smc-1024-11-arccos Graphs

Trigonometry, EXT1 T1 2024 HSC 4 MC

What are the domain and range of the function \(y = 2 \cos^{-1}(2x) + 2 \sin^{-1}(2x)\)?

Domain: \([-0.5, 0.5]\) and Range: \(\{\pi\}\)
Domain: \([-0.5, 0.5]\) and Range: \([-\pi, 3 \pi ]\)
Domain: \([-2, 2]\) and Range: \(\{\pi\}\)
Domain: \([-2, 2]\) and Range: \([-\pi, 3\pi]\)

Show Answers Only

\(A\)

Show Worked Solution

\(\text{Domain:}\ \ -1 \leqslant 2x \leqslant 1 \ \ \Rightarrow\ \ -\dfrac{1}{2} \leqslant x \leqslant \dfrac{1}{2} \)

\(\text{Range:}\ \ 2\Big(\cos^{-1}(2x)+ \sin^{-1}(2x)\Big) = 2 \times \dfrac{\pi}{2} = \pi\)

\(\Rightarrow A\)

Trigonometry, EXT1 T1 EQ-Bank 4

Sketch the graph \(y=2 \cos ^{-1}(x+1)\). (3 marks)

--- 10 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

\(\text{Domain:}\)

\(-1 \leqslant x+1 \leqslant 1 \ \Rightarrow \ -2 \leqslant x \leqslant 0\)

\(\text{Range:}\)

\(0 \leqslant \cos ^{-1}(x) \leqslant \pi\ \ \Rightarrow \ \ 0\leqslant 2 \cos ^{-1}(x) \leqslant 2 \pi\)

Trigonometry, EXT1 T1 SM-Bank 4

Sketch `y=3cos^(-1)(2x-1)` (3 marks)

Show Answers Only

Show Worked Solution

`text{Domain:}`

`-1`	`<=(2x-1)<=1`
`0`	`<=2x<=2`
`0`	`<=x<=1`

`text{Range of}\ \ cos^(-1)(x)=[0,pi]`

`=>\ text{Range of}\ \ 3cos^(-1)(x)=[0,3pi]`

`text{At}\ \ x=0,\ \ y=3cos^(-1)(-1)=3pi`

`text{Sketch}\ \ y=3cos^(-1)(2x-1):`

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?

`y = cos^(-1)((x-5)/2)`
`y = cos^(-1)((x-8)/2)`
`y = cos^(-1)(2x-1)`
`y = cos^(-1)(2x-2)`

Show Answers Only

`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`

Trigonometry, EXT1 T1 2019 HSC 9 MC

Which graph best represents `y = cos^(-1) (-sin x)`, for `-pi/2 <= x <= pi/2`?

A.		B.
C.		D.

Show Answers Only

`D`

Show Worked Solution

`y`	`= cos^(-1)(-sin x)`
	`= cos^(-1)(cos(x + pi/2))`
	`= x + pi/2`

`=> D`

Trigonometry, EXT1 T1 2017 HSC 11d

Sketch the graph of the function `y = 2cos^(-1)x`. (2 marks)

Show Answers Only

Show Worked Solution

`y = 2cos^(-1)x`

`=> text(Domain)\ -1 <= x <= 1`

`text(Range)\ \ y = 2cos^(-1)x\ \ text(is)\ \ 0 <= y <= pi`

`=> text(Range)\ \ y = 2cos^(-1)x\ \ text(is)\ \ 0 <= y <= 2pi`

Trigonometry, EXT1 T1 2007 HSC 2b

Let `f(x) = 2 cos^(-1)x`.

Sketch the graph of `y = f(x)`, indicating clearly the coordinates of the endpoints of the graph. (2 marks)

--- 8 WORK AREA LINES (style=lined) ---
State the range of `f(x)`. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(See Worked Solutions.)`
`0 ≤ y ≤ 2pi`

Show Worked Solution

i. `y= 2\ cos^(-1)x`

`text(Domain:)\ -1 ≤ x ≤ 1`

`text{Range:}\ \0 ≤`	`y/2 ≤ pi`
`0 ≤`	`y ≤ 2pi`

ii. `text(Range:)\ \ 0 ≤ y ≤ 2pi`

Trigonometry, EXT1 T1 2011 HSC 2d

Sketch the graph of the function `f(x) = 2arccos x`. Clearly indicate the domain and range of the function. (2 marks)

Show Answers Only

Show Worked Solution

`y`	`= 2 cos^(-1) x`
`y/2`	`= cos^(-1)x`

`text(Domain)\ -1 <= x <= 1`

`text(S)text(ince)\ \ 0 <= y/2 <= pi`

`text(Range)\ 0 <= y <= 2pi`