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Graphs, MET2 2025 VCAA 1 MC

A function that has a range of \([6,12]\) is

  1. \(f: R \rightarrow R, \ f(x)=6+3 \cos (9 x)\)
  2. \(f: R \rightarrow R, \ f(x)=6+6 \cos (3 x)\)
  3. \(f: R \rightarrow R, \ f(x)=9-3 \cos (6 x)\)
  4. \(f: R \rightarrow R, \ f(x)=9-6 \cos (3 x)\)
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\(C\)

Show Worked Solution

\(\text{By trial and error,}\)

\(\text{Consider option C:}\ \  f(x)=9-3 \cos (6 x)\)

\(\text{Since}\ \ -1 \leqslant \cos (6 x) \leqslant 1\)

\(\text{Range is} \ \ [-3+9,3+9]=[6,12]\)

\(\Rightarrow C\)

Graphs, MET1 2025 VCAA 3

Let  \(f:[0,2 \pi] \rightarrow R, f(x)=2 \cos (2 x)+1\).

  1. State the range of \(f\).   (1 mark)

  2. Solve  \(f(x)=0\)  for \(x\).   (3 marks)

  3. Sketch the graph of  \(y=f(x)\)  for  \(x \in\left[\dfrac{\pi}{2}, \dfrac{3 \pi}{2}\right]\) on the axes below.
  4. Label the endpoints with their coordinates.   (2 marks)

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a.    \(\text{Range of } f(x):-1 \leqslant y \leqslant 3\)

b.    \(x=\dfrac{\pi}{3}, \dfrac{2 \pi}{3}, \dfrac{4 \pi}{3}, \dfrac{5 \pi}{3}\)

c.   

   

Show Worked Solution

a.    \(\text{Amplitude}=2 \ \ \text{about} \ \ y=1.\)

\(\text{Range of } f(x):\ -1 \leqslant y \leqslant 3\)
 

b.     \(2 \cos (2 x)+1\) \(=0\)
  \(\cos (2 x)\) \(=-\dfrac{1}{2}\)

 
\(\text{Base angle}=\dfrac{\pi}{3}\)

\(2x=\pi-\dfrac{\pi}{3}, \pi+\dfrac{\pi}{3}, \cdots\)

\(2x=\dfrac{2 \pi}{3}, \dfrac{4 \pi}{3}, \dfrac{8 \pi}{3}, \dfrac{10 \pi}{3}\)

  \(x=\dfrac{\pi}{3}, \dfrac{2 \pi}{3}, \dfrac{4 \pi}{3}, \dfrac{5 \pi}{3}\)
  

c.   

   

♦ Mean mark (c) 49%.

Functions, MET1 2021 VCAA 3

Consider the function  `g: R -> R, \ g(x) = 2sin(2x).`

  1. State the range of `g`.   (1 mark)

  2. State the period of `g`.   (1 mark)

  3. Solve  `2 sin(2x) = sqrt3`  for  `x ∈ R`.   (3 marks)

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  1. `[-2,2]`
  2. `pi`
  3. `x= pi/6 + npi, pi/3 + npi\ \ \ (n in ZZ)`
Show Worked Solution

a.   `text(S)text(ince)  -1<sin(2x)<1,`

`text(Range)\  g(x) = [-2,2]`
 

b.   `text(Period) = (2pi)/n = (2pi)/2 = pi`
 

c.    `2sin(2x)` `=sqrt3`
  `sin(2x)` `=sqrt3/2`
  `2x` `=pi/3, (2pi)/3, pi/3 + 2pi, (2pi)/3 + 2pi, …`
  `x` `=pi/6, pi/3, pi/6+pi, pi/3+pi, …`

 
`:.\ text(General solution)`

`= pi/6 + npi, pi/3 + npi\ \ \ (n in ZZ)`

Graphs, MET2-NHT 2019 VCAA 2 MC

The diagram below shows one cycle of a circular function.
 

The amplitude, period and range of this function are respectively

  1. `3, 2 \ \ text(and)\ \ [-2, 4]`
  2. `3, (pi)/(2) \ \ text(and)\ \ [-2, 4]`
  3. `4, 4 \ \ text(and) \ \ [0, 4]`
  4. `4, (pi)/(4) \ \ text(and) \ \ [-2, 4]`
  5. `3, 4 \ \ text(and) \ \ [-2, 4]`
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`E`

Show Worked Solution

`text(Graph centres around)\ \ y = 1`

`text(Amplitude) \ = 3`

`:. \ text(Range) \ = [1 – 3, 1 + 3] = [-2, 4]`

`text(Period:) = 4`

`=> E`

Graphs, MET2 2019 VCAA 1 MC

Let  `f: R -> R,\ \ f(x) = 3 sin ((2x)/5) - 2`.

The period and range of  `f`  are respectively

  1. `5 pi`  and  `[-3, 3]`
  2. `5 pi`  and  `[-5, 1]`
  3. `5 pi`  and  `[-1, 5]`
  4. `(5 pi)/2`  and  `[-5, 1]`
  5. `(5 pi)/2`  and  `[-3, 3]`
Show Answers Only

`B`

Show Worked Solution
`text(Period)` `= (2pi)/n`
  `= (2 pi)/(2/5)`
  `= 5 pi`
   
`text(Range)` `= [-2 -3, -2 + 3]`
  `= [-5, 1]`

 
`=>   B`

Graphs, MET2 2017 VCAA 1 MC

Let  `f : R → R, \ f (x) = 5sin(2x) - 1`.

The period and range of this function are respectively

  1. `π\ text(and)\ [−1, 4]`
  2. `2π\ text(and)\ [−1, 5]`
  3. `π\ text(and)\ [−6, 4]`
  4. `2π\ text(and)\ [−6, 4]`
  5. `4π\ text(and)\ [−6, 4]`
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`C`

Show Worked Solution

`text(Period) = (2pi)/2 = pi`

`text(Range)` `= [−1 – 5, −1 + 5]`
  `= [−6 ,4]`

`=> C`

Graphs, MET2 2016 VCAA 2 MC

Let  `f: R -> R,\ f(x) = 1 - 2 cos ({pi x}/2).`

The period and range of this function are respectively

  1. `4 and [−2, 2]`
  2. `4 and [−1, 3]`
  3. `1 and [−1, 3]`
  4. `4 pi and [−1, 3]`
  5. `4 pi and [−2, 2]`
Show Answers Only

`B`

Show Worked Solution
`text(Period)` `= (2 pi)/n = (2pi)/(pi/2)=4`
   

`text(Amplitude = 2 and median is)\ \ y=1.`

`text(Range)` `= [1 – 2, quad 1 + 2]`
  `= [−1, 3]`

`=>   B`

Functions, MET1 2011 VCAA 3a

State the range and period of the function

`h: R -> R,\ \ h(x) = 4 + 3 cos ((pi x)/2)`.   (2 marks)

Show Answers Only

`text(Range) = [1, 7];\ \ \ text(Period) = 4`

Show Worked Solution
  `-1` `<cos ((pi x)/2)<1`
  `-3` `<3cos ((pi x)/2)<3`
  `1` `< 4+ 3cos ((pi x)/2)<7`

 

`:.\ text(Range:)\  [1, 7]`

`text(Period) = (2pi)/n = (2 pi)/(pi/2) = 4`

Graphs, MET2 2015 VCAA 1 MC

Let `f: R -> R,\ f(x) = 2sin(3x) - 3.`

The period and range of this function are respectively

  1. `text(period) = (2 pi)/3 and text(range) = text{[−5, −1]}`
  2. `text(period) = (2 pi)/3 and text(range) = text{[−2, 2]}`
  3. `text(period) = pi/3 and text(range) = text{[−1, 5]}`
  4. `text(period) = 3 pi and text(range) = text{[−1, 5]}`
  5. `text(period) = 3 pi and text(range) = text{[−2, 2]}`
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`A`

Show Worked Solution

`text(Range:)\ [−3 – 2, −3 + 2]`

`= [−5,−1]`

`text(Period) = (2pi)/n = (2pi)/3`

`=>   A`