(x) =
𝟏 𝒙+𝟐
( )
𝟐
f(x) = 𝒂𝒃
h= 0
a = 1, b =
y>0
(0,1)
No x-intercept
y=0
𝟏
( ),
𝟐
h = -2
(0,
f(x) = 𝒂𝒃𝒙−𝒉
a = 1, b =
𝟏
( ),
𝟐
h=3
y>0
y>0
𝟏
( )
𝟒
f(x) =
𝟏 𝒙−𝟑
( )
𝟐
)
(0, 8)
No x-intercept
No x-intercept
y=0
y=0
Given an exponential function, can we determine the
characteristics without actually graphing it?
Remember:
Graphing an Exponential
Horizontal Shift
Function
with
a
An exponential function of the form f(x) = ab x – h is an exponential function
with a horizontal shift. The constant h is what causes the horizontal shift to
occur. A horizontal shift is when the graph of the function is moved to the left
or right a fixed distance, h. When a horizontal shift is applied to an exponential
function, what features of the graph are affected? The only feature of the
graph that is affected when a = 1 is the y-intercept. The y-intercept
will move to the left or right a fixed amount, h. This new point will NO longer be
the y-intercept, but it will be a point on the graph that we can use as a
reference point when visualizing the graph. The horizontal asymptote will still
be at the x-axis or y = 0.
Exponential Function of the Form f(x)
x-h
= ab + k
xab
Exponential Function of the Form f(x) =
h + k have both a vertical shift of k units and
a horizontal shift of h units.
Clue : h – horizontal shift
Therefore k - vertical shift
Consider the f(x) = abx-h + k where a = 1 and b > 1
f(x) =
x-1
2
-2
Exponential Growth
2x-1 – 2 of the y
x
Characteristics
graph
15
(-2) -1 – 2
-2
2
All
Real
- orNumbers
-1.875
Domain
8
-2
-1Range
2(-1) -1 – 2 - 7 y
or>– 1.75
4
30, -1.5 )
(
y-intercept
0
20 -1 – 2
- or – 1. 5
2
x-intercept
( 2,
-1 0 )
1
21-1 – 2
Asymptote
y 0= -2
2
22-1 - 2
3
23-1 - 2
2
f(x) = 2x-1 - 2
Consider the f(x) = abx-h + k where a = 1
f(x) =
x-1
2
-2
Graph the function by
transformation using the
parent function.
h - Horizontal shift
k - Vertical shift
Parent
function
f(x) = 2x
f(x) = 2x-1 - 2
Consider the f(x) = abx-h + k where a = 1
f(x) =
x+4
2
+3
Parent
function
f(x) = 2x
Graph the function by
transformation using the
parent function.
h - Horizontal shift
k - Vertical shift
f(x) = 2x+4 + 3
Consider the f(x) = abx-h + k where a = 1
f(x) =
x-5
2
-6
Parent
function
f(x) = 2x
Graph the function by
transformation using the
parent function.
h - Horizontal shift
k - Vertical shift
f(x) = 2x-5 - 6
Have an advance reading on the following:
What will be the effect to the graph of f(x) = abx-h + k
when a > 1 or a < 1?
How can we determine the domain and range of the
function if we are not to draw the graph?
How can we determine the x and y– intercept of the
function if we are not to draw the graph?
How can we determine the horizontal asymptote of
the function if we are not to draw the graph?