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Functions, 2ADV’ F2 2019 HSC 4 MC

The diagram shows the graph of  `y = f(x)`.
 


 

Which equation best describes the graph?

A.     `y = x/(x^2 - 1)`

B.     `y = x^2/(x^2 - 1)`

C.     `y = x/(1 - x^2)`

D.     `y = x^2/(1 - x^2)`

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`B`

Show Worked Solution

`text(By elimination:)`

`text(Graph is an even function)`

`=> f(x) = f(-x)`

`:.\ text(Eliminate A and C)`
 

`text(When)\ -1 < x < 1,\ \ y <= 0`

`:.\ text(Eliminate D)`

`=>  B`

Functions, 2ADV’ F2 2012 HSC 13b

  1. Find the horizontal asymptote of the graph  `y=(2x^2)/(x^2 + 9)`.   (1 mark)

  2. Without the use of calculus, sketch the graph  `y=(2x^2)/(x^2 + 9)`, showing the asymptote found in part (i).    (2 marks)

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  1. `text(Horizontal asymptote at)\ y = 2`
  2.  
    Geometry and Calculus, EXT1 2012 HSC 13b Answer
Show Worked Solution
i.    `y` `= (2x^2)/(x^2 +9)`
    `= 2/(1 + 9/(x^2))`

 

`text(As)\ \ x -> oo,\ y ->2`

`text(As)\ \ x -> – oo,\ y -> 2`

`:.\ text(Horizontal asymptote at)\ y = 2`

 

ii.    `text(At)\ \ x = 0,\ y = 0`

`f(x) = (2x^2)/(x^2 + 9) >= 0\ text(for all)\ x`

`f(–x) = (2(–x)^2)/((–x)^2 + 9) = (2x^2)/(x^2 + 9) = f(x)`

`text(S)text(ince)\ \ f(x) = f(–x) \ \ =>\ text(EVEN function)`

Geometry and Calculus, EXT1 2012 HSC 13b Answer

Functions, 2ADV’ F2 2017 HSC 5 MC

Which graph best represents the function  `y = (2x^2)/(1 - x^2)`?
 

A. B.
       
C. D.
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`D`

Show Worked Solution
`y` `= (2x^2)/((1 – x^2))`
  `= −((2 – 2x^2 – 2))/((1 – x^2))`
  `= −(2(1 – x^2))/((1 – x^2)) – 2/((1 – x^2))`
  `= −2 – 2/((1 – x^2))`

 

`text(As)\ \ x -> oo,\ \ y -> −2`

`:. text(Horizontal asymptote at)\ \ y = −2`

`⇒D`