Now we consider changes to the inside of a function. When we multiply a function’s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a
horizontal stretch ; if the constant is greater than 1, we get a
horizontal compression of the function.
Given a function
the form
results in a horizontal stretch or compression. Consider the function
Observe
[link] . The graph of
is a horizontal stretch of the graph of the function
by a factor of 2. The graph of
is a horizontal compression of the graph of the function
by a factor of 2.
Horizontal stretches and compressions
Given a function
a new function
where
is a constant, is a
horizontal stretch or
horizontal compression of the function
If
then the graph will be compressed by
If
then the graph will be stretched by
If
then there will be combination of a horizontal stretch or compression with a horizontal reflection.
Given a description of a function, sketch a horizontal compression or stretch.
Write a formula to represent the function.
Set
where
for a compression or
for a stretch.
Graphing a horizontal compression
Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population,
will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.
Symbolically, we could write
See
[link] for a graphical comparison of the original population and the compressed population.
(a) Original population graph (b) Compressed population graph
Finding a horizontal stretch for a tabular function
A function
is given as
[link] . Create a table for the function
2
4
6
8
1
3
7
11
The formula
tells us that the output values for
are the same as the output values for the function
at an input half the size. Notice that we do not have enough information to determine
because
and we do not have a value for
in our table. Our input values to
will need to be twice as large to get inputs for
that we can evaluate. For example, we can determine
We do the same for the other values to produce
[link] .
4
8
12
16
1
3
7
11
[link] shows the graphs of both of these sets of points.
The graph of
looks like the graph of
horizontally compressed. Because
ends at
and
ends at
we can see that the
values have been compressed by
because
We might also notice that
and
Either way, we can describe this relationship as
This is a horizontal compression by
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?
![Two side-by-side graphs. The first graph has function for original population whose domain is [0,7] and range is [0,3]. The maximum value occurs at (3,3). The second graph has the same shape as the first except it is half as wide. It is a graph of transformed population, with a domain of [0, 3.5] and a range of [0,3]. The maximum occurs at (1.5, 3).](/ocw/mirror/col11758/m51266/CNX_Precalc_Figure_01_05_029ab.jpg)
