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Functions, MET2 2023 VCAA 9 MC

The function \(f\) is given by
 

\(f(x) = \begin {cases}
\tan\Bigg(\dfrac{x}{2}\Bigg)         &\ \ 4 \leq x \leq 2\pi \\
\sin(ax) &\ \ \ 2\pi\leq x\leq 8
\end{cases}\)

 
The value of \(a\) for which \(f\) is continuous and smooth at  \( x\) = \(2\pi\)  is

  1. \(-2\)
  2. \(-\dfrac{\pi}{2}\)
  3. \(-\dfrac{1}{2}\)
  4. \(\dfrac{1}{2}\)
  5. \(2\)
Show Answers Only

\(C\)

Show Worked Solution

\(\text{Solve for}\ a\ \text{given}\ \ x=2\pi:\)

\(\tan\Bigg(\dfrac{2\pi}{2}\Bigg)=\sin(a2\pi)=0\)

\(a=\pm \dfrac{1}{2}\)
 

\(\text{For smoothness, solve for}\ a\ \text{given}\ \ x=2\pi:\)

\(\dfrac{d}{dx}\tan\Bigg(\dfrac{x}{2}\Bigg)=\dfrac{d}{dx}\sin(ax)\)

\(\therefore a=-\dfrac{1}{2}\)

 
\(\Rightarrow C\)


♦ Mean mark 46%.

Graphs, MET1 2020 VCAA 3

Shown below is part of the graph of a period of the function of the form  `y = tan(ax + b)`.
 


 

The graph is continuous for  `x = ∈ [-1, 1]`.

Find the value of `a` and the value of `b`, where  `a > 0`  and  `0 < b < 1`.   (3 marks)

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`a = (7pi)/24, b = pi/24`

Show Worked Solution

`y = tan(ax + b)`

`text(Substitute)\ \ (1, sqrt3), (-1,-1)\ \ text(into equation:)`

`tan(a + b)` `= sqrt3`
`tan(b-a)` `=-1`
`a + b` `= pi/3 \ …\ (1)`
`b-1` `=-pi/4 \ …\ (2)`

 
`text{Add (1) + (2):}`

`2b` `= pi/3-pi/4`
`b` `= pi/24`

 
`text{Substitute into (1):}`

`a + pi/24` `= pi/3`
`a` `= (7pi)/24`

Graphs, MET2 2018 VCAA 11 MC

The graph of  `y = tan(ax)`, where  `a ∈ R^+`, has a vertical asymptote  `x = 3 pi`  and has exactly one `x`-intercept in the region  `(0, 3 pi)`.

The value of `a` is

  1. `1/6`
  2. `1/3`
  3. `1/2`
  4. `1`
  5. `2`
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`C`

Show Worked Solution

`y = tan(ax)`

`tan x ->\ text(period of)\ pi,\ text(asymptotes at)\ \ x = pi/2, (3 pi)/2`

`tan(x/2) ->\ text(period of)\ 2 pi,\ text(asymptotes at)\ \ x=pi, 3 pi`

`tan(x/2) -> text(has one)\ x text(-intercept of)\ 2 pi\ \ {x: (0, 3 pi)}`

`:. a = 1/2`
 

`=>   C`

Graphs, MET2 2012 VCAA 6 MC

A section of the graph of  `f` is shown below.

VCAA 2012 6mc

The rule of `f` could be

  1. `f (x) = tan (x)`
  2. `f (x) = tan (x - pi/4)`
  3. `f (x) = tan(2 (x - pi/4))`
  4. `f (x) = tan(2 (x - pi/2))`
  5. `f (x) = tan(1/2 (x - pi/2))`
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`C`

Show Worked Solution

`text(Period) = pi/2`

`=>\ text(must be)\ C\ text(or)\ D`

`text(Shift)\ \ y = tan(x)\ \ text(right)\ \ pi/4.`

`=>   C`