# 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

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

First remember

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

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

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

First remember

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

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

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

First remember

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

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

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

First remember

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

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

To translate 6 to the right and 4 down

Remember

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