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Functions, 2ADV EQ-Bank 08

Solve for \(x\), giving your answers in the simplest form  \(a+b\sqrt{c}\)  where \(a, b\) and \(c\) are real:

\(5 x^2-20 x+4=0\)   (2 marks)

Show Answers Only

\(x=2 \pm \dfrac{4}{5} \sqrt{5}\)

Show Worked Solution

\(5 x^2-20 x+4=0\)

\(x\) \(=\dfrac{-b \pm \sqrt{b^2-4 a c}}{2 a}\)
  \(=\dfrac{20 \pm \sqrt{20^2-4 \times 5 \times 4}}{2 \times 5}\)
  \(=\dfrac{20 \pm \sqrt{320}}{10}\)
  \(=2 \pm \dfrac{8 \sqrt{5}}{10}\)
  \(=2 \pm \dfrac{4}{5} \sqrt{5}\)

Functions, 2ADV EQ-Bank 7 MC

What are the solutions to  \(3x^2+2x-4=0\)?

  1. \(x=\dfrac{-1 \pm \sqrt{13}}{3}\)
  2. \(x=\dfrac{1 \pm \sqrt{13}}{3}\)
  3. \(x=\dfrac{-1 \pm \sqrt{5}}{2}\)
  4. \(x=\dfrac{1 \pm \sqrt{5}}{2}\)
Show Answers Only

\(A\)

Show Worked Solution

\(3 x^2+2 x-4=0\)

\(x\) \(=\dfrac{-b \pm \sqrt{b^2-4 a c}}{2a}\)
  \(=\dfrac{-2 \pm \sqrt{2^2-4 \times 3 \times-4}}{2 \times 3}\)
  \(=\dfrac{-2 \pm \sqrt{52}}{6}\)
  \(=\dfrac{-1 \pm \sqrt{13}}{3}\)

 

\(\Rightarrow A\)

Functions, 2ADV EQ-Bank 10

  1. Express  \(y=x^2-4 x+6\)  in the form  \(y=(x-a)^2+c\).   (1 mark)

  2. Graph the parabola, labelling its vertex and \(y\)-intercept.   (2 marks)

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a.   \(y=(x-2)^2+2\)

b.   \(\text {Vertex}=(2,2)\)

\(y \text{-intercept}=(0,6)\)
 

Show Worked Solution
a.     \(y\) \(=x^2-4 x+6\)
    \(=x^2-4 x+4+2\)
    \(=(x-2)^2+2\)

 
b.    \(\text {Vertex}=(2,2)\)

\(y \text{-intercept}=(0,6)\)

Functions, 2ADV EQ-Bank 09

Using the discriminant, or otherwise, justify why the graph of  \(f(x)=-x^2+2 x-2\)  lies entirely below the \(x\)-axis.   (2 marks)

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\(\Delta=b^2-4 a c=2^2-4(-1)(-2)=-4\)

\(\text{Since \(\ \Delta<0, \ y=-x^2+2 x-2\ \) does not intersect the \(x\)-axis.}\)

\(\text{Since \(\ a=-1<0, f(x)\) is an upside down parabola.}\)

\(\Rightarrow f(x)\ \text{must lie entirely below} \ x\text{-axis.}\)

Show Worked Solution

\(\Delta=b^2-4 a c=2^2-4(-1)(-2)=-4\)

\(\text{Since \(\ \Delta<0, \ y=-x^2+2 x-2\ \) does not intersect the \(x\)-axis.}\)

\(\text{Since \(\ a=-1<0, f(x)\) is an upside down parabola.}\)

\(\Rightarrow f(x)\ \text{must lie entirely below} \ x\text{-axis.}\)

Functions, 2ADV F1 2025 HSC 11

The graph of a quadratic function represented by the equation  \(h=t^2-8 t+12\)  is shown.
 

  1. Find the values of \(t\) and \(h\) at the turning point of the graph.   (2 marks)

  2. The graph shows  \(h=12\)  when  \(t=0\).
  3. What is the other value of \(t\) for which  \(h=12\)?   (1 mark)

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a.   \(\text{Turning point at} \ \ (4,-4)\)

b.   \(t=8\)

Show Worked Solution

a.    \(\text{Strategy 1 (no calculus)}\)

\(\text{Axis of quadratic occurs when}\ \ t= \dfrac{2+6}{2} = 4\)

\(\text{At} \ \ t=4:\)

\(h=4^2-8 \times 4+12=-4\)

\(\therefore \ \text{Turning point at} \ \ (4,-4)\)
 

\(\text{Strategy 2 (using calculus)}\)

\(h=t^2-8 t+12\)

\(h^{\prime}=2 t-8\)

\(\text{Find \(t\) when} \ \ h^{\prime}=0:\)

\(2 t-8=0 \ \Rightarrow \ t=4\)
 

b.    \(\text {When} \ \ h=12:\)

\(t^2-8 t+12\) \(=12\)
\(t(t-8)\) \(=0\)

 
\(\therefore \ \text{Other value:} \ \ t=8\)

Functions, 2ADV F1 EQ-Bank 1 MC

The graph of a quadratic function  \(f(x)=a x^2+b x+c\)  is drawn below.
 

Which of the following are true?

  1. \(a<0, c=0\)  and  \(b^2-4 a c=0\)
  2. \(a>0, c=0\)  and  \(b^2-4 a c=0\)
  3. \(a>0, c>0\)  and  \(b^2-4 a c>0\)
  4. \(a<0, c>0\)  and  \(b^2-4 a c=0\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Quadratic touches } x \text{-axis once only} \ \ \Rightarrow b^2-4 a c=0\ \ \text{(eliminate C)}\)

\(\text{Quadratic is inverted} \Rightarrow a<0 \ \ \text{(eliminate B)}\)

\(\text{If} \ \ c=0, f(x)=a x^2+b x+0=x(a x+b) \Rightarrow \text{cuts twice (Eliminate A)}\)

\(\Rightarrow D\)

Functions, 2ADV F1 EQ-Bank 20

Find the value of \(k\)  if  \(4kx^2-(3-4k) x+k=0\)  has one root.   (2 marks)

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\(k=\dfrac{3}{8}\)

Show Worked Solution

\(4kx^2-(3-4k) x+k=0\)

\(\text{1 root}\ \Rightarrow \Delta=0\)

  \(\Delta\) \(=b^2-4 a c\)
  \(0\) \(=\left[ -\left( 3-4k \right)\right]^2-4\times 4k \times k\)
  \(0\) \(=9-24k + 16k^2-16k^2\)
  \(24k\) \(=9\)
  \(k\) \(=\dfrac{9}{24}=\dfrac{3}{8}\)

Functions, 2ADV F1 EQ-Bank 19

\(R\left(r, r^2\right), S\left(s, s^2\right)\) and \(T\left(t, t^2\right)\) are points on the parabola  \(y=x^2\).

Given \(RT\) is parallel to \(SO\), show  \(r+t=s\)   (2 marks)
 

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\(R\left(r, r^2\right), S\left(s, s^2\right), T\left(t, t^2\right)\)

\(m_{S O}=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{s^2-0}{s-0}=s\)

\(m_{R T}=\dfrac{t^2-r^2}{t-r}=\dfrac{(t-r)(t+r)}{(t-r)}=t+r\)

\(\text{Given}\ R T \ \| \  SO \ \Rightarrow \ m_{SO}=m_{R T}\)

\(\therefore s=r+t\ \ …\ \text{as required} \)

Show Worked Solution

\(R\left(r, r^2\right), S\left(s, s^2\right), T\left(t, t^2\right)\)

\(m_{S O}=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{s^2-0}{s-0}=s\)

\(m_{R T}=\dfrac{t^2-r^2}{t-r}=\dfrac{(t-r)(t+r)}{(t-r)}=t+r\)

\(\text{Given}\ R T \ \| \  SO \ \Rightarrow \ m_{SO}=m_{R T}\)

\(\therefore s=r+t\ \ …\ \text{as required} \)

Functions, 2ADV F1 EQ-Bank 17

The tangent to the parabola  \(y=x^2+2 x-4\)  is  \(y=px-5\)  where  \(p>0\).

Find the value of \(p\).   (2 marks)

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\(p=4\)

Show Worked Solution

\(\text{Intersection occurs when:}\)

\(x^2+2x-4\) \(=px-5\)  
\(x^2+(2-p)x+1\) \(=0\)  

 
\(\text{Tangent touches once}\ \Rightarrow\ \text{Discriminant}\ \Delta=0\)

\((2-p)^2-4 \times 1 \times 1\) \(=0\)  
\(4-4p+p^2-4\) \(=0\)  
\(p(p-4)\) \(=0\)  
\(p\) \(=4\ \ \ (p\gt 0)\)  
COMMENT: Key is to recognise this is a discriminant question, not a calculus application.

Functions, 2ADV F1 EQ-Bank 45

Determine whether the function  \(f(x)=2x^3-5x\)  is even, odd or neither. Show all working.   (2 marks)

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\(f(x)=2x^{3}-5x\)

\(\text{Function is odd if:}\ \ f(-x)=-f(x) \)

\(f(-x)\) \(=2(-x)^{3}-5(-x) \)  
  \(=-2x^{3}+5x \)  
  \(=-(2x^{3}-5x)\)  
  \(=-f(x)\)  

 
\(\therefore f(x)\ \text{is odd.}\)

Show Worked Solution

\(f(x)=2x^{3}-5x\)

\(\text{Function is odd if:}\ \ f(-x)=-f(x) \)

\(f(-x)\) \(=2(-x)^{3}-5(-x) \)  
  \(=-2x^{3}+5x \)  
  \(=-(2x^{3}-5x)\)  
  \(=-f(x)\)  

 
\(\therefore f(x)\ \text{is odd.}\)

Functions, 2ADV F1 SM-Bank 25

Show that the parabola  \(2x^2-kx+k-2\)  has at least one real root.  (3 marks)

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\(2x^2-kx+k-2=0\)

\(\Delta=b^2-4ac=(-k)^2-4 \times 2(k-2) = k^{2}-8k+16\)

\(\text{Real roots:}\ \ \Delta \geqslant 0\)

\(k^2-8k+16\) \(\geqslant 0\)  
\((x-4)^2\) \(\geqslant 0\)  

 
\(\therefore\ \text{At least one root exists for all}\ k\)

Show Worked Solution

\(2x^2-kx+k-2=0\)

\(\Delta=b^2-4ac=(-k)^2-4 \times 2(k-2) = k^{2}-8k+16\)

\(\text{Real roots:}\ \ \Delta \geqslant 0\)

\(k^2-8k+16\) \(\geqslant 0\)  
\((x-4)^2\) \(\geqslant 0\)  

 
\(\therefore\ \text{At least one root exists for all}\ k\)

Functions, 2ADV F1 2023 HSC 10 MC

The graph  \(y = x^2\)  meets the line  \(y = k\)  (where \(k>0\)) at points \(P\) and \(Q\) as shown in the diagram. The length of the interval \(PQ\) is \(L\).
 

Let \(a\) be a positive number. The graph  \(y=\dfrac{x^2}{a^2}\)  meets the line  \(y=k\)  at points \(S\) and \(T\).

What is the length of \(ST\)?

  1. \(\dfrac{L}{a}\)
  2. \(\dfrac{L}{a^2}\)
  3. \(aL\)
  4. \(a^2L\)
Show Answers Only

\(C\)

Show Worked Solution

\(\text{Intersection of}\ \ y=x^2\ \ \text{and}\ \ y=k:\)

\(x^2=k\ \ \Rightarrow\ \ x=\pm \sqrt k\)

\(\therefore L=2\sqrt k\)

\(\text{Intersection of}\ \ y=\dfrac{x^2}{a^2}\ \ \text{and}\ \ y=k:\)

\(\dfrac{x^2}{a^2} \) \(=k\)  
\(x^2\) \(=a^2k\)  
\(x\) \(=\pm a\sqrt k\)  

\(\therefore ST=a \times 2\sqrt k = aL \)

\(\Rightarrow C\)

♦♦♦ Mean mark 24%.

Functions, 2ADV F1 2023 HSC 9 MC

Let \(f(x)\) be any function with domain all real numbers.

Which of the following is an even function, regardless of the choice of \(f(x)\)?

  1. \(2 f(x)\)
  2. \(f(f(x))\)
  3. \((f(-x))^2\)
  4. \(f(x) f(-x)\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Even function}\ \rightarrow \ f(x)=f(-x)\)

\(\text{Consider the function}\ \ f(x) = x-2\)

\( 2f(1)=-2,\ \ 2f(-1)=-6\ \ \text{(not even)}\)

\( f(f(1))=f(-1)=-3,\ \ f(f(-1))=f(-3)=-5\ \ \text{(not even)}\)

\( (f(-1))^2=(-3)^2=9,\ \ (f(1))^2=(-1)^2=1\ \ \text{(not even)}\)

\( f(1)f(-1)=-1 \times -3=3,\ \ f(-1)f(1)=-3 \times -1=3 \ \text{(possibly even)}\)

\(=>D\)

♦♦♦ Mean mark 24%.

Functions, 2ADV F1 2023 HSC 4 MC

The graph of a polynomial is shown.
 


  

Which row of the table is correct for this polynomial?
 

Show Answers Only

`B`

Show Worked Solution

`text{By elimination:}`

`text{Double root where graph SP touches}\ xtext{-axis}`

`:.\ C and D\ text{incorrect}`

`(x-c)^2\ text{occurs where}\ x>0`

`:. c>0\ (A\ text{is incorrect})`

`=>B`

Functions, 2ADV F1 2021 HSC 8 MC

The graph of  `y = f(x)`  is shown.

Which of the following could be the equation of this graph?

  1. `y = (1 - x)(2 + x)^3`
  2. `y = (x + 1)(x - 2)^3`
  3. `y = (x + 1)(2 - x)^3`
  4. `y = (x - 1)(2 + x)^3`
Show Answers Only

`C`

Show Worked Solution

`text(By elimination:)`

`text(A single negative root occurs when)\ \ x =–1`

`->\ text(Eliminate A and D)`

`text(When)\ \ x = 0, \ y > 0`

`->\ text(Eliminate B)`

`=> C`

Functions, 2ADV F1 SM-Bank 23

Find the values of `k` for which the expression  `x^2-3x + (4-2k)`  is always positive.  (3 marks)

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`k < 7/8`

Show Worked Solution

`x^2-3x + (4-2k) > 0`

`x^2-3x + (4-2k) = 0\ \ text(is a concave up parabola)`

`=>\ text{Always positive (no roots) if}\ \ Delta < 0`
 

`b^2-4ac < 0`

`(−3)^2-4 · 1 · (4-2k)` `< 0`
`9-16 + 8k` `< 0`
`8k` `< 7`
`k` `< 7/8`

Functions, 2ADV F1 2017 HSC 2 MC

Which expression is equal to  `3x^2-x-2`?

  1. `(3x-1) (x + 2)`
  2. `(3x + 1) (x-2)`
  3. `(3x-2) (x + 1)`
  4. `(3x + 2) (x-1)`
Show Answers Only

`D`

Show Worked Solution

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

`=>  D`

Functions, 2ADV F1 2016 HSC 11e

Find the points of intersection of  `y=-5-4x`  and  `y=3-2x-x^2.`  (3 marks)

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`(4, – 21) and (– 2, 3)`

Show Worked Solution

`y = 3 – 2x – x^2`

`text(Substitute)\ \ y = -5 – 4x\ \ text(into equation)`

`-5 – 4x` `= 3 – 2x – x^2`
`x^2 – 2x – 8` `= 0`
`(x – 4) (x + 2)` `= 0`

  
`:. x = 4 or -2`
 

`text(When)\ \ x = 4,\ \ y = -5 – 4(4) = -21`

`text(When)\ \ x = -2,\ \ y = -5 – 4 (-2) = 3`  
 

`:.\ text(Intersection at)\ \ (4, – 21) and (– 2, 3)`

Functions, 2ADV F1 2016 HSC 4 MC

Which diagram shows the graph of an odd function?
 

hsc-2016-4mci

hsc-2016-4mcii

Show Answers Only

`A`

Show Worked Solution

`text(Odd functions occur when:)`

♦ Mean mark 38%.

`f(x) = – f(x)`

`text(Graphically, this occurs when a function has)`

`text(symmetry when rotated 180° about the origin.)`

`=>  A`

Functions, 2ADV F1 2014 HSC 6 MC

Which expression is a factorisation of  `8x^3 + 27`? 

  1. `(2x - 3)(4x^2 + 12x - 9)`
  2. `(2x + 3)(4x^2 - 12x + 9)`
  3. `(2x - 3)(4x^2 + 6x - 9)`
  4. `(2x + 3)(4x^2 - 6x + 9)`
Show Answers Only

`D`

Show Worked Solution

`8x^3 + 27`

COMMENT: Factorising a cubic is only examinable with scaffolding, as provided here by expanding the answer options.

`= (2x)^3 + 3^3`

`= (2x + 3)(4x^2 – 6x + 9)`

 
`=>  D`

Functions, 2ADV F1 2010 HSC 1a

Solve  `x^2 = 4x`.   (2 marks)

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 `x = 0\ text(or)\ 4`

Show Worked Solution
`x^2` `= 4x`
`x^2-4x` `= 0`
`x(x-4)` `= 0`

 

`:.\ x = 0\ text(or)\ 4`

Functions, 2ADV F1 2013 HSC 1 MC

What are the solutions of   `2x^2-5x-1 = 0`? 

  1. `x = (-5 +-sqrt17)/4` 
  2. `x = (5 +-sqrt17)/4`
  3. `x = (-5 +-sqrt33)/4`
  4. `x = (5 +-sqrt33)/4`
Show Answers Only

`D`

Show Worked Solution

`2x^2-5x-1 = 0`

`text(Using)\ x = (-b +- sqrt( b^2-4ac) )/(2a)`

`x` `= (5 +- sqrt{\ \ (-5)^2-4 xx 2 xx(-1) })/ (2 xx 2)`
  `= (5 +- sqrt(25 + 8) )/4`
  `= (5 +- sqrt(33) )/4`

 
`=>  D`