Problem-solving strategy: plotting a curve in polar coordinates
Create a table with two columns. The first column is for
and the second column is for
Create a list of values for
Calculate the corresponding
values for each
Plot each ordered pair
on the coordinate axes.
Connect the points and look for a pattern.
Watch this
video for more information on sketching polar curves.
Graphing a function in polar coordinates
Graph the curve defined by the function
Identify the curve and rewrite the equation in rectangular coordinates.
Because the function is a multiple of a sine function, it is periodic with period
so use values for
between 0 and
The result of steps 1–3 appear in the following table.
[link] shows the graph based on this table.
0
0
0
0
This is the graph of a circle. The equation
can be converted into rectangular coordinates by first multiplying both sides by
This gives the equation
Next use the facts that
and
This gives
To put this equation into standard form, subtract
from both sides of the equation and complete the square:
This is the equation of a circle with radius 2 and center
in the rectangular coordinate system.
The graph in
[link] was that of a circle. The equation of the circle can be transformed into rectangular coordinates using the coordinate transformation formulas in
[link] .
[link] gives some more examples of functions for transforming from polar to rectangular coordinates.
Transforming polar equations to rectangular coordinates
Rewrite each of the following equations in rectangular coordinates and identify the graph.
Take the tangent of both sides. This gives
Since
we can replace the left-hand side of this equation by
This gives
which can be rewritten as
This is the equation of a straight line passing through the origin with slope
In general, any polar equation of the form
represents a straight line through the pole with slope equal to
First, square both sides of the equation. This gives
Next replace
with
This gives the equation
which is the equation of a circle centered at the origin with radius 3. In general, any polar equation of the form
where
k is a positive constant represents a circle of radius
k centered at the origin. (
Note : when squaring both sides of an equation it is possible to introduce new points unintentionally. This should always be taken into consideration. However, in this case we do not introduce new points. For example,
is the same point as
Multiply both sides of the equation by
This leads to
Next use the formulas
This gives
To put this equation into standard form, first move the variables from the right-hand side of the equation to the left-hand side, then complete the square.
This is the equation of a circle with center at
and radius 5. Notice that the circle passes through the origin since the center is 5 units away.
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
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miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities.
before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue..
sometimes due to these physical problems..the diagnosis may be misdiagnosed.
treatment for autism.
well it depends on the severity.
since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision.
play therapy.
speech therapy.
Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game.
string and beads game.
the more you interact with the child the more effective
Jharna
results you'll get..
please consult a therapist to know what suits best on your child.
and last as a parent. I know sometimes it's overwhelming to guide a special kid.
but trust the process and be strong and patient as a parent.
Jharna
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