Understanding Oblique Asymptotes and Graphing Rational Functions

This video tutorial covers the process of finding and graphing oblique asymptotes in rational functions. It explains the conditions under which oblique asymptotes occur, specifically when the degree of the numerator is one more than the degree of the denominator. The tutorial demonstrates how to use synthetic and long division to find the equation of the oblique asymptote, ignoring the remainder. It also provides steps for graphing these functions, including finding x-intercepts, vertical asymptotes, and using a graphing calculator to ensure accuracy.

What is the condition for a rational function to have an oblique asymptote?

• The degree of the numerator is equal to the degree of the denominator.

• The degree of the numerator is two more than the degree of the denominator.

• The degree of the numerator is one less than the degree of the denominator.

• The degree of the numerator is one more than the degree of the denominator.

What is the equation of an oblique asymptote derived from the division of polynomials?

• y = ax^2 + bx + c

• y = c

• y = mx + b

• y = ax + b

Which division method is preferred when the divisor is linear with a leading coefficient of one?

• Synthetic division

• Long division

• Matrix division

• Polynomial division

Why is the remainder ignored when finding the equation of an oblique asymptote?

• It affects the slope

• It affects the y-intercept

• It does not affect the location of the asymptote

• It is always zero

When graphing rational functions, what is the first step after identifying oblique asymptotes?

• Determine the domain

• Find the y-intercept

• Use a graphing calculator

• Factor the function to see if anything cancels

What is the domain of a rational function with a vertical asymptote at x = 0?

• All real numbers

• All x except x = 0

• All x greater than 0

• All x less than 0

What tool is recommended to ensure accurate placement of hyperbolic curves when graphing?

• A protractor

• A compass

• A ruler

• A graphing calculator

What is the range of a rational function with hyperbolic curves extending infinitely?

• All y less than 0

• All y greater than 0

• All y except y = 0

• All real numbers

In problem-solving, what happens if the degree of the numerator equals the degree of the denominator?

• There is a slant asymptote

• There is a horizontal asymptote

• There is no asymptote

• There is a vertical asymptote

What should be done if the divisor in synthetic division is not linear with a leading coefficient of one?

• Multiply the divisor by a constant

• Use synthetic division anyway

• Use long division

• Ignore the divisor