To find the equation that represents the translation of the line y = 5x with given asymptotes, we start by identifying how translation works in mathematical terms.
The original equation, y = 5x, is a straight line passing through the origin with a slope of 5. However, to incorporate the asymptotes x = 6 and y = 7, we need to translate this line accordingly.
First, the vertical asymptote x = 6 indicates that our equation will not be defined for x = 6, suggesting we need to express the equation in such a way that it won’t be defined at this point. This typically means introducing a fraction where x = 6 results in division by zero.
Similarly, the horizontal asymptote y = 7 indicates that as we move toward infinity or negative infinity in the y-direction, the function approaches 7.
Putting this all together, we can create a rational function that fits these criteria. One suitable equation could be:
f(x) = 7 + (5(x – 6)) / (x – 6)
In this equation:
- The term (x – 6) in the denominator creates the asymptote at x = 6.
- As x approaches infinity, the term (5(x – 6)) / (x – 6) approaches 5, and then we add 7 to give us the horizontal asymptote at y = 7.
Thus, we have successfully translated the line y = 5x into a form that meets the specified asymptotes!