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Question 1108989: Hi, can you please show me how to find the domain and range of a circle without sketching it? x^2 + (y-2)^2 = 25.
Also after you've sketched the graph of a circle and you have the centre and radius, how do you find the other coordinates of the circle. (is there any other method than just counting from the graph).
Thankyou so much.
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52864)   (Show Source):
You can put this solution on YOUR website! .
From your post, it looks like you need to gain your knowledge about the subject from the scratch.
See the lessons
- Standard equation of a circle
- General equation of a circle
- Transform general equation of a circle to the standard form by completing the squares
- Identify elements of a circle given by its general equation
in this site.
After reading these lessons, it will be clear to you that the citcle has the center at the point (x,y) = (0,2) and the radius of 5 units.
After reading these lessons, it will be clear to you also that the domain is the set of x
-5 <= x <= 5, or, which is the same, the segment [-5,5],
while the range is the set of y
-5 <= y-2 <= 5, or which is the same, -3 <= y <= 7, the segment [-3,7].
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Also, you have this free of charge online textbook in ALGEBRA-II in this site
ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic
"Conic sections: Ellipses. Definition, major elements and properties. Solved problems".
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Comment from student: Thank you. So if it was like (x+1)^2 + (y-4)^2 = 169, domain = [-14,12] and range = [-9, 17] right?
My response: Right, correct.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! here's the graph.
the dashed lines are only there to allow me to show you the coordinate points of interest.
you can see that:
the center is at (0,2).
the value of x goes from -5 to 5.
that's the domain.
the value of y goes from -3 to 7.
that's the range.
the radius of the circle is 5.
you can see that from the graph.
you can also determine all of this from the equation itself, without looking at the graph.
the standard form of the equation of a circle is (x-h)^2 + (y-k)^2 = r^2.
(h,k) is the center of the circle.
r is the radius of the circle.
your equation is x^2 + (y-2)^2 = r^2
that makes h = 0 and k = 2, which means the center of the circle is at (h,k) = (0,2).
the radius of the circle is sqrt(r^2) which gets you plus or minus 5.
the radius has to be positive, so the radius is 5.
from the center and the radius, you can find the domain and the range.
to find the domain, add and subtract the radius from h.
since h = 0, the domain is all values of x from -5 to 5.
to find the range, add and subtract the radius from k.
since k = 2, the range is all value of y from -3 to 7.
the graph confirms this to be true.
if you have the equation of the circle in standard form, then finding domain and range from the equation is fairly simple, as shown above.
if the equation is not in standard form, then you need to convert it to standard form by using the completing the square method.
here's a reference on how to do that.
http://www.purplemath.com/modules/sqrcircle.htm
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