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] .
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
