Question

Using two function notations, describe the transformation.

x^2+y^2=1 --> (x+1)^2 + (y-4)^2 = 25

HELPPPP

Answer 1

5f(x + 1) + 4

Explanation:

x^2+y^2=1

centre: (0,0)

radius = 1

(x+1)^2 + (y-4)^2 = 25

centre: (-1,4)

radius = sqrt(25) = 5

Stretch the circle by factor 5

5f(x)

Then translate by vector < -1 , 4 >

5f(x - -1) + 4

5f(x + 1) + 4

Answer 2

5f(x + 1) + 4

Explanation:

These two functions are both of circles because circles have the equation:
`(x-h)^2+(y-k)^2=r^2`, where (h, k) is the centre and r is the radius.

In the original function, the centre is (0, 0) and the radius is √1 = 1. In the new function, however, the centre is (-1, 4) and the radius is √25 = 5. Let's focus on the translation transformation first.

We see that the centre moved from (0, 0) to (-1, 4). That basically just means the circle moved 1 unit to the left (because it's negative 1 and 1 is the x-coordinate, indicating a horizontal transformation) and 4 units up (because it's positive 4 and 4 is the y-coordinate, indicating a vertical transformation).

So, if the original function was f(x), we can write the new function so far as the following: f(x + 1) + 4 (notice that we have "x + 1" and not "x - 1" because horizontal transformations are "backwards" like that so "-" means right and "+" means left).

Now, look at the change in radius: it goes from 1 to 5, which is essentially a dilation. We can say that the original circle was dilated by a factor of 5, or, in function notation, stretched vertically by a factor of 5 (vertically because the whole function was changed).

We finally have: 5f(x + 1) + 4