Functions of the form
are known as
hyperbolic functions. The general form of the graph of this function is shown in
[link] .
Investigation : functions of the form
On the same set of axes, plot the following graphs:
Use your results to deduce the effect of
.
On the same set of axes, plot the following graphs:
Use your results to deduce the effect of
.
You should have found that the value of
affects whether the graph is located in the first and third quadrants of Cartesian plane.
You should have also found that the value of
affects whether the graph lies above the
-axis (
) or below the
-axis (
).
These different properties are summarised in
[link] . The axes of symmetry for each graph are shown as a dashed line.
Table summarising general shapes and positions of functions of the form
. The axes of symmetry are shown as dashed lines.
Domain and range
For
, the function is undefined for
. The domain is therefore
.
We see that
can be re-written as:
This shows that the function is undefined at
. Therefore the range of
is
.
For example, the domain of
is
because
is undefined at
.
We see that
is undefined at
. Therefore the range is
.
Intercepts
For functions of the form,
, the intercepts with the
and
axis is calculated by setting
for the
-intercept and by setting
for the
-intercept.
The
-intercept is calculated as follows:
which is undefined because we are dividing by 0. Therefore there is no
-intercept.
For example, the
-intercept of
is given by setting
to get:
which is undefined.
The
-intercepts are calculated by setting
as follows:
For example, the
-intercept of
is given by setting
to get:
Asymptotes
There are two asymptotes for functions of the form
. Just a reminder, an asymptote is a straight or curved line, which the graph of a function will approach, but never touch. They are determined by examining the domain and range.
We saw that the function was undefined at
and for
. Therefore the asymptotes are
and
.
For example, the domain of
is
because
is undefined at
. We also see that
is undefined at
. Therefore the range is
.
From this we deduce that the asymptotes are at
and
.
Sketching graphs of the form
In order to sketch graphs of functions of the form,
, we need to determine four characteristics:
domain and range
asymptotes
-intercept
-intercept
For example, sketch the graph of
. Mark the intercepts and asymptotes.
We have determined the domain to be
and the range to be
. Therefore the asymptotes are at
and
.
There is no
-intercept and the
-intercept is
.
Draw the graph of
.
The domain is:
and the range is:
.
We look at the domain and range to determine where the asymptotes lie. From the domain we see that the function is undefined when
, so there is one asymptote at
. The other asymptote is found from the range. The function is undefined at
and so the second asymptote is at
There is no y-intercept for graphs of this form.
The x-intercept occurs when
. Calculating the x-intercept gives:
So there is one x-intercept at
.
Putting all this together gives us the following graph:
Does the point (-2; 3) lie on the graph ? Give a reason for your answer.
Why is the point (-2; -3) not on the graph ?
If the
-value of a point on the drawn graph is 0,25, what is the corresponding
-value ?
What happens to the
-values as the
-values become very large ?
With the line
as line of symmetry, what is the point symmetrical to (-2; 3) ?
Draw the graph of
.
How would the graph
compare with that of
? Explain your answer fully.
Draw the graph of
on the same set of axes.
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?