Key Features of Rational Functions

x-intercept

• Set numerator equal to zero after simplifying

• Find the slope of the line

• Calculate the y-intercept

• Determine the domain of the function

y-intercept

• Plug in zero for x after simplifying

• Find the slope of the line

• Calculate the derivative of the function

• Evaluate the function at its maximum point

If degree of numerator is EQUAL TO degree of denominator, the horizontal asymptote is...

• y = the ratio of leading coefficients

• y = the sum of leading coefficients

• y = the difference of leading coefficients

• y = zero

Vertical asymptote

• Set the numerator equal to zero after simplifying

• Set the denominator equal to zero after simplifying

• Find the limit as x approaches infinity

• Identify the x-intercepts of the function

horizontal asymptote

• Compare leading coefficients of numerator and denominator

• Evaluate the limit as x approaches infinity

• Find the roots of the polynomial

• Determine the vertical asymptotes

If degree of numerator is LESS THAN degree of denominator, the horizontal asymptote is...

• y = 0

• y = 1

• y = ∞

• y = -1

x-value of hole

• Set cancelled out factor equal to zero

• Find the derivative of the function

• Evaluate the limit as x approaches the hole

• Factor the polynomial completely

y-value of hole

• Plug x-value of hole into simplified equation

• Calculate the derivative at the hole

• Evaluate the limit as x approaches the hole

• Find the y-intercept of the function

If degree of numerator is GREATER THAN degree of denominator, the horizontal asymptote is...

• y = 0

• y = infinity

• N/A - there isn't one

• y = the ratio of leading coefficients