1.3 Identifying asymptotes of and holes in the graphs of rational functions

• What is a rational function expressed as?

  Ratio of two polynomials
• Holes in rational functions occur when the denominator of the function is equal to zero
• Match the type of asymptote with its definition or characteristic:

  Vertical Asymptote ↔️ Vertical line the graph approaches

  Horizontal Asymptote ↔️ Horizontal line the graph approaches as x approaches ±∞
• Holes in rational functions occur when a common factor in both the numerator and denominator cancels out.


  True
• Holes occur in rational functions when the numerator and denominator both equal zero
• Vertical asymptotes occur when the denominator equals zero and the numerator does not.


  True
• What is the horizontal asymptote of f(x) = (3x + 1) / (x - 2)?

  y = 3
• What is a hole in a rational function?

  Undefined point on the graph
• Vertical asymptotes occur when the denominator of a rational function equals zero
• Holes in rational functions occur when both the numerator and denominator equal zero
• To identify holes, look for common factors in the numerator and denominator that cancel
• Match the feature with its type in rational functions:

  Holes ↔️ Points where numerator and denominator both equal zero and factors cancel

  Vertical Asymptotes ↔️ Vertical lines where denominator equals zero but numerator does not
• Steps to calculate horizontal asymptotes:

  1️⃣ Compare the degrees of the numerator and denominator

  2️⃣ If degree of numerator < degree of denominator, then y = 0

  3️⃣ If degrees are equal, then y = (leading coefficient of numerator) / (leading coefficient of denominator)

  4️⃣ If degree of numerator > degree of denominator, there is no horizontal asymptote
• What is a vertical asymptote in a rational function?

  A line the graph approaches
• An oblique asymptote occurs when the degree of the numerator is greater than the degree of the denominator.


  True
• Rational functions can have three types of asymptotes
• What is an oblique asymptote in a rational function?

  Non-zero slope
• How are vertical asymptotes calculated in a rational function?

  Denominator equals zero
• Holes occur in rational functions when common factors in the numerator and denominator cancel
• Holes in rational functions occur at specific x-values where both the numerator and denominator equal zero.


  True
• A vertical asymptote is a vertical line the graph of the function touches but never crosses.

  False
• When do vertical asymptotes occur in rational functions?

  Denominator equals zero
• Vertical asymptotes occur when the denominator of a rational function equals zero
• What is the definition of a hole in a rational function?

  Undefined point due to cancellation
• What is a rational function expressed as?

  P(x) / Q(x)
• Where does the vertical asymptote of f(x) = 1 / (x - 2) occur?

  x = 2
• Oblique asymptotes are diagonal lines the graph approaches as x approaches infinity.


  True
• In f(x) = (x - 2) / (x - 2)(x + 3), there is a hole at x = 2.

  True
• Factors that cancel from the numerator and denominator can indicate the presence of a vertical asymptote.

  True
• Holes occur at x-values where common factors in the numerator and denominator cancel out.

  True
• Holes in rational functions occur when common factors in the numerator and denominator cancel
• In the function f(x) = (x^2 - 4) / (x^2 - 1), the hole occurs at x = ±1
• What is the horizontal asymptote of the function f(x) = (2x^2 + 3x) / (x^2 + 1)?

  y = 2
• The function f(x) = (x+1) / (x-2) has a vertical asymptote at x = 2
• What condition must be true for a rational function to have a hole?

  Common factors must cancel
• What is a vertical asymptote in a rational function?

  Graph approaches but never touches
• Rational functions can have holes, which are points where the function is undefined because both the numerator and denominator equal zero
• The rational function f(x) = 1 / (x - 2) has a vertical asymptote at x = 2.


  True
• Match the feature with its description in rational functions:

  Holes ↔️ Common factors cancel

  Vertical Asymptotes ↔️ Denominator equals zero
• What are holes in rational functions?

  Points where numerator and denominator both equal zero