Answer:
5f(x + 1) + 4
Step-by-step explanation:
These two functions are both of circles because circles have the equation: [tex](x-h)^2+(y-k)^2=r^2[/tex], 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
