To find the equation of the translated function, we can start with the basic function y = 4x. The goal is to translate this function so that it has vertical and horizontal asymptotes at x = 7 and y = 6, respectively.
The given asymptotes indicate that the graph of the function must be shifted to the right by 7 units and up by 6 units. We can express the transformation using the general formula for translating a function:
- If we translate a function f(x) horizontally to the right by h units, the new function is f(x – h).
- If we translate it vertically by k units, the new function is f(x) + k.
Applying these transformations to our function:
- Translating y = 4x to the right by 7 units:
- We replace x with (x – 7), giving us y = 4(x – 7).
- Translating y = 4(x – 7) up by 6 units:
- We add 6 to the entire function, resulting in y = 4(x – 7) + 6.
Now, let’s combine these steps:
- Start with y = 4(x – 7).
- Add 6: y = 4(x – 7) + 6.
Finally, we can simplify the equation:
y = 4x – 28 + 6
y = 4x – 22
Thus, the equation of the translated function that has the required asymptotes is y = 4x – 22.