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Functions, 2ADV EQ-Bank 4

  1. Without calculus, sketch the graph of \(g(x)\), where  \(g(x)=-2-\dfrac{3}{x}\)
  2. In your answer label all asymptotes and any intercepts.   (3 marks)

  3. Express the range of \(g(x)\) in set notation.   (1 mark)

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a.    
       

b.    \(\text{Range} \ \ y \in(-\infty,-2) \cup(-2, \infty)\)

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a.    \(\text{As} \ \ x \rightarrow \infty, g(x) \rightarrow-2^{-}; \text{As} \ \ x \rightarrow-\infty, g(x) \rightarrow-2^{+}\)

\(\text{As} \ \ x \rightarrow 0^{+}, g(x) \rightarrow-\infty; \text{As} \ \ x \rightarrow 0^{-}, g(x) \rightarrow \infty\)

\(\text{Intercept} \ (x \text {-axis}):-2-\dfrac{3}{x}=0 \ \Rightarrow \ 2 x=-3 \ \Rightarrow \ x=-\dfrac{3}{2}\)
 

 
b.    \(\text{Range} \ \ y \in(-\infty,-2) \cup(-2, \infty)\)

Functions, 2ADV EQ-Bank 3

Without calculus, sketch the graph of  \(y=\dfrac{3 x-2}{x}\), labelling all asymptotes and intercepts.   (3 marks)

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Show Worked Solution

\(y=\dfrac{3 x-2}{x} \ \Rightarrow \ y=3-\dfrac{2}{x}\)

\(\text{As} \ \ x \rightarrow \infty, y \rightarrow 3^{-} ; \ \text{As} \ \ x \rightarrow-\infty, y \rightarrow 3^{+}\)

\(\text{As} \ \ x \rightarrow 0^{+}, y \rightarrow-\infty ; \ \text{As} \ \ x \rightarrow 0^{-}, y \rightarrow \infty \)
 

Functions, 2ADV F2 EQ-Bank 3 MC

The graph  \(y=\dfrac{2}{x-2}\)  undergoes the following transformations:

  • translated 3 units to the left
  • dilated vertically by a factor of 2

Determine which of the following is the new function.

  1.  \(2 y=\dfrac{2}{x+1}\)
  2.  \(\dfrac{y}{2}=\dfrac{2}{x-5}\)
  3.  \(2 y=\dfrac{2}{x-5}\)
  4.  \(\dfrac{y}{2}=\dfrac{2}{x+1}\)
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\(\Rightarrow D\)

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\(\text{Translate 3 units to the left:}\)

\(y=\dfrac{2}{x-2} \ \Rightarrow \ y^{′}=\dfrac{2}{(x+3)-2}=\dfrac{2}{x+1}\)
 

\(\text{Dilate vertically by a factor of 2:}\)

\(y^{′}=\dfrac{2}{x+1} \ \Rightarrow \ \dfrac{y^{″}}{2}=\dfrac{2}{x+1}\)

\(\Rightarrow D\)

Functions, 2ADV F2 2024 MET1 3*

Let  \(f(x)=\dfrac{1}{(x+3)^2}-2\).

On the axes below, sketch the graph of  \(y=f(x)\),  labelling all asymptotes with their equations and axis intercepts with their coordinates.   (4 marks)



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Show Worked Solution

\(\text{Find asymptotes:}\)

\((x+3) \neq 0\ \ \Rightarrow\ \ \text{Asymptote at}\ x=-3\)

\(\text{As}\ x \rightarrow \infty, \ \dfrac{1}{(x+3)^2} \rightarrow 0\ \ \Rightarrow \ \ \text{Asymptote at}\ y=-2\)
 

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

\(y=\dfrac{1}{(0+3)^2}-2=-\dfrac{17}{9}\)

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

\(\dfrac{1}{(x+3)^2}-2\) \(=0\)
\((x+3)^2\) \(=\dfrac{1}{2}\)
\(x+3\) \(=\pm\dfrac{1}{\sqrt{2}}\)
\(x\) \(=-3\pm\dfrac{1}{\sqrt{2}}\)

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 2023 MET1 3

  1. Sketch the graph of  \(f(x)=2-\dfrac{3}{x-1}\) on the axes below, labelling all asymptotes with their equation and axial intercepts with their coordinates.   (3 marks)
     


  2. Find the values of \(x\) for which \(f(x)\leq1\).   (1 mark)
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a.   

b.    \(1<x\leq4\ \ \ \text{or}\ \ \big(1,4\big]\)

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a.    \(\text{Vertical asymptote when}\ \ x=1\)

\(y\text{-int:}\ y=2-(-3)\ \ \Rightarrow\ \ y=5\)

\(x\text{-int:}\ 2-\dfrac{3}{x-1}=0\ \ \Rightarrow\ \ x=\dfrac{5}{2} \)

\(\text{As}\ \ x \rightarrow \infty, \ \ y \rightarrow 2^{-}; \ \ x \rightarrow -\infty, \ \ y \rightarrow 2^{+} \)

b.    \(\text{From the graph:}\)

\(f(x)=1\ \text{when }x=4\)

\(x>1\ \text{to the right of the vertical asymptote}\)

\(\therefore\ f(x)\leq1\ \text{when}\ \ 1<x\leq4\ \ \ \text{or}\ \ \big(1,4\big]\)


♦♦ Mean mark (b) 38%.
MARKER’S COMMENT: Many students did not use their graph from part (a).

Functions, 2ADV F2 SM-Bank 9 MC

The graph of the function  `f(x)=(3x+2)/(5-x)`, has asymptotes at

  1. `x=-5,y=(3)/(2)`
  2. `x=(2)/(3),y=-3`
  3. `x=5,y=3`
  4. `x=5,y=-3`
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`D`

Show Worked Solution
`f(x)` `=(3x+2)/(5-x)`  
  `=(-(15-3x)+17)/(5-x)`  
  `=-3+17/(5-x)`  

 
`text(Vertical asymptote:)\ \ x=5`

`text(Horizontal asymptote:)\ \ y=-3`

`=>D`

Functions, 2ADV F2 SM-Bank 20

  1. Sketch the graph of  `y = 1 - 2/(x - 2)`  on the axes below. Label asymptotes with their equations and axis intercepts with their coordinates.  (3 marks)
     

     
  2. Find the values of  `x`  for which  `1 - 2/(x - 2) >= 3`.  (1 mark)

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a. 

b. `x in [1, 2)`

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a.   `text(Asymptotes:)`

`x=2`

`text(As)\ \ x→ +-oo, \ y→1\ \ =>\ text(Asymptote at)\ \ y=1`

`ytext(-intercept at)\ (0,2)`

`xtext(-intercept at)\ (4,0)`


 

b.   `text(By inspection of the graph:)`

`1-2/(x-2) >=3\ \ text(for)\ \ x in [1, 2)`

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 F1 EQ-Bank 6

The graph of `f(x)` is shown below. It has asymptotes at  `y = 3`  and  `x = 2`.
 


 

Using interval notation, state the domain and range of `f(x)`.   (2 marks)

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`text(Domain:)\ [−4, 2) ∪ (2, ∞)`

`text(Range:)\ (−∞, ∞)`

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`text(Domain:)\ [−4, 2) ∪ (2, ∞)`

`text(Range:)\ (−∞, ∞)`

Functions, 2ADV F2 EQ-Bank 9

Consider the function  `f(x) = 1/(4x - 1)`.

  1. Find the domain of  `f(x)`.  (1 mark)

  2. Sketch  `f(x)`, showing all asymptotes and intercepts?  (2 marks)

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  1. `{text(all real)\ x,  x!=1/4}`
  2.   
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i.    `4x – 1` `!= 0`
  `x` `!= 1/4`

 
`:.\ text(Domain:)\ {text(all real)\ x,  x!=1/4}`

 

ii.   `text(When)\ \ x = 0, \ y = −1`

`text(As)\ \ x -> ∞, \ y -> 0^+`

`text(As)\ \ x -> −∞, \ y -> 0^−`

Functions, 2ADV F2 SM-Bank 12

Sketch the graph of  `f(x) = (2x+1)/(x-1)`. Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation.  (4 marks)

  

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

COMMENT: Manipulation of the equation makes graphing much easier.

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

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

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

`text(As)\ \ x->-1 (-),\ \ y->-oo`

`text(As)\ \ x->-1 (+),\ \ y->oo`

 

 

Functions, 2ADV F2 EQ-Bank 11

On the axes below, sketch the graph of   `f(x) = (2x-2)/(x + 1)`.

Label all axis intercepts. Label each asymptote with its equation.   (4 marks)

 
met1-2008-vcaa-q2

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met1-2008-vcaa-q2-answer1

Show Worked Solution
`(2x-2)/(x + 1)` `=(2(x+1) – 4)/(x+1)`
  `=2- 4/((x+1)`

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

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

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

`text(As)\ \ x->-1 (-),\ \ y->oo`

`text(As)\ \ x->-1 (+),\ \ y->-oo`

 

met1-2008-vcaa-q2-answer1

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`
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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.`