Understanding Horizontal Asymptotes

The video tutorial explores the function f(x) = (-x^2 + ax + b) / (x^2 + cx + d), where a, b, c, and d are unknown constants. The focus is on determining the possible graph of y = f(x) by analyzing horizontal asymptotes as x approaches positive or negative infinity. The function approaches -1, indicating a horizontal asymptote at y = -1. The tutorial evaluates four graph choices, ruling out those with incorrect asymptotes, and concludes that choice A is correct based on the given information.

What is the main challenge presented in identifying the graph of the function f(x)?

• Understanding the behavior of the function at x = 0.

• Determining the coefficients a, b, c, and d.

• Finding the roots of the function.

• Identifying the vertical asymptotes.

What is the role of the constants a, b, c, and d in the function f(x)?

• They determine the horizontal asymptote.

• They determine the vertical asymptote.

• They are unknown and do not affect the asymptote.

• They determine the roots of the function.

As x approaches positive or negative infinity, what does f(x) approximate to?

• Positive infinity

• Negative infinity

• Negative one

• Zero

What is the significance of the highest degree terms in determining the horizontal asymptote?

• They dominate the behavior of f(x) as x becomes very large.

• They determine the vertical asymptote.

• They are irrelevant to asymptotes.

• They help find the roots of the function.

What is the horizontal asymptote of the function f(x) as x approaches infinity?

• y = 1

• y = -1

• y = 0

• y = 2

Why can choice B be ruled out as the correct graph for f(x)?

• It has a horizontal asymptote at y = 2.

• It intersects the x-axis.

• It has a vertical asymptote at x = 0.

• It has no asymptotes.

Which graph choice has a horizontal asymptote at y = 0?

• Choice B

• Choice A

• Choice D

• Choice C

Which of the following is NOT a reason for ruling out a graph choice?

• Incorrect number of roots.

• Incorrect vertical asymptote.

• Incorrect intersection with the y-axis.

• Incorrect horizontal asymptote.

What is the key information given to determine the correct graph of f(x)?

• The vertical asymptote.

• The number of roots of the function.

• The coefficients a, b, c, and d.

• The horizontal asymptote.

Why is choice A considered the correct graph for f(x)?

• It has no asymptotes.

• It has a horizontal asymptote at y = -1.

• It intersects the y-axis at y = 0.

• It has a vertical asymptote at x = 0.