Write an equation for the translation of y=4/x that has the asymptotes x=7 and y=6
The given function y=4/x has a vertical asymptote at x=0 and a horizontal asymptote at y=0. To translate the function to have asymptotes x=7 and y=6, we can use the following form of the equation:
y = a * (4 / (x - h)) + k
where h and k are the horizontal and vertical shift respectively, and a is a scaling factor.
To make the vertical asymptote at x=7, we need to shift the function horizontally by 7 units to the right. So, h=7.
To make the horizontal asymptote at y=6, we need to shift the function vertically by 6 units upwards. So, k=6.
The scaling factor a can be chosen based on the desired behavior of the function around the asymptotes. Since the original function y=4/x approaches zero as x goes to infinity or negative infinity, and since we want the same behavior for the translated function, we can choose a=4/7.
Putting these values into the equation, we get:
y = (4/7) * (4 / (x - 7)) + 6
Simplifying this equation, we get:
y = (16 / (7x - 49)) + 6
This is the equation for the translation of y=4/x that has asymptotes x=7 and y=6.
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Yes, I am sure of my answer.
To translate the equation y=4/x so that it has asymptotes x=7 and y=6, we can start by identifying the translation amounts needed for the x and y directions.
For the x-direction, we need to shift the graph horizontally to the right by 7 units to match the asymptote at x=7. To achieve this, we replace x in the equation with (x-7).
In the y-direction, we need to shift the graph vertically upwards by 6 units to match the asymptote at y=6. To achieve this, we add 6 to the equation.
Combining these translation steps, the equation for the translated graph is:
y = 4 / (x - 7) + 6