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] .
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?