Mammoth Memory

Translating parabolas

If you translate (slide) a parabola there are four elements you need to remember:

Start with the most famous parabola.

y=x2

 

1.  y=x2+1          Goes up  ↑         (Shifts the parabola up)

2.  y=x2-1          Goes down  ↓    (Shifts the parabola down)

3.  y=(x+1)2       Goes left  ←      (Shifts the parabola to the left)

4.  y=(x-1)2       Goes right  →    (Shifts the parabola to the right)

 

How do you remember translating parabolas:

Draw a diagonal line as in a graph

To start with draw a diagonal line of a graph

Then write

Write up above down

Then write

Up and left is positive, down and right is minus

This should remind you.

UP +1   y=x2+1
Down -1   y=x2-1
Right (-1)   y=(x-1)2
Left (+1)   y=(x+1)2

 

The results would be as the following examples:

 

Example 1

y=x2+1                or  Up +1

And we can prove this or further help to remember this by plotting the graph out as follows:

x=3   y=32+1 =9+1 =10
x=2   y=22+1 =4+1 =5
x=1   y=12+1 =1+1 =2
x=0   y=02+1 =0+1 =1
x=-1   y=(-1)2+1 =1+1 =2
x=-2   y=(-2)2+1 =4+1 =5
x=-3   y=(-3)2+1 =9+1 =10

Transition a parabola down example 1

 

 The parabola has been translated by (01)

UP +1  ↑

 

 

Example 2

y=x2-1                   or  Down -1

Again we can prove this or further help to remember this by plotting the graph as follows:

x=3   y=32-1 =9-1 =8
x=2   y=22-1 =4-1 =3
x=1   y=12-1 =1-1 =0
x=0   y=02-1 =0-1 =-1
x=-1   y=(-1)2-1 =1-1 =0
x=-2   y=(-2)2-1 =4-1 =3
x=-3   y=(-3)2-1 =9-1 =8

Transition a parabola down example 2

 

 

The parabola has been translated by (01)

Down -1  ↓

 

 

Example 3

y=(x+1)2                     or  Left ←

 

NOTE:

See how we added brackets around the x  value and added +1

 

We can prove this or further help remember this by plotting the graph as follows:

x=3   y=(3+1)2 =42 =16
x=2   y=(2+1)2 =32 =9
x=1   y=(1+1)2 =22 =4
x=0   y=(0+1)2 =12 =1
x=-1   y=(-1+1)2 =02 =0
x=-2   y=(-2+1)2 =(-1)2 =1
x=-3   y=(-3+1)2 =(-2)2 =4
x=-4   y=(-4+1)2 =(-3)2 =9

Transition a parabola to the left example 3

The parabola has been translated by (-10)

Left -1  ←

 

 

Example 4

y=(x-1)2                    or  Right →

 

NOTE:

See how we added brackets around the x  value and subtracted 1

 

We can prove this or further help remember this by plotting the graph as follows:

x=4   y=(4-1)2 =32 =9
x=3   y=(3-1)2 =22 =4
x=2   y=(2-1)2 =12 =1
x=1   y=(1-1)2 =02 =0
x=0   y=(0-1)2 =(-1)2 =1
x=-1   y=(-1-1)2 =(-2)2 =4
x=-2   y=(-2-1)2 =(-3)2  =9
x=-3   y=(-3-1)2 =(-4)2 =16

 Transition a parabola to the right example 4

The parabola has been translated by (10)

Right +1  →