Rational and power functions
Find the equations of the asymptotes of the graph of the function.
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The graph of the function y = 1/x is a hyperbola. It has two branches and two lines that it keeps getting closer to but never touches — the asymptotes. For y = 3/(x − 2) + 1 the whole hyperbola is shifted 2 to the right and 1 up, so the asymptotes are x = 2 and y = 1.
What does this generator do?
At the easy level you read the asymptotes from the formula y = k/(x − m) + n. At the medium level you match a formula to the graph of a hyperbola. At the hard level you alternately find the asymptotes of a function written as (ax + b)/(x − m) and decide on the properties of power functions y = x², x³, x⁻¹, x⁻² …
How to use it
1) The vertical asymptote is where the denominator is zero: x − m = 0, so x = m. 2) The horizontal asymptote is the number n that y gets close to for large x. 3) Divide the form (ax + b)/(x − m) first: you get a + k/(x − m), and the horizontal asymptote is y = a. 4) For y = xⁿ, what matters is whether n is even or odd and whether it is positive or negative.