Mammoth Memory

Manipulating parabolas further examples

 

Example 1

An equation of a parabola is `y=x^2-2x-2`

translate this equation upwards by `3`.

To answer this, first plot `y=x^2-2x-2`

`x=3`   `y=3^2-2times3-2` `=9-6-2` `=1`
`x=2`   `y=2^2-2times2-2` `=4-4-2` `=-2`
`x=1`   `y=1^2-2times1-2` `=1-2-2` `=-3`
`x=0`   `y=0^2-2times0-2` `=0-0-2` `=-2`
`x=-1`   `y=(-1)^2-2times-1-2` `=1+2-2` `=1`

Translate this parabola up by 3

To translate `y=x^2-2x-2` upwards by `3`

First remember
Up and left is positive, down and right is minus

Upwards is therefore add `+3` to the whole equation.

`y=x^2-2x-2+3`

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

Now plot this new parabola.

`x=3`   `y=3^2-2times3+1` `=9-6+1` `=4`
`x=2`   `y=2^2-2times2+1` `=4-4+1` `=1`
`x=1`   `y=1^2-2times1+1` `=1-2+1` `=0`
`x=0`   `y=0^2-2times0+1` `=0-0+1` `=1`
`x=-1`   `y=(-1)^2-2times(-1)+1` `=1+2+1` `=4`

Move the parabola up by plus 3

 

Example 2

An equation of a parabola is `y=-1(x^2)-2x+2`

translate this equation downwards by `3`.

To answer this, first plot `y=-1(x^2)-2x+2`

`x=3`   `y=-(3)^2-2times3+2` `=-9-6+2` `=-13`
`x=2`   `y=-(2)^2-2times2+2` `=-4-4+2` `=-6`
`x=1`   `y=-(1)^2-2times1+2` `=-1-2+2` `=-1`
`x=0`   `y=0^2-2times0+2` `=0-0+2` `=2`
`x=-1`   `y=-(-1)^2-2times(-1)+2` `=-1+2+2` `=3`
`x=-2`   `y=-(-2)^2-2times(-2)+2` `=-4+4+2` `=2`
`x=-3`   `y=-(-3)^2-2times(-3)+2` `=-9+6+2` `=-1`
`x=-4`   `y=-(-4)^2-2times(-4)+2` `=-16+8+2` `=-6`

Move this parabola downwards by 3

To translate `y=-x^2-2x+2` downwards by `3`

First remember
Up and left is positive, down and right is minus

Upwards is therefore subtract `3`  from the whole equation.

`y=-x^2-2x+2-3`

`y=x^2-2x-1`

Now plot the new parabola

`x=3`   `y=-(3)^2-2times3-1` `=-9-6-1` `=-16`
`x=2`   `y=-(2)^2-2times2-1` `=-4-4-1` `=-9`
`x=1`   `y=-(1)^2-2times1-1` `=-1-2-1` `=-4`
`x=0`   `y=0^2-2times0-1` `=0-0-1` `=-1`
`x=-1`   `y=-(-1)^2-2times(-1)-1` `=-1+2-1` `=0`
`x=-2`   `y=-(-2)^2-2times(-2)-1` `=-4+4-1` `=-1`
`x=-3`   `y=-(-3)^2-2times(-3)-1` `=-9+6-1` `=-4`
`x=-4`   `y=-(-4)^2-2times(-4)-1` `=-16+8-1` `=-9`

The new parabola drawn out translated downwards by 3

  

Example 3

An equation of a parabola is `y=2x^2+3x-2`

translate this equation to the right by `4`.

To answer this, first plot `y=2x^2+3x-2`

`x=2`   `y=2times2^2+3times2-2` `=8+6-2` `=12`
`x=1`   `y=2times1^2+3times1-2` `=2+3-2` `=3`
`x=0`   `y=2times0^2+3times0-2` `=0+0-2` `=-2`
`x=-1`   `y=2times(-1)^2+3times(-1)-2` `=2-3-2` `=-3`
`x=-2`   `y=2times(-2)^2+3times(-2)-2` `=8-6-2` `=0`
`x=-3`   `y=2times(-3)^2+3times(-3)-2` `=18-9-2` `=7`

Move this parabola to the right minus 4

To translate `y=2x^2+3x-2`  to the right by `4`

First remember
Up and left is positive, down and right is minus

To the right is therefore `(-4)` from each `x`

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

Plot this on the original curve.

`x=6`   `y=2(6-4)^2+3(6-4)-2` `=8+6-2` `=12`
`x=5`   `y=2(5-4)^2+3(5-4)-2` `=2+3-2` `=3`
`x=4`   `y=2(4-4)^2+3(4-4)-2` `=0+0-2` `=-2`
`x=3`   `y=2(3-4)^2+3(3-4)-2` `=2-3-2` `=-3`
`x=2`   `y=2(2-4)^2+3(2-4)-2` `=8-6-2` `=0`
`x=1`   `y=2(1-4)^2+3(1-4)-2` `=18-9-2` `=7`

The new parabola drawn out translated to the right minus 4

 

 

Example 4

An equation of a parabola is `y=x^2-3x+4`

translate this equation to the left by `3`.

To answer this, first plot `y=x^2-3x+4`

`x=4`   `y=4^2-3xx4+4` `=16-12+4` `=8`
`x=3`   `y=3^2-3xx3+4` `=9-9+4` `=4`
`x=2`   `y=2^2-3xx2+4` `=4-6+4` `=2`
`x=1`   `y=1^2-3xx1+4` `=1-3+4` `=2`
`x=0`   `y=0^2-3xx0+4` `=0-0+4` `=4`
`x=-1`   `y=(-1)^2-3times(-1)+4` `=1+3+4` `=8`

 Move this parabola to the left by 3

To translate `y=x^2-3x+4` to the left by `3`

First remember
Up and left is positive, down and right is minus

To the left is therefore`(+3)` for each `x`

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

 

`x=1`   `y=(1+3)^2-3(1+3)+4` `=16-12+4` `=8`
`x=0`   `y=(0+3)^2-3(0+3)+4` `=9-9+4` `=4`
`x=-1`   `y=(-1+3)^2-3(-1+3)+4` `=4-6+4` `=2`
`x=-2`   `y=(-2+3)^2-3(-2+3)+4` `=1-3+4` `=2`
`x=-3`   `y=(-3+3)^2-3(-3+3)+4` `=0-0+4` `=4`
`x=-4`   `y=(-4+3)^2-3(-4+3)+4` `=1+3+4` `=8`

 The new parabola drawn out translated to the left plus 3

 

 

Example 5

An equation of a parabola is `y=-1(x^2) -3x+5` translate this equation 6 to the right and 4 down.

 

To answer this first plot `y=-1(x^2)-3x+5`

`x=2`   `y=-(2)^2-3times2+5` `=-4-6+5` `=-5`
`x=1`   `y=-(1)^2-3times1+5` `=-1-3+5` `=1`
`x=0`   `y=-(0)^2-3times0+5` `=-0-0+5` `=5`
`x=-1`   `y=-(-1)^2-3times-1+5` `=-1+3+5` `=7`
`x=-2`   `y=-(-2)^2-3times-2+5` `=-4+6+5` `=7`
`x=-3`   `y=-(-3)^2-3times-3+5` `=-9+9+5` `=5`
`x=-4`   `y=-(-4)^2-3times-4+5` `=-16+12+5` `=1`
`x=-5`   `y=-(-5)^2-3times-5+5` `=-25+15+5` `=-5`

Move this parabola to the right by 6 and 4 down 

To translate 6 to the right and 4 down

Remember 
Up and left is positive, down and right is minus

`6`  to the right is therefore `(-6)`  from each `x`

`4` down is therefore `-4` from whole equation.

 

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

Becomes

`y=-1(x-6)^2-3(x-6)+5-4`

`y=-(x-6)^2-3(x-6)+1`

 

`x=8`   `y=-1times(8-6)^2-3(8-6)+1=-4-6+1` `=-9`
`x=7`   `y=-1times(7-6)^2-3(7-6)+1=-1-3+1` `=-3`
`x=6`   `y=-1times(6-6)^2-3(6-6)+1=0-0+1` `=1`
`x=5`   `y=-1times(5-6)^2-3(5-6)+1=-1+3+1` `=3`
`x=4`   `y=-1times(4-6)^2-3(4-6)+1=-4+6+1` `=3`
`x=3`   `y=-1times(3-6)^2-3(3-6)+1=-9+9+1` `=1`
`x=2`   `y=-1times(2-6)^2-3(2-6)+1=-16+12+1` `=-3`
`x=1`   `y=-1times(1-6)^2-3(1-6)+1=-25+15+1` `=-9`

 The new parabola drawn out 6 places to the right and 4 places down

 

 

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