Summarizing translations of the exponential function
Now that we have worked with each type of translation for the exponential function, we can summarize them in
[link] to arrive at the general equation for translating exponential functions.
Translations of the Parent Function
Translation
Form
Shift
Horizontally
units to the left
Vertically
units up
Stretch and Compress
Stretch if
Compression if
Reflect about the
x -axis
Reflect about the
y -axis
General equation for all translations
Translations of exponential functions
A translation of an exponential function has the form
Where the parent function,
is
shifted horizontally
units to the left.
stretched vertically by a factor of
if
compressed vertically by a factor of
if
shifted vertically
units.
reflected about the
x- axis when
Note the order of the shifts, transformations, and reflections follow the order of operations.
Writing a function from a description
Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.
is vertically stretched by a factor of
, reflected across the
y -axis, and then shifted up
units.
We want to find an equation of the general form
We use the description provided to find
and
We are given the parent function
so
The function is stretched by a factor of
, so
The function is reflected about the
y -axis. We replace
with
to get:
The graph is shifted vertically 4 units, so
Substituting in the general form we get,
The domain is
the range is
the horizontal asymptote is
General Form for the Translation of the Parent Function
Key concepts
The graph of the function
has a
y- intercept at
domain
range
and horizontal asymptote
See
[link] .
If
the function is increasing. The left tail of the graph will approach the asymptote
and the right tail will increase without bound.
If
the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote
The equation
represents a vertical shift of the parent function
The equation
represents a horizontal shift of the parent function
See
[link] .
Approximate solutions of the equation
can be found using a graphing calculator. See
[link] .
The equation
where
represents a vertical stretch if
or compression if
of the parent function
See
[link] .
When the parent function
is multiplied by
the result,
is a reflection about the
x -axis. When the input is multiplied by
the result,
is a reflection about the
y -axis. See
[link] .
All translations of the exponential function can be summarized by the general equation
See
[link] .
Using the general equation
we can write the equation of a function given its description. See
[link] .
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost
6% more than previous year? Provide with reasoning.
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Good day
How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200.
Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho
ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9
no Mpc is greater than 1
Y=100+.9Y+50
Y-.9Y=150
0.1Y/0.1=150/0.1
Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question
If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output?
discuss your answer using appropriate graph.