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