Question

Help as fast as possible. This is about the reciprocal of a quadratic function \(y = 1/(x^2-9)\). What is its end behavior?

a) Intervals (-[infinity]/+[infinity])
b) Intervals of slope (+/-)
c) Intervals of slope (increasing/decreasing)
d) Intervals (-3/3)

Answer 1

Intervals (-∞/+∞)

The correct option is a.

Step-by-step explanation:

The reciprocal of the quadratic function
`\(y = \frac{1}{{x^2-9}}\)` has end behavior characterized by the intervals (-∞/+∞). To understand this, let's analyze the given quadratic function
`\(y = x^2-9\).` The roots of this quadratic equation are
`\(x = -3\) and \(x = 3\). As \(x\)`approaches these roots from either side, the denominator
`\(x^2-9\)` approaches zero, causing the reciprocal
`\(\frac{1}{{x^2-9}}\)`to approach infinity. This behavior signifies vertical asymptotes at
`\(x = -3\) and \(x = 3\)`. Therefore, as x goes to negative or positive infinity, the reciprocal function approaches zero, giving us the end behavior intervals of (-∞/+∞).

Understanding the end behavior involves examining the behavior of the function as x approaches both positive and negative infinity. In this case, as x becomes infinitely large (positive or negative), the quadratic term dominates the expression, and the function approaches zero. The vertical asymptotes at x = -3 and x = 3 play a crucial role in determining this behavior, ensuring that the function approaches zero as x reaches infinity. This understanding is essential in analyzing the overall shape and characteristics of the reciprocal of the given quadratic function.

The correct option is a.