Question

Consider the equations y = x , y = 2 3 x , and y = 2 3 x + 1 . Use transformations to complete the table of values.

X y=x y=2/3 x y=2/3x+1
-6
-3
0
3
6

Answer 1

The table:

y = x y = 2/3 x y = 2/3 x + 1

-6 → -4 → -3

-3 → -2 → -1

0 → 0 → 1

3 → 2 → 3

6 → 4 → 5

Explanation:

- We have three equations:

1. y = x

2. y =
`(2)/(3)` x

3. y =
`(2)/(3)` x + 1

∵ Equation 2 =
`(2)/(3)` × equation 1

- That means we multiply the value of y by
`(2)/(3)`

∵
`(2)/(3)` < 1

∴ Equation 2 is the image of equation 1 by vertical compression with

scale factor
`(2)/(3)`

∵ Equation 3 = equation 2 + 1

- That means we add the value of y by 1

∴ Equation 3 is the image of equation 2 by translation 1 unit up

- To fill the table we will multiply each y value by
`(2)/(3)` to

find the values of the equation 2 and then add 1 to find the values of

equation 3

→ The 1st equation y = x

∵ x = -6 , -3 , 0 , 3 , 6

∴ y = -6 , -3 , 0 , 3 , 6

→ The 2nd equation y =
`(2)/(3)` x

∵ x = -6 , -3 , 0 , 3 , 6

∴ y = -4 , -2 , 0 , 2 , 4

→ The 3rd equation y =
`(2)/(3)` x + 1

∵ x = -6 , -3 , 0 , 3 , 6

∴ y = -3 , -1 , 1 , 3 , 5

The table:

x → y = x → y = 2/3 x → y = 2/3 x + 1

-6 → -6 → -4 → -3

-3 → -3 → -2 → -1

0 → 0 → 0 → 1

3 → 3 → 2 → 3

6 → 6 → 4 → 5