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Functions, 2ADV F2 SM-Bank 3

 

The diagram below shows part of the graph of the function with rule

`f (x) = k log_e (x + a) + c`, where `k`, `a` and `c` are real constants.
 

    • The graph has a vertical asymptote with equation  `x = –1`.
    • The graph has a y-axis intercept at 1.
    • The point `P` on the graph has coordinates  `(p, 10)`, where `p` is another real constant.
       

      VCAA 2010 1b

  1. State the value of `a`.  (1 mark)

  2. Find the value of `c`.  (1 mark)

  3. Show that  `k = 9/(log_e (p + 1)`.  (2 marks)

Show Answers Only
  1. `1`
  2. `1`
  3. `text(Proof)\ \ text{(See Worked Solutions)}`
Show Worked Solution

i.   `text(Vertical Asymptote:)`

`x = – 1`

`:. a = 1`
 

  ii.   `text(Solve)\ \ f(0) = 1\ \ text(for)\ \ c,`

`c = 1`
 

iii.  `f(x)= k log_e (x + 1) + 1`

`text(S)text(ince)\ \ f(p)=10,`

`k log_e (p + 1) + 1` `= 10`
`k log_e (p + 1)` `= 9`
`:. k` `= 9/(log_e (p + 1))\ text(… as required)`

Functions, 2ADV F2 SM-Bank 1

  1.  Draw the graph  `y = ln x`.  (1 mark)

  2.  Explain how the above graph can be transformed to produce the graph
     
             `y = 3ln(x + 2)`
     
    and sketch the graph, clearly identifying all intercepts.  (3 marks)

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  1.  

  2.  
Show Worked Solution

i.

 

ii.   `text(Transforming)\ \ y = ln x => \ y = ln(x + 2)`

`y = ln x\ \ =>\ text(shift 2 units to left.)`
 

`text(Transforming)\ \ y = ln(x + 2)\ \ text(to)\ \ y = 3ln(x + 2)`

`=>\ text(increase each)\ y\ text(value by a factor of 3)`