Question

Write an equation for the translation of -1/x that has the asymptotes x = –2 and y = 4.

Answer 1

Notice that y = -1/x has a horizontal asymptote of y = 0 and vertical asymptote of x = 0. If the translation leads to the the horizontal asymptote being shifted 4 units up and the vertical asymptote being shifted 2 units to the left then that must mean that we must translate -1/x 2 units to the left and 4 units up. First to shift the function 2 units to the left we "add" 2 to the denominator:\[\bf y=-\frac{ 1 }{ x+2 }\]Adding '2' means that the function has now been shifted 2 units to the left and we can confirm this because \(\bf x + 2 = 0 \implies x = -2\) is our new vertical asymptote, just like the question wants. But our horizontal asymptote is still at 0 but we want it y = 4 hence we must shift the function 4 units, i.e. Add +4 to the whole function so each value that comes out gets shifted 4 units up:\[\bf y=-\frac{1}{x+2}+4\]As you can see, now as 'x' approaches postive infinity or negative infinity, our horizontal asymptote will be at y = 4 not y = 0. Now to put all of this as a single fraction we equalise the denominators:\[\bf y=-\frac{1}{x+2}+\frac{4(x+2)}{x+2}=\frac{ -1+4(x+2) }{ x+2 }=\frac{ 4x+7 }{ x+2 }\]And there it is. H.A at y = 4 and V.A at x = -2.

Answer 2

@ichibanstunna

Answer 3

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