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Quadratics, SMB-013

Factorise the parabola described by the equation  `y=-x^2-x+12`  and find its vertex.  (3 marks)

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`y=(3-x)(x+4)`

`text{Vertex}\ = (-1/2,12 1/4)`

Show Worked Solution
`y` `=-x^2-x+12`  
  `=-(x^2+x-12)`  
  `=-(x+4)(x-3)`  
  `=(3-x)(x+4)`  

 

`text{Solutions at}\ \ x=3, -4`

 `text{Line of symmetry at mid-point of solutions.}`

  `x=(3+(-4))/2=-1/2`
 

`text{Substitute}\ \ x=-1/2\ \ text{into}\ \ y=-x^2-x+12`

`y=-1/4+1/2+12=12 1/4`

`:.\ text{Vertex}\ = (-1/2,12 1/4)`

Quadratics, SMB-012

  1. Factorise  `y=x^2-8x+15`  (1 mark)

  2. Find the vertex of the parabola with equation  `y=x^2-8x+15`  (2 marks)

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  1. `y=(x-5)(x-3)`
  2. `text{Vertex}\ = (4,-1)`
Show Worked Solution
i.   `y` `=x^2-8x+15`  
  `=(x-5)(x-3)`  

 

ii.    `text{Solutions at}\ \ x=3,5.`

 `text{Line of symmetry at mid-point of solutions.}`

  `x=(3+5)/2=4`
 

`text{Substitute}\ \ x=4\ \ text{into}\ \ y=x^2-8x+15`

`y=4^2-8xx4+15=-1`

`:.\ text{Vertex}\ = (4,-1)`

Quadratics, SMB-011

  1. Factorise  `y=2x^2+5x-3`  (1 mark)

  2. Find the vertex of the parabola with equation  `y=2x^2+5x-3`  (2 marks)

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  1. `y=(2x-1)(x+3)`
  2. `text{Vertex}\ = (-5/4,-49/8)`
Show Worked Solution
i.   `y` `=2x^2+5x-3`  
  `=(2x-1)(x+3)`  

 

ii.    `text{Solutions at}\ \ x=1/2,-3.`

 `text{Line of symmetry at mid-point of solutions.}`

  `x=(1/2+(-3))/2=-5/4`
 

`text{Substitute}\ \ x=-5/4\ \ text{into}\ \ y=2x^2+5x-3`

`y=2xx(-5/4)^2-5xx5/4-3=-49/8`

`:.\ text{Vertex}\ = (-5/4,-49/8)`

Quadratics, SMB-010

  1. Factorise  `y=6-x-x^2`  (2 marks)

  2. Find the vertex of the parabola with equation  `y=6-x-x^2`  (2 marks)

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  1. `y=(2-x)(x+3)`
  2. `text{Vertex}\ = (-1/2,6 1/4)`
Show Worked Solution
i.   `y` `=6-x-x^2`  
  `=-(x^2+x-6)`  
  `=-(x-2)(x+2)`  
  `=(2-x)(x+3)`  

 

ii.    `text{Solutions at}\ \ x=2,-3.`

 `text{Line of symmetry at mid-point of solutions.}`

  `x=(2+(-3))/2=-1/2`
 

`text{Substitute}\ \ x=-1/2\ \ text{into}\ \ y=6-x-x^2`

`y=6+1/2-1/4=6 1/4`

`:.\ text{Vertex}\ = (-1/2,6 1/4)`

Quadratics, SMB-009

By completing the square, find the coordinates of the vertex of the parabola with equation

  `y=x^2-3x+1`  (3 marks)

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`(3/2,-5/4)`

Show Worked Solution
`y` `=x^2-3x+1`  
  `=x^2-3x+9/4-5/4`  
  `=(x-3/2)^2-5/4`  

 
`:.\ text{Vertex}\ = (3/2,-5/4)`

Quadratics, SMB-008

By completing the square, find the coordinates of the vertex of the parabola with equation

  `y=x^2+8x+9`  (3 marks)

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`(-4,-7)`

Show Worked Solution
`y` `=x^2+8x+9`  
  `=x^2+8x+16-7`  
  `=(x+4)^2-7`  

 
`:.\ text{Vertex}\ = (-4,-7)`

Quadratics, SMB-007

By completing the square, find the coordinates of the vertex of the parabola with equation

  `y=x^2-6x-4`  (3 marks)

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`(3,-13)`

Show Worked Solution
`y` `=x^2-6x-4`  
  `=x^2-6x+9-13`  
  `=(x-3)^2-13`  

 
`:.\ text{Vertex}\ = (3,-13)`