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Functions and Graphs, SMB-020

A circle has centre `(5,3)` and radius 3.

  1.  Describe, with inequalities, the region that consists of the interior of the circle and more than 2 units above the `x`-axis.  (2 marks)

  2.  Sketch the region.  (1 mark)

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i.    `(x-5)^2 + (y-3)^2 < 9\ ∩\ y > 2`

ii.    
       

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i.   `text(Equation of circle:)`

`(x-5)^2 + (y-3)^2 = 3^2`
 

`:.\ text(Region is:)`

`(x-5)^2 + (y- 3)^2 < 9\ ∩\ y > 2`

COMMENT: The broken line on the graph represents an excluded boundary.

 

ii.   

Functions, EXT1* F1 2009 HSC 3c

Shade the region in the plane defined by  `y >= 0`  and  `y <= 4-x^2`.   (2 marks)

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`text(Shaded area is region where)`

`y >= 0\ text(and)\ y >= 4-x^2`

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COMMENT: This past “Advanced” HSC question now fits into the Ext1 (new) syllabus.

`text(Shaded area is region where)`

`y >= 0\ \ text(and)\ \ y >= 4-x^2`

Functions, EXT1* F1 2012 HSC 8 MC

The diagram shows the region enclosed by  `y = x- 2`  and  `y^2 = 4-x`. 
  

Which of the following pairs of inequalities describes the shaded region in the diagram? 

  1. `y^2 <= 4-x\ \ and\ \ y <= x-2`  
  2. `y^2 <= 4-x\ \ and\ \ y >= x-2`  
  3. `y^2 >= 4-x\ \ and\ \ y<= x-2`  
  4. `y^2 >= 4-x\ \ and\ \ y >= x-2`
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`A`

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♦  Mean mark 44%.

`text(Using information from diagram)`

`(3,0)\ text(is in the shaded region)`

`text{Substituting (3,0) into}\ \ \ y^2<=4-x,\ \ \ 0 <= 4-3 => text(true)`

`:.\ text(Cannot be)\ C\ text(or)\ D`
 

`text(Similarly)`

`(3,0)\ text(must satisfy other inequality)`

`text(i.e.)\ \ y <= x-2\ \ text(becomes)\ \ 0<= 3-2 =>\ text(true)`

`=>  A`