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Functions, 2ADV F2 EQ-Bank 2

A graph of the hyperbola  \(y=\dfrac{1}{x+p}+q\)  is shown, where \(p\) and \(q\) are constants.
 

Find the values of \(p\) and \(q\) and hence the graphs \(x\)-axis intercept.   (2 marks)

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\(\text {Vertical asymptote at } \ x=-1 \ \ \Rightarrow p=1\)

\(\text {Horizontal asymptote at } \ y=\dfrac{3}{2} \ \  \Rightarrow q=\dfrac{3}{2}\)

\(y=\dfrac{1}{x+1}+\dfrac{3}{2}\)
 

\(x\text{-intercept when} \ \ y=0:\)

  \(\dfrac{1}{x+1}+\dfrac{3}{2}\)  \(=0\)
  \(\dfrac{1}{x+1}\)  \(=-\dfrac{3}{2}\)
  \(3 x+3\)  \(=-2\)
  \(3x\)  \(=-5\)
  \(x\)  \(=-\dfrac{5}{3}\)

Show Worked Solution

\(\text {Vertical asymptote at } \ x=-1 \ \ \Rightarrow p=1\)

\(\text {Horizontal asymptote at } \ y=\dfrac{3}{2} \ \  \Rightarrow q=\dfrac{3}{2}\)

\(y=\dfrac{1}{x+1}+\dfrac{3}{2}\)
 

\(x\text{-intercept when} \ \ y=0:\)

  \(\dfrac{1}{x+1}+\dfrac{3}{2}\)  \(=0\)
  \(\dfrac{1}{x+1}\)  \(=-\dfrac{3}{2}\)
  \(3 x+3\)  \(=-2\)
  \(3x\)  \(=-5\)
  \(x\)  \(=-\dfrac{5}{3}\)

Functions, 2ADV F2 2022 SPEC2 3 MC

The graph of  `y=\frac{x^2+2x+c}{x^2-4}`, where `c \in R`, will always have

  1. two vertical asymptotes and one horizontal asymptote.
  2. a vertical asymptote with equation `x=-2` and one horizontal asymptote with equation `y=1`.
  3. one horizontal asymptote with equation `y=1` and only one vertical asymptote with equation `x=2`.
  4. a horizontal asymptote with equation `y=1` and at least one vertical asymptote.
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`D`

Show Worked Solution

`y=\frac{x^2+2x+c}{x^2-4}\ \ =>\ \ y=\frac{x^2+2x+c}{(x-2)(x+2)}`

`text{Vertical asymptotes:}\ x=2, \ x=-2`

`->\ text{However, if}\ c=0,\ \text{only 1 vertical asymptote at}\ \ x=2.`

`text{Horizontal asymptote:}\ y=1`

`=>D`


♦♦ Mean mark 38%.
41% of students chose `A` and did not consider when `c = 0`

Functions, 2ADV F2 2021 HSC 19

Without using calculus, sketch the graph of  `y = 2 + 1/(x + 4)`, showing the asymptotes and the `x` and `y` intercepts.  (3 marks)

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Show Worked Solution

`text(Asymptotes:)\ x = -4`

`text(As)\ \ x -> ∞, y -> 2`

`ytext(-intercept occurs when)\ \ x = 0:`

`y = 2.25`

`xtext(-intercept occurs when)\ \ y = 0:`

`2 + 1/(x + 4) = 0 \ => \ x = -4.5`
 

Functions, 2ADV F2 SM-Bank 10 MC

The graph of the function  `f(x) = (x - 3)/(2 - x)`  has asymptotes

  1. `x = 3,\ \ \ \ \ \ \ \ \ y = 2`
  2. `x = -2,\ \ \ \ \ y = 1`
  3. `x = 2,\ \ \ \ \ \ \ \ \ y = -1`
  4. `x = 2,\ \ \ \ \ \ \ \ \ y = 1`
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`D`

Show Worked Solution
`y` `=-((3-x)/(2-x))`
  `=-((2-x)/(2-x) + 1/(2-x))`
  `=-1 -1/(2-x)`

 
`:.\ text(Asymptotes:)\ \ x = 2, \ y = – 1`

`=>   C`

Functions, 2ADV’ F2 2007 HSC 3b

  1. Find the vertical and horizontal asymptotes of the hyperbola

     

    `qquad y = (x − 2)/(x − 4)`  and hence sketch the graph of  `y = (x − 2)/(x − 4)`.  (3 marks)

  2. Hence, or otherwise, find the values of  `x`  for which  `(x − 2)/(x − 4) ≤ 3`.  (2 marks)

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  1. `text(See Worked Solutions.)`
  2. `x < 4\ text(and)\ x ≥ 5`
Show Worked Solution

i.    `y = (x − 2)/(x − 4)`

`text(Vertical asymptote at)\ \ x = 4`

`lim_(x → ∞) (x − 2)/(x − 4)` `= lim_(x → ∞) (1 − 2/x)/(1 − 4/x)=1`

`ytext(–intercept)\ = 1/2`

`xtext(–intercept)\ = 2`

 

Geometry and Calculus, EXT1 2007 HSC 3b Answer

 

ii.  `text(Find)\ \ x\ \ text(so that)\ \ (x − 2)/(x − 4) ≤ 3`

`(x − 2)/(x − 4)` `= 3`
`x − 2` `= 3x − 12`
`2x` `= 10`
`x` `= 5`

 
`=>(5, 3)\ \ text(is the intersection of)\ \ y = 3\ and\ y = (x − 2)/(x − 4)`
 

`:. (x − 2)/(x − 4) ≤ 3\ \ text(when)\ \ x < 4\ \ text(and)\ \ x ≥ 5.`

Functions, 2ADV’ F2 2015 HSC 5 MC

What are the asymptotes of `y = (3x)/((x + 1)(x + 2))`

A.    `y = 0,` `x = −1,` `x = −2`
B.    `y = 0,` `x = 1,` `x = 2`
C.    `y = 3,` `x = −1,` `x = −2`
D.    `y = 3,` `x = 1,` `x = 2`
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`A`

Show Worked Solution

`y = (3x)/((x + 1)(x + 2))`

`text(Asymptotes at)\ \ x = −1\ \ text(and)\ \ x = −2`

`text(As)\ \ x → ∞, y → 0^+`

`text(As)\ \ x → −∞, y → 0^−`

`:.\ text(Horizontal asymptote at)\ \ y = 0`

`⇒ A`