Algebra 94 - Rational Functions with Oblique or Curvilinear Asymptotes

The video tutorial by Professor von Schmohawk explores the concept of asymptotes in rational functions. It distinguishes between vertical and non-vertical asymptotes, explaining that rational functions can have multiple vertical asymptotes but only one non-vertical asymptote. The tutorial covers horizontal, oblique, and curvilinear asymptotes, detailing how to determine their shapes and positions using polynomial functions. Examples are provided to illustrate the calculation of these asymptotes. The video concludes with a preview of upcoming topics on logarithmic and exponential functions.

What is a vertical asymptote in a rational function?

• A line that the graph approaches as x becomes very large

• A line where the function's value becomes undefined

• A line that the graph never touches

• A line that the graph crosses at infinity

How many non-vertical asymptotes can a rational function have?

• Infinite

• None

• One

• Two

What determines the type of non-vertical asymptote in a rational function?

• The coefficients of the leading terms

• The degree of the numerator and denominator

• The constant term in the numerator

• The x-intercepts of the function

When does a rational function have an oblique asymptote?

• When the degree of the numerator is two more than the degree of the denominator

• When the degree of the numerator is one more than the degree of the denominator

• When the degree of the numerator is equal to the degree of the denominator

• When the degree of the numerator is less than the degree of the denominator

What is the result of dividing the numerator by the denominator in a rational function?

• An undefined expression

• A constant value

• A polynomial that describes the asymptote

• A new rational function

What type of asymptote is described by a second-degree polynomial?

• Cubic

• Oblique

• Horizontal

• Parabolic

How is a curvilinear asymptote determined when the denominator has more than one term?

• By graphing the function

• By using simple division

• By using polynomial long division

• By factoring the numerator

What is the degree of the polynomial that describes a cubic asymptote?

• First degree

• Fourth degree

• Second degree

• Third degree

What happens to terms with x in the denominator as x becomes very large?

• They remain constant

• They approach zero

• They become negative

• They approach infinity

What will be introduced in the next lectures following the discussion on asymptotes?

• Trigonometric functions

• Logarithmic and exponential functions

• Quadratic equations

• Complex numbers