I'm having some troubles. I can't think of an example to this question.
Give an exmaple of a quadratic whose reciprocal has no vertical asymptotes
I think it's 1/x^2. The question says vertical asymptotes so I'm assuming they mean there will only be one. But I don't think I'm right.
Consider the quadratic function
f(x) = x^2 + 5
the reciprocal function would be g(x) = 1/(x^2 + 5)
there is no real number which would make the denominator zero, the condition for vertical asymptotes.
So there is your example, there would be countless of others
Thank you.
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To find a quadratic whose reciprocal has no vertical asymptotes, we need to consider the factors that lead to vertical asymptotes in the reciprocal function.
Recall that vertical asymptotes occur when the denominator of a function approaches zero. For the reciprocal of a quadratic function, the denominator will be the quadratic function itself. Therefore, we need to avoid having any roots in the quadratic function for it to not have vertical asymptotes when we take its reciprocal.
Here's the step-by-step process to find a suitable example:
1. Start with a general quadratic function: y = ax^2 + bx + c
2. Determine the discriminant: The discriminant is found using the formula b^2 - 4ac. This value helps in determining the nature of the roots.
3. Since we want the quadratic to have no real roots, the discriminant should be negative (because imaginary roots will not contribute to vertical asymptotes).
4. Once you choose values for a, b, and c that satisfy the condition, you can take the reciprocal of the resulting quadratic to see if it has any vertical asymptotes.
5. Simplify the reciprocal function as much as possible to identify the presence of any vertical asymptotes.
By following these steps, you can find a quadratic that meets the criteria specified in the question.