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Functions, 2ADV F2 2024 MET2 12 MC

The graph of \(y=f(x)\) is shown below.

Which of the following options best represents the graph of \(y=f(2 x+1)\) ?
 

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\(A\)

Show Worked Solution

\(\text{By elimination:}\)

\(\text{Graph has been dilated by a factor of}\ \dfrac{1}{2}\ \text{from}\ y\text{axis.}\)

→ \(\text{Eliminate C and D.}\)

\(\text{Graph is then translated}\ \dfrac{1}{2}\ \text{unit to the left.}\)

\(\text{Consider the turning point}\ (2, 1)\ \text{after translation:}\)

\(\left(2, 1\right)\ \rightarrow \ \left(2\times \dfrac{1}{2}-\dfrac{1}{2}, 1\right)=\left(\dfrac{1}{2}, 1\right)\)

\(\therefore\ \text{Option A is the only possible solution.}\)

\(\Rightarrow A\)

♦ Mean mark 47%.

Functions, 2ADV F2 2024 HSC 7 MC

The diagram shows the graph  \(y = f(x)\).
 

Which of the following best represents the graph  \(y = f(2x-1)\)?
 

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\(C\)

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\(\text{At}\ \ x=0:\)

\(f(2x-1)=f(-1)\ \ \Rightarrow\ \ \text{Eliminate}\ A\ \text{and}\ B.\)
 

\(\text{Consider the transformations of}\ f(x) \rightarrow\ f(2x-1) \)

\(\rightarrow\ \text{Shift}\ f(x)\ \text{1 unit to the right.}\)

\(\rightarrow\ \text{Dilate}\ f(x-1)\ \text{by a factor of}\ \dfrac{1}{2}\ \text{from the}\ y\text{-axis.}\)

\(\Rightarrow C\)

♦ Mean mark 47%.

Functions, 2ADV F2 2024 HSC 4 MC

The parabola  \(y=(x-3)^2-2\)  is reflected about the \(y\)-axis. This is then reflected about the \(x\)-axis.

What is the equation of the resulting parabola?

  1. \(y=(x+3)^2+2\)
  2. \(y=(x-3)^2+2\)
  3. \(y=-(x+3)^2+2\)
  4. \(y=-(x-3)^2+2\)
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\( C \)

Show Worked Solution

\(y=(x-3)^2-2\)

\(\text{Reflect in the}\ y\text{-axis}\ (f(-x)):\)

\(y=(-x-3)^2-2\)

\(\text{Reflect in the}\ x\text{-axis}\ (-f(-x)):\)

\(y\) \(=-\left[(-x-3)^2-2\right]\)  
  \(=-(x+3)^2+2\)  

 
\( \Rightarrow C \)

♦ Mean mark 54%.

Functions, 2ADV F2 2022 HSC 19

The graph of the function  `f(x)=x^2`  is translated `m` units to the right, dilated vertically by a scale factor of `k` and then translated 5 units down. The equation of the transformed function is  `g(x)=3 x^2-12 x+7`.

Find the values of `m` and `k`.  (3 marks)

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`m=2, \ \ k=3`

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`text{Horizontal translation}\ m\ text{units to the right:}`

`x^2\ → \ (x-m)^2`

`text{Dilated vertically by scale factor}\ k:`

`(x-m)^2\ →\ k(x-m)^2`

`text{Vertical translation 5 units down:}`

`k(x-m)^2\ →\ k(x-m)^2-5`

`y` `=k(x-m)^2-5`  
  `=k(x^2-2mx+m^2)-5`  
  `=kx^2-2kmx+(km^2-5)`  

 
`:.k=3`

`-2km` `=-12`  
`:.m` `=2`  

♦ Mean mark 51%.

Functions, 2ADV F2 SM-Bank 16

Let  `f(x) = x^2 - 4`

Let the graph of `g(x)` be a transformation of the graph of `f(x)` 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

Find `g(x)` and the coordinates of the horizontal axis intercepts of the graph of `g(x)`.  (3 marks)

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

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`text(1st transformation)`

`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 \ => \ g(x) = 4(x-2)^2 – 4`
 

`xtext(-axis intercept of)\ g(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, 2ADV F2 2020 HSC 2 MC

The function  `f(x) = x^3`  is transformed to  `g(x) = (x - 2)^3 + 5`  by a horizontal translation of 2 units followed by a vertical translation of 5 units.

Which row of the table shows the directions of the translations?
 

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`B`

Show Worked Solution

`text(Horizontal translation: 2 units to the right)`

`x^3 -> (x – 2)^3`

`text(Vertical translation: 5 units up`

`(x – 2)^3 -> (x – 2)^3 + 5`

`=>\ B`

Functions, 2ADV F2 EQ-Bank 13

The curve  `y = kx^2 + c`  is subject to the following transformations

    • Translated 2 units in the positive `x`-direction
    • Dilated in the positive `y`-direction by a factor of 4
    • Reflected in the `y`-axis

The final equation of the curve is  `y = 8x^2 + 32x - 8`.

  1.  Find the equation of the graph after the dilation.  (1 mark)

  2.  Find the values of  `k`  and  `c`.  (2 marks)

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  1. `y = 4k(x – 2)^2 + 4c`
  2. `k = 2, c = −10`
Show Worked Solution

i.    `y = kx^2 + c`

`text(Translate 2 units in positive)\ xtext(-direction.)`

`y = kx^2 + c \ => \ y = k(x – 2)^2 + c`

`text(Dilate in the positive)\ ytext(-direction by a factor of 4.)`

`y = k(x – 2)^2 + c \ => \ y = 4k(x – 2)^2 + 4c`

 

ii.    `y` `= 4k(x^2 – 4x + 4) + 4c`
    `= 4kx^2 – 16kx + 16k + 4c`

 

 
`text(Reflect in the)\ ytext(-axis.)`

COMMENT: Using “swap” terminology for reflections in the y-axis is simpler and more intelligible for students in our view.

`=>\ text(Swap:)\ \ x →\ – x`

`y` `= 4k(−x)^2 – 16k(−x) + 16k + 4c`
  `= 4kx^2 + 16kx + 16k + 4c`

 

 
`text(Equating co-efficients:)`

`4k` `=8`  
`:. k` `=2`  

 

`16k + 4c` `= −8`
`4c` `= −40`
`:. c` `=-10`

Functions, 2ADV F2 EQ-Bank 14

List a set of transformations that, when applied in order, would transform  `y = x^2`  to the graph with equation  `y = 1 - 6x - x^2`.  (3 marks)

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`text(T1: Translate 3 units in negative)\ xtext(-direction)`

`text(T2: Translate 10 units in negative)\ ytext(-direction)`

`text(T3: Reflect in the)\ xtext(-axis)`

Show Worked Solution

`y = x^2`

`text(Transformation 1:)`

`text(Translate 3 units in negative)\ xtext(-direction)`

`y = (x + 3)^2`

`y = x^2 + 6x + 9`
 

`text(Transformation 2:)`

`text(Translate 10 units in negative)\ ytext(-direction)`

`y = x^2 + 6x – 1`
 

`text(Transformation 3:)`

`text(Reflect in the)\ xtext(-axis)`

`y` `= −(x^2 + 6x – 1)`
  `= 1 – 6x – x^2`

Functions, 2ADV F2 EQ-Bank 2 MC

Which diagram best shows the graph

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

A. B.
C. D.
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`A`

Show Worked Solution

`text(Transforming)\ \ y = x^2 :`

`text(Translate 1 unit left)\ \ => \ y = (x + 1)^2`

`text(Dilate from)\ xtext(-axis by a factor of 2)\ => \ y = 2(x + 1)^2`

`text(Reflect in)\ xtext(-axis)\ \ => \ y= −2(x + 1)^2`

`text(Translate 1 unit up)\ \ => \ y = 1 – 2(x + 1)^2`

`:.\ text(Transformations describe graph)\ A.`

`=>\ A`

Functions, 2ADV F2 EQ-Bank 16

`y = -(x + 2)^4/3`  has been produced by three successive transformations: a translation, a dilation and then a reflection.

  1. Describe each transformation and state the equation of the graph after each transformation.  (2 marks)

  2. Sketch the graph.  (1 mark)

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  1. `text(See Worked Solutions)`
  2.  
Show Worked Solution

i.   `text(Transformation 1:)`

`text(Translate)\ \ y = x^4\ \ 2\ text(units to the left.)`

`y = x^4 \ => \ y = (x + 2)^4`
  

`text(Transformation 2:)`

`text(Dilate)\ \ y = (x + 2)^4\ \ text(by a factor of)\ 1/3\ text(from the)\ xtext(-axis)`

`y = (x + 2)^4 \ => \ y = ((x + 2)^4)/3`
 

`text(Transformation 3:)`

`text(Reflect)\ \ y = ((x + 2)^4)/3\ \ text(in the)\ xtext(-axis).`

`y = ((x + 2)^4)/3 \ => \ y = −(x + 2)^4/3`

 

ii.   

Functions, 2ADV F1 SM-Bank 35

  1. Sketch the function  `y = f(x)`  where  `f(x) = (x - 1)^3`  on a number plane, labelling all intercepts.  (1 mark)

  2. On the same graph, sketch  `y = −f(−x)`. Label all intercepts.  (2 marks)

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  1.   
  2.   
Show Worked Solution

i.   `y = (x – 1)^3 => y = x^3\ text(shifted 1 unit to the right.)`
 

 

ii.   `y = −f(x) \ => \ text(reflect)\ \ y = (x – 1)^3\ \ text(in)\ xtext(-axis).`

`y = −f(−x) \ => \ text(reflect)\ \ y = −f(x)\ \ text(in)\ ytext(-axis).`