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Functions, EXT1* F1 2019 HSC 11g

The parabola  `y = x^2`  meets the line  `y = x + 2`  at the points  `(-1, 1)`  and  `(2, 4)`. Do NOT prove this.

By first sketching the graphs of  `y = x^2`  and  `y = x + 2`, shade the region which simultaneously satisfies the two inequalities  `y >= x^2`  and  `y >= x + 2`.  (2 marks)

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Functions, EXT1 F1 SM-Bank 3

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

  2.  

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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 2017 HSC 8 MC

The region enclosed by  `y = 4 - x,\ \ y = x`  and  `y = 2x + 1`  is shaded in the diagram below.
 

Which of the following defines the shaded region?

A.   `y <= 2x + 1, qquad` `y <= 4-x, qquad` `y >= x`
B.   `y >= 2x + 1, qquad` `y <= 4-x, qquad` `y >= x`
C.   `y <= 2x + 1, qquad` `y >= 4-x, qquad` `y >= x`
D.   `y >= 2x + 1, qquad` `y >= 4-x, qquad` `y >= x`
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`A`

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`text(Consider)\ \ y = 2x + 1,`

`text(Shading is below graph)`

`=> y <= 2x + 1`

`text(Consider)\ \ y = 4-x,`

`text(Shading is below graph)`

`=> y <= 4-x`

`=>  A`

Functions, EXT1 F1 2004 HSC 1a

Indicate the region on the number plane satisfied by  `y ≥ |\ x + 1\ |.`  (2 marks) 

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`text(See Worked Solution)`

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 Real Functions, EXT1 2004 HSC 1a Answer

`y ≥ |\ x + 1\ |`

`text(Test)\ (0, 0)`

`0 ≥ |\ 0 + 1\ |`

`0 ≥ 1\ \ \ \ \ \ ⇒\ text(False)`

 

`:.\ text(Shaded area represents)`

`y ≥ |\ x + 1\ |`

Functions, EXT1* F1 2015 HSC 13b

  1. Find the domain and range for the function  `f(x) = sqrt (9 - x^2)`.  (2 marks)

  2. On a number plane, shade the region where the points `(x, y)` satisfy both of the inequalities
     
      `qquad y <= sqrt (9 - x^2)`  and  `y >= x` .  (2 marks)

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  1. `text(Domain)\ -3 <= x <= 3\ \ \ \ \ \ \ \ text(Range)\ 0<= y <= 3`
  2. `text(See Worked Solutions)`
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i.  `f(x) = sqrt(9 – x^2)`

`text(Domain)`

`9 – x^2` `>= 0`
`x^2` `<= 9`

`-3 <= x <= 3`

`text(Range)`

`0 <= y <= 3`

 

♦ Mean mark 34%.
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`

Functions, EXT1* F1 2013 HSC 11g

Sketch the region defined by  `(x-2)^2 + ( y-3)^2 >= 4`.    (3 marks)

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`text(The region is the exterior of a circle,)`

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

`text(centre)\ text{(2,3)}\ text(and radius 2.)`