Consider f(x)=x^2/x^2+a, a>0. Determine the effect on the graph of f if a is varied?

Question

consider f(x)=x^2/x^2+a, a>0. Determine the effect on the graph of f if a is varied Does this mean that...

A) Each y value is multiplied by a

B) As a increases, the vertical tangent lines move further from the origin

C) The graph of the curve is shifted √a units to the left

D) As a increases, the curve approaches its asymptote more slowly

E) none of these

Can you explain how you got the answer and why it's right. Thank you

Answer 1

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Answer 2

f(x) = (x^2 + a - a)/(x^2 + a) = 1 - a/(x^2 + a)

There is a horizontal asymptote y = 1. As a increases, a/(x^2 + a) = 1/[(x^2 /a) + 1] approaches zero more slowly. Thus, the curve approaches its asymptote more slowly.

Answer: D)