Consider f(x)=x^2/(x^2+a), a>0. Determine the effect on the graph of f if a is varied.
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 sqrt(a) units to the left.
D. As a increases, the curve approaches its asymptote more slowly
E. None of these
have some fun at Wolfram .
Change the value of the constant to see what effect it has on your graph.
http://www.wolframalpha.com/input/?i=plot+y+%3D+%3Dx%5E2%2F%28x%5E2%2B5%29+
It appears as if the shape does not change, but if you look at the scale , what Wolfram is doing is to change the scale on the x-axis
Or, you can plot multiple curves to compare them:
http://www.wolframalpha.com/input/?i=plot+y+%3D+%3Dx^2%2F%28x^2%2B5%29+%2C+y%3Dx^2%2F%28x^2%2B20%29
To determine the effect on the graph of f(x) if a is varied, let's analyze the given function f(x) = x^2 / (x^2 + a).
We can start by considering what happens when a = 0:
If a = 0, the function becomes f(x) = x^2 / (x^2 + 0) = x^2 / x^2 = 1.
Thus, when a = 0, the graph of f(x) is a horizontal line at y = 1.
Now, let's consider what happens as a increases:
As a increases, the denominator of the function also increases, which leads to changes in the behavior of the function and its graph.
Option A: Each y value is multiplied by a - This is not true. As a increases, the y-values do not change by a constant multiplication factor.
Option B: As a increases, the vertical tangent lines move further from the origin - This is not true either. The position of the vertical tangent lines does not change as a varies.
Option C: The graph of the curve is shifted √a units to the left - This is also not true. The graph is not shifted to the left by any constant value as a changes.
Option D: As a increases, the curve approaches its asymptote more slowly - This is the correct option. As a increases, the function approaches the horizontal asymptote y = 1 more slowly. The closer the value of a is to zero, the faster the function approaches the asymptote.
Therefore, the correct answer is option D: As a increases, the curve approaches its asymptote more slowly.
To summarize, as the value of a increases, the graph gradually approaches its horizontal asymptote more slowly, while none of the other options accurately describe the effect on the graph.