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Topics
Circles and Ellipses
Identify Characteristics of Circles

Domain and Range of Circles

Lesson

Practice
Circles at Origin

Circles with translations

Circles (mixed set)

Circles (Mixed and Semi-Circles)

Circles (manipulation of equation, graphing and applications)

Applications of Circles

Solve Inequalities involving Circles

Introduction to Ellipses

Identify Characteristics of Ellipses

Domain and Range of Ellipses

Finding the Equation of Ellipse

Graphing Ellipses

Manipulate Ellipse Equations

Applications of Ellipses

Solve Inequalities Involving Ellipses

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India
Class XI



DOMAIN AND RANGE OF CIRCLES

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The diagram below shows that the domain of a circle consists of all $$x-values
within the interval $$a−r≤x≤a+r and the range of of a circle consists of all
$$y-values within the interval $$b−r≤y≤b+r.



We can also write the domain in interval notation as $$[a−r,a+r], and the range
in interval notation as $$[b−r,b+r].

The circle itself is not a function, but can be split into two semicircles, each
of which are functions. We can rearrange the equation to make this clear:

$$(x−a)2+(y−b)2 $$= $$r2 $$(y−b)2 $$= $$r2−(x−a)2 $$y−b $$= $$±√r2−(x−a)2 $$y
$$= $$y=b±√r2−(x−a)2

So, for example, in the new form the circle whose centre is located at $$(2,5)
and has radius $$r=3 has the equation given by $$y=5±√9−(x−2)2. It can be split
into the two functions, say $$f and $$g, where $$f(x)=5+√9−(x−2)2 and
$$g(x)=5−√9−(x−2)2.

Each of these functions are semicircles. Both functions will have the same
domain; that of the original circle, given by $$−1≤x≤5. The range of $$f becomes
$$5≤y≤8 and the range of $$g becomes $$2≤y≤5 as shown here.



 

PRACTICE QUESTIONS

QUESTION 1



Consider the graph of the circle shown below.

0,0

-10
-5
5
10
x
-10
-5
5
10
y

Loading Graph...



 1. State the domain of the graph in interval notation.

 2. State the range of the graph in interval notation.

Reveal SolutionWatch video



QUESTION 2



Consider the equation $$(x+5)2+(y+3)2=16.

 1. Plot the graph described by the equation.
    
    
    
    0,0
    
    -10
    -5
    5
    10
    x
    -10
    -5
    5
    10
    y
    
    Loading Graph...
    
    

 2. State the domain of the graph, using interval notation.

 3. State the range of the graph, using interval notation.

Reveal SolutionWatch video



QUESTION 3



The top of a semicircle has a domain of $$[−10,2] and a range of $$[−2,4].

 1. Plot the semicircle.
    
    
    
    0,0
    
    -10
    -5
    5
    10
    x
    -10
    -5
    5
    10
    y
    
    Loading Graph...
    
    

 2. State the equation for the semicircle in the form $$y=±√r2−(x−h)2+k.

Reveal SolutionWatch video




OUTCOMES


11.SF.RF.2

Definition of relation, pictorial diagrams, domain, co-domain and range of a
relation. Function as a special kind of relation from one set to another.
Pictorial representation of a function, domain, co-domain and range of a
function. Real valued function of the real variable, domain and range of these
functions, constant, identity, polynomial, rational, modulus, signum and
greatest integer functions with their graphs. Sum, difference, product and
quotients of functions.




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