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Trigonometry, EXT1 T1 2024 HSC 5 MC

Consider the function  \(g(x) = 2 \sin^{-1}(3x)\).

Which transformations have been applied to \(f(x) = \sin^{-1}(x)\)  to obtain \(g(x)\)?

  1. Vertical dilation by a factor of \(\dfrac{1}{2}\) and a horizontal dilation by a factor of \(\dfrac{1}{3}\)
  2. Vertical dilation by a factor of \(\dfrac{1}{2}\) and a horizontal dilation by a factor of 3
  3. Vertical dilation by a factor of 2 and a horizontal dilation by a factor of \(\dfrac{1}{3}\)
  4. Vertical dilation by a factor of 2 and a horizontal dilation by a factor of 3
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\(C\)

Show Worked Solution

\(\text{A vertical dilation of factor 2:}\)

\(f(x) = \sin^{-1}(x)\ \ \rightarrow \ \ f_1(x) = 2\sin^{-1}(x)\)

\(\text{A horizontal dilation of factor}\ \dfrac{1}{3}:\)

\(f_1(x) = 2\sin^{-1}(x)\ \ \rightarrow \ \ f_2(x) = 2\sin^{-1}(3x)\)

\(\Rightarrow C\)

Trigonometry, EXT1 T1 EQ-Bank 5 MC

Which equation describes the graph shown?
 

  1. \(y=2 \cos ^{-1}(x+1)\)
  2. \(y= \sin ^{-1}(x-1)\)
  3. \(y=2 \cos ^{-1}(x-1)\)
  4. \(y=\dfrac{1}{2} \sin ^{-1}(x+1)\)
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\(C\)

Show Worked Solution

\(\text{Graph shape (general)}\ \ → \cos^{-1}(x) \)

\(\text{Graph translated 1 unit to the right}\ → \cos^{-1}(x-1) \)

\(\text{Range:}\ 0 \leq \cos^{-1}(x-1) \leq \pi\ \ →\ 0 \leq 2\cos^{-1}(x-1) \leq 2\pi \)

\(\Rightarrow C\)

Trigonometry, EXT1 T1 EQ-Bank 4 MC

The graph of the function  `y = sin^(-1)(x-2)`  is transformed by being dilated by a factor of 3 from the `y`-axis and then translated to the right by 2.

What is the equation of the transformed graph?

  1. `y = sin^(-1)((x-8)/3)`
  2. `y = sin^(-1)((x-12)/3)`
  3. `y = sin^(-1)(3x-4)`
  4. `y = sin^(-1)(3x-8)`
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`A`

Show Worked Solution

`y = sin^(-1)(x-2)`
 

`text(Dilate by factor 3 from the)\ ytext(-axis)`

`text(Swap:)\ \ x -> x/3`

`y_1` `= sin^(-1)(x/3-2)`
  `= sin^(-1)((x-6)/3)`

 

`text(Translate to the right by 2)`

`text(Swap:)\ \ x -> x-2`

`y_2` `= sin^(-1)(((x-2)-6)/3)`
  `= sin^(-1)((x-8)/3)`

 
`=>A`

Trigonometry, EXT1 T1 EQ-Bank 3 MC

The graph of the function  `y = arccos(x-3)`  is transformed by being dilated horizontally with a scale factor of `1/2` and then translated to the left by 1.

What is the equation of the transformed graph?

  1. `y = cos^(-1)((x-5)/2)`
  2. `y = cos^(-1)((x-8)/2)`
  3. `y = cos^(-1)(2x-1)`
  4. `y = cos^(-1)(2x-2)`
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`C`

Show Worked Solution

`y = cos^(-1)(x-3)`

`text(Dilate horizontally with scale factor)\ 1/2`

`text(Swap:)\ \ x -> 2x`

`y_1 = cos^(-1)(2x-3)`

 

`text(Translate to the left by 1)`

`text(Swap:)\ \ x -> x + 1`

`y_2` `= cos^(-1)(2 (x + 1)-3)`
  `= cos^(-1)(2x + 2-3)`
  `= cos^(-1)(2x-1)`

 
`=>C`