How can we write the equation for the translation of the function y = 4x, considering the asymptotes x = 7 and y = 6?

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Understanding the Translation of the Function

The original function given is y = 4x. To translate this function with respect to the given asymptotes at x = 7 and y = 6, we first need to understand what translating a function means. Translating a function involves shifting it horizontally and/or vertically without changing its shape.

Finding the Transformed Equation

The vertical and horizontal translations can be implemented by adjusting the x and y coordinates in the equation:

  • To translate horizontally by 7 units to the right, we replace x with (x – 7).
  • To translate vertically by 6 units upwards, we replace y with (y – 6).

Constructing the Translated Equation

Starting with the original equation y = 4x, we apply these translations:

y - 6 = 4(x - 7)

This equation reflects the desired translations:

  • The term (x – 7) represents the horizontal shift to the right by 7 units.
  • The term (y – 6) represents the vertical shift upwards by 6 units.

Simplifying the Equation

Now, we can simplify the equation:

y - 6 = 4x - 28

By adding 6 to both sides, we get:

y = 4x - 22

Final Answer

Thus, the equation for the translation of the function y = 4x with asymptotes located at x = 7 and y = 6 is:

y = 4x - 22

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