To find the equation that describes the translation of the function y = 6x while incorporating the asymptotes x = 4 and y = 5, we will begin by understanding what these asymptotes represent.
The line x = 4 indicates a vertical asymptote, which means that our function will approach this vertical line but never actually touch or cross it. The line y = 5 acts as a horizontal asymptote, suggesting that as x approaches infinity or negative infinity, the value of y will approach 5.
To incorporate these translations into our equation, we first need to adjust our base function y = 6x to align with the asymptotes. We can transform the given linear function in the following manner:
y = 6(x - 4) + 5
In this equation:
- (x – 4) shifts the function to the right by 4 units, aligning the vertical asymptote to x = 4.
- +5 shifts the function upward by 5 units, placing the horizontal asymptote at y = 5.
Thus, the finalized equation that represents the translation of y = 6x with the specified asymptotes would be:
y = 6(x - 4) + 5
This transforms to:
y = 6x - 24 + 5
Or simplifying further:
y = 6x - 19
In conclusion, the equation capturing the translation of the original function with the desired asymptotes is y = 6x – 19.