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Exploration 1 :-
Initially, on our first attempt on the questions we decided to try something else and discover a bit more from the apps provided and turned out we basically found that a quick temporary workaround for the negative theta range problem for those who need it is to replace theta with "theta + a2pi" or "theta + 360a" where "a" is a scaling factor. in the slider for a (or whichever letter you choose), write something like "a = [-50, -49, -48,..., 50]" to set your range in multiples of 2pi. it's basically drawing a bunch of different graphs of different ranges that look like one graph, which means a bigger range will take longer, but my computer can barely open ms word yet handles 600 items on that list pretty well.

here's an example with a log function, which decays pretty fast. notice i've changed a2pi to a12pi because that's where the function usually cuts off for me that way it saves time.

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On the other hand, these were what we obtained from our exploration onwards throughout the questions below, and by using the apps we could saw it clearly as we can justify yet clarify the shapes, length of radius, number of petals of the graphs etc.
We were very satisfied with the findings as it improves more on our understanding not just our skills to sketch graphs but also to actually predict the shape that it will produce just by looking at the equations given, and it kind of very interesting, thanks to the apps especially for our Dr.'s guidence on how to use it wisely.

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Ho Pei Yi, Cheong Wei Yi, Cheong Kai Xuan, Chin Look Yee

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Ho Pei Yi, Cheong Wei Yi, Cheong Kai Xuan, Chin Look Yee

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When r=a+b cos x or r=a+b sin x, we found that a determine the size of the graph, the shape of the value will change depend on the value of a. When the value of a increases, the size of the graph will also increase.

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Pua Lei Wen, Yeoh Shir Ling, Tan Li Teng, Lois Lo

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Pua Lei Wen, Yeoh Shir Ling, Tan Li Teng, Lois Lo

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Exploration 1:
If r=2a cos x, a determine the radius of the graph. The larger the a, the bigger the graph.
When there is 2a,the length of radius is doubled.
https://www.desmos.com/calculator/kkblltxrza

Exploration 2:
If r=a+b cos x or r=a+b sin x, sin and cos determine the symmetry of graph.
a determines the size of graph.
When there is a+b in the equation, the graph will has a inner circle of the value of a is small.
When a=b in the equation, the graph is a circle.
http://www.desmos.com/calculator/sap92m2bus

Exploration 3:
r=a cos 2x
a determine the size of the graph
The number of petals is determine by 2x

Exploration 4:
r= a cos n x
a determines the size of the graph
When n is odd there are n petals and
When n is even there are 2n petals.

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We decided to have some fun and try out the equation r= 11 cos θ + 4 sin 6θ which, we realized, is symmetrical along the x axis and somewhat resembles a mermaid shell :D

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We found out new solution to solve mathematical problems instead of mathematical working. By using desmos we can easily determined shape of the graph based on the equation given. 🙂

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we find that the petals is determined by the coefficient of theta

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We found through exploration 1 is that a determines the diameter of the circle in tge graph and 2a doubles the length of diameter. In exploration 2, sin and cos the determines the intersection of graph on y or x axis, a+b will determine the shape of the graph and a=b determines the position of the curve wether its above x axis or not. In exploration 3, a determines the diameter of petals and the number of petals is determined by value of n. In exploration 4, if n is even then the number of petals=n but if n is odd the number of petals=2n. While a determins the diameter of petals. So when n is odd there are 2n petals and when n is even there are n petals. About our finding in exploration 5, we can conclude that relationships of polar equations with trigonometry and cartesian equation that we've learn before plays an important roles in sketching the polar curve so that an overall picture of the equations can be visualized. Polar graphs implies 2 variables as analogies to dimensions same as Cartesian and Trigonometry graph but every entities of dimensions are different from each other in which polar graph implies magnitude and direction as an analogy to vector (r and theta) while Cartesian uses magnitude of x and y axis. On the other hand, trigonometry uses y and theta as variables. Thus, y=sin theta in trigonometry known as periodic function is not equal to r=sin theta in which sometines polar repeats the same points of location because of their unique coordinates taht represent the same location but in Cartesian one coordinate represent one specific location..So, thus majorith of polar graphs form curve because theta is changing and sometimes can be determined by symmetrical test without finding all values of theta.

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For r=2a cos x
We found out that a determines the position and the shape of the graph while 2a determines the radius of circle which increase twice. The larger the a value, the larger the circle.
For r=a+b cos x and r= a+b sin x
We learned that sin and cos determine the orientation of the angle of the graph. Sin graph will be symmetry to the verticalous while cos graph will be symmetry to the horizontal AXIS. a determines the size( a increase, size increase). a+b determines shape a<b, will have inner loop and a>:, will have dimpled limacon. a=b determines the shape.

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we've tried to explore this equation r=2 -5sin theta and this what we've got.

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LEE SEE TING, HUE SEOW YIE, JANE SENG , YAP YONG ZI

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LEE SEE TING, HUE SEOW YIE, JANE SENG , YAP YONG ZI

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From graph r=2cosx and r=6sinx we can conclude that the value of a determines the diameter of circle. When a value is positive the circle will be on right hand side but when a value is negative the circle will be on the left hand side. Whereas, 2a determines the diameter of circle by doubling the value. When a value increases,the diameter increases and vice versa. 🕊

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We found that when the 'theta' is double, more petals formed. Besides, when the 'alpha' increased, the radius of the petals increase.

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LEE SEE TING, HUE SEOW YIE, JANE SENG , YAP YONG ZI

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LEE SEE TING, HUE SEOW YIE, JANE SENG , YAP YONG ZI

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IT'S VERY INTERESTING

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Koh Zi Ning,Hon Hui Nee,Kek Jazz Yee,Hong Siew Yi

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Koh Zi Ning,Hon Hui Nee,Kek Jazz Yee,Hong Siew Yi

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We found that a determines the radius of curve, which is the size of the petals,and the number of petals is determined by the number of times of x.

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We learned that for r = a cos  nθ , the petals are even numbered if n is even and odd if n is odd. a determines the size of the petals.

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We learned that adjusting a for r = a +b cos θ determines the number of times the graph intersects the x axis. For -3<a<3, the graph intersects the x axis three times. Otherwise it intersects the x axis twice.

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I learned that r=c*theta will gives a spiral look. If c is positive, my spiral moves in clockwise direction. If c is negative, my spiral moves in anticlockwise direction.