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] .
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits