smc-753-40-Combinations

Graphs, MET2 2020 VCAA 13 MC

The transformation `T:R^(2)rarrR^(2)` that maps the graph of `y=cos(x)` onto the graph of `y=cos(2x+4)` is

`T([[x],[y]])=[[(1)/(2),0],[0,1]]([[x],[y]]+[[-4],[0]])`
`T([[x],[y]])=[[(1)/(2),0],[0,1]][[x],[y]]+[[-4],[0]]`
`T([[x],[y]])=[[(1)/(2),0],[0,1]]([[x],[y]]+[[-2],[0]])`
`T([[x],[y]])=[[2,0],[0,1]]([[x],[y]]+[[2],[0]])`
`T([[x],[y]])=[[2,0],[0,1]][[x],[y]]+[[2],[0]]`

Show Answers Only

`A`

Show Worked Solution

`y=cos(2x+4)=cos(2(x+2))`

♦♦♦ Mean mark 26%.

`text{Dilation of factor} \ 1/2 \ text{from} \ y text{-axis}.`

`text{Translation of 2 units to the left.}`

`T([[x],[y]])=[[(1)/(2),0],[0,1]][[x],[y]]+[[-2],[0]]=[[(1)/(2),0],[0,1]]([[x],[y]]+[[-4],[0]])`

`=> A`

Functions, MET2 2021 VCAA 5 MC

Consider the following four functional relations.

`f(x) = f(-x)\ \ \ \ \ -f(x) = f(-x)\ \ \ \ \ f(x) = -f(x)\ \ \ \ \ (f(x))^2 = f(x^2)`

The number of these functional relations that are satisfied by the function `f : R -> R, \ f(x) = x` is

Show Answers Only

`C`

Show Worked Solution

`text{Consider each relation}`

`f(x) = x \ ,`	`f(–x) = –x \ \ ✘`
`–f(x) = –x \ ,`	` f(-x) = –x \ \ ✓`
`f(x) = x \ ,`	`–f(x) = –x \ \ ✘`
`(f(x))^2 = x^2 \ ,`	`f(x^2) = x^2 \ \ ✓`

`=> C`

Algebra, MET1 2021 VCAA 5

Let `f:R -> R, \ f(x) = x^2 - 4` and `g:R -> R, \ g(x) = 4(x - 1)^2 - 4`.

The graphs of `f` and `g` have a common horizontal axis intercept at `(2, 0)`.
Find the coordinates of the other horizontal axis intercept of the graph of `g`. (2 marks)
Let the graph of `h` be a transformation of the graph of `f` where the transformations have been applied in the following order:
• dilation by a factor of `1/2` from the vertical axis (parallel to the horizontal axis)
• translation by two units to the right (in the direction of the positive horizontal axis
State the rule of `h` and the coordinates of the horizontal axis intercepts of the graph of `h`. (2 marks)

Show Answers Only

`(0,0)`
`(1,0) and (3,0)`

Show Worked Solution

a. `xtext(-axis intercept of)\ g(x):`

`4(x-1)^2-4`	`=0`
`(x-1)^2`	`=1`
`x-1`	`=+-1`

`text(When)\ \ x-1=-1\ \ =>\ \ x=0`

`:.\ text{Other horizontal intercept occurs at (0, 0)}`

b. `text(1st transformation)`

♦♦ Mean mark part (b) 25%.

`text(Dilation by a factor of)\ 1/2\ text(from the)\ ytext(-axis:)`

`x^2 – 4 \ → \ (x/(1/2))^2 -4 = 4x^2-4`

`text(2nd transformation)`

`text(Translation by 2 units to the right:)`

`4x^2-4 \ → \ h(x) = 4(x-2)^2 – 4`

`xtext(-axis intercept of)\ h(x):`

`4(x-2)^2-4`	`=0`
`(x-2)^2`	`=1`
`x-2`	`=+-1`

`x-2=1 \ => \ x=3`

`x-2=-1 \ => \ x=1`

`:.\ text(Horizontal axis intercepts occur at)\ (1,0) and (3,0).`

Functions, MET2-NHT 2019 VCAA 17 MC

The graph of the function `g` is obtained from the graph of the function `f` with rule `f(x) = cos(x) - (3)/(8)` by a dilation of factor `(4)/(pi)` from the `y`-axis, a dilation of factor `(4)/(3)` from the `x`-axis, a reflection in the `y`-axis and a translation of `(3)/(2)` units in the positive `y` direction, in that order.

The range and period of `g` are respectively

`[–(1)/(3) , (7)/(3)] \ text(and) \ 2`
`[–(1)/(3) , (7)/(3)] \ text(and) \ 8`
`[–(7)/(3) , (1)/(3)] \ text(and) \ 2`
`[–(7)/(3) , (1)/(3)] \ text(and) \ 8`
`[–(4)/(3) , (4)/(3)] \ text(and) \ (pi^2)/(2)`

Show Answers Only

`B`

Show Worked Solution

`text(Dilation of) \ \ (4)/(pi) \ \ text(from) \ y text(-axis:)`

`cos x -(3)/(8) \ => \ cos ((pi x)/(4)) – (3)/(8)`

`text(Dilation of) \ \ (4)/(3) \ \ text(from) \ x text(-axis:)`

`cos((pi x)/(4)) – (3)/(8) \ => \ (4)/(3) (cos({pi x}/{4}) -(3)/(8)) = (4)/(3) \ cos((pi x)/(4)) – (1)/(2)`

`text(Reflection in) \ y text(-axis and translation up) \ (3)/(2) :`

`(4)/(3) cos((pi x)/(4)) – (1)/(2) \ => \ (4)/(3) \ cos((–pi x)/(4)) + 1`

`:. y = (4)/(3) \ cos ((–pi x)/(4)) + 1`

`text(Range:) \ [1 – (4)/(3) , 1 + (4)/(3)] = [–(1)/(3) , (7)/(3)]`

`text(Period:) \ (2 pi)/(n) = (pi)/(4) \ => \ n = 8`

`=> \ B`

Graphs, MET2 2019 VCAA 13 MC

The graph of the function `f` passes through the point `(-2, 7)`.

If `h(x) = f(x/2) + 5`, then the graph of the function `h` must pass through the point

`(-1, -12)`
`(-1, 19)`
`(-4, 12)`
`(-4, -14)`
`(3, 3.5)`

Show Answers Only

`C`

Show Worked Solution

`h(x) = f(x/2) + 5`

`text(Dilate by a factor 2 from)\ y text(-axis)`

`(-2, 7) -> (-2 xx 2, 7) -= (-4, 7)`

`text(Translate up 5 vertically):`

`(-4, 7) -> (-4, 7 + 5) -= (-4, 12)`

`=> C`

Graphs, MET2 2018 VCAA 4 MC

The point `A (3, 2)` lies on the graph of the function `f`. A transformation maps the graph of `f` to the graph of `g`,

where `g(x) = 1/2 f(x - 1)`. The same transformation maps the point `A` to the point `P`.

The coordinates of the point `P` are

`(2, 1)`
`(2, 4)`
`(4, 1)`
`(4, 2)`
`(4, 4)`

Show Answers Only

`C`

Show Worked Solution

`g(x) = 1/2 f(x – 1),\ A(3, 2)`

♦ Mean mark 48%.

`text(Dilate by a factor of)\ 1/2\ text(from)\ x text(-axis:)`

`A(3, 2) -> A′(3, 1)`

`text(Translate 1 unit to right:)`

`A′(3, 1) -> P(4, 1)`

`=> C`

Graphs, MET2 2007 VCAA 15 MC

The graph of the function `f: [0, oo) -> R` where `f(x) = 3x^(5/2)` is reflected in the `x`-axis and then translated 3 units to the right and 4 units down.

The equation of the new graph is

`y = 3(x - 3)^(5/2) + 4`

`y = -3 (x - 3)^(5/2) - 4`

`y = -3 (x + 3)^(5/2) - 1`

`y = -3 (x - 4)^(5/2) + 3`

`y = 3(x - 4)^(5/2) + 3`

Show Answers Only

`B`

Show Worked Solution

`text(Let)\ \ y= 3x^(5/2)`

`text(Reflect in the)\ x text(-axis:)`

`y= – 3x^(5/2)`

`text(Translate 3 units to the right:)`

`y=- 3(x-3)^(5/2)`

`text(Translate 4 units down:)`

`y=- 3(x-3)^(5/2) – 4`

`=> B`

Graphs, MET2 2008 VCAA 18 MC

Let `f: [0, pi/2] -> R,\ f(x) = sin(4x) + 1.` The graph of `f` is transformed by a reflection in the `x`-axis followed by a dilation of factor 4 from the `y`-axis.

The resulting graph is defined by

`g: [0, pi/2] -> R\ \ \ \ \ \ g(x) = -1 - 4 sin (4x)`
`g: [0, 2 pi] -> R\ \ \ \ \ \ \ g(x) = -1 - sin (16x)`
`g: [0, pi/2] -> R\ \ \ \ \ \ g(x) = 1 - sin (x)`
`g: [0, 2 pi] -> R\ \ \ \ \ \ \ g(x) = 1 - sin (4x)`
`g: [0, 2 pi] -> R\ \ \ \ \ \ \ g(x) = -1 - sin (x)`

Show Answers Only

`E`

Show Worked Solution

`f(x) = sin(4x)+1`

♦♦ Mean mark 36%.

`text(Reflecting in the)\ x text(-axis,)`

`=> h(x) = – sin (4x) – 1`

`text(Dilation by a factor of 4 from the)\ \ y text(-axis,)`

`=> g(x) = -sin(x)-1`

`text(Dilation factor: adjust the domain from)\ \ [0, pi/2]`

`text(to)\ \ [0xx4, pi/2 xx 4]=[0,2pi].`

`=> E`

Graphs, MET2 2009 VCAA 19 MC

The graph of a function `f`, with domain `R`, is as shown.

The graph which best represents `1 - f (2x)` is

Show Answers Only

`E`

Show Worked Solution

`text(Determine transformations)`

`text(from)\ \ f -> – f (2x) + 1`

`text(- Dilate by factor)\ 1/2\ text(from)\ y text(-axis.)`

`text(- Reflect in)\ x text(-axis.)`

`text(- Shift up 1 unit.)`

`=> E`

Graphs, MET2 2016 VCAA 12 MC

The graph of a function `f` is obtained from the graph of the function `g` with rule `g(x) = sqrt (2x - 5)` by a reflection in the `x`-axis followed by a dilation from the `y`-axis by a factor of `1/2`.

Which one of the following is the rule for the function `f`?

`f(x) = sqrt (5 - 4x)`
`f(x) = - sqrt (x - 5)`
`f(x) = sqrt (x + 5)`
`f(x) = −sqrt (4x - 5)`
`f(x) = −sqrt (4x - 10)`

Show Answers Only

`D`

Show Worked Solution

`text(Let)\ \ y=sqrt(2x-5)`

`text(1st transformation:)`

`y = – sqrt(2x-5)`

`text(2nd transformation:)`

`y`	`=-sqrt(2(2x)-5)`
	`=- sqrt(4x-5)`
`:. f(x)`	`= −sqrt(4x – 5)`

`=> D`

Graphs, MET2 2014 VCAA 1 MC

The point `P\ text{(4, −3)}` lies on the graph of a function `f`. The graph of `f` is translated four units vertically up and then reflected in the `y`-axis.

The coordinates of the final image of `P` are

`text{(−4, 1)}`
`text{(−4, 3)}`
`text{(0, −3)}`
`text{(4, −6)}`
`text{(−4, −1)}`

Show Answers Only

`A`

Show Worked Solution

`text(Using mapping notation:)`

`P(4,−3)\ {:(y + 4),(vec(quad(1)quad)):}\ (4,1)\ {:(−x),(vec(\ (2)\ )):}\ pprime(−4,1)`

`=> A`