Askiitians Tutor Team
Last Activity: 18 Days ago
To analyze the function , we need to break it down step by step. This function is a rational function, meaning it is the ratio of two polynomials. Let's explore its components and behavior.
Understanding the Components
The function consists of a numerator and a denominator:
Analyzing the Numerator
The numerator is a quadratic expression. We can rewrite it in vertex form to understand its properties better. Completing the square:
This shows that the parabola opens upwards, has a vertex at , and does not cross the x-axis, indicating it is always positive.
Examining the Denominator
Now, let's look at the denominator:
Rearranging gives us . To find its roots, we can use the quadratic formula:
Here, , , and . Plugging in these values:
Since the discriminant is negative, the denominator has no real roots, meaning it does not cross the x-axis and is always negative.
Behavior of the Function
Now that we understand both parts, we can analyze the overall function:
- The numerator is always positive.
- The denominator is always negative.
Thus, the function will always be negative for all real values of . This means that the graph of the function will lie below the x-axis.
Finding Asymptotes
Since the denominator does not equal zero for any real , there are no vertical asymptotes. However, we can consider horizontal asymptotes by examining the degrees of the polynomials:
Both the numerator and denominator are quadratic, so we can find the horizontal asymptote by taking the leading coefficients:
This indicates that as approaches infinity or negative infinity, approaches -1.
Summary of Key Points
- The function is always negative.
- It has no real roots and no vertical asymptotes.
- There is a horizontal asymptote at .
In conclusion, this function exhibits interesting behavior as a rational function, and understanding its components helps us predict its overall characteristics. If you have any further questions or need clarification on any part, feel free to ask!