Definition of a Rational Function Any function of the form Where N(x) and D(x) are polynomials and D(x) is not the zero polynomial Examples.

x=1 is a vertical asymptote because _____ Vertical asymptote at x = 1

x=1 is a vertical asymptote because_____ Vertical asymptote at x = 1

vertical asymptote ( set denominator = 0 of reduced fraction) x + 5 = 0 x = -5 There is a vertical asymptote at x = -5 There is a hole at x = 5 ( the zeros of common factors)

Graph of

Find all vertical asymptotes and holes of x = 2 is a vertical asymptote because 2 is a zero of the denominator in the reduced form. there are no common factorsno holes

Find all vertical asymptotes and holes of

y=2 is a horizontal asymptote because___ horizontal asymptote at y = 2

Horizontal Asymptotes y=0 is a horizontal asymptote because ____ horizontal asymptote at y = 0

(let n = degree of numerator and d = degree of denominator ) a.If n < d, then y = 0 is the horizontal asymptote Horizontal asymptote at y = 0 Examples:

Leading coefficient of numerator Leading coefficient of denominator b. If n = d, then is the horizontal asymptote Horizontal asymptote at y = 3/2 Horizontal asymptote at y = 10/5 = 2 Examples:

c. If n > d, then there is no horizontal asymptote No horizontal asymptotes

Identify any vertical or horizontal asymptotes, and any holes in the graph

Slant Asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator of the reduced fraction Slant Asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator of the reduced fraction. To find the equation of a slant asymptote use long division To find the equation of a slant asymptote use long division. Equation of the horizontal asymptote Equation of the horizontal asymptote is Ex. Find the equation of the slant asymptote of the equation

Find the equation of the slant asymptote Equation of the horizontal asymptote Equation of the horizontal asymptote is y = x-2

Find the equation of the slant asymptote Equation of the horizontal asymptote Equation of the horizontal asymptote is y = x

 State the domain of the function  Find and plot the y-intercept by evaluating f(0)  Find and plot the x-intercepts by finding the zeros of the numerator  Sketch the vertical asymptotes using dashed vertical lines, and holes using open circles.  Find and sketch any horizontal asymptote using dashed lines.  Find and sketch any slant asymptote using dashed lines.  Plot at least one point between and one point beyond each x-intercept and vertical asymptote.  Use smooth curves to complete the graph

The cost c of producing x units of a product is given by And the average cost per unit is given by Graph the average cost function, and estimate the number of units that should be produced to minimize the average cost per unit. Average Cost

The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is given by a.Determine the horizontal asymptote of the function and interpret its meaning in the context of the problem b.Graph the function and approximate the time when the bloodstream concentration is the greatest. c.When is the concentration less than 0.345? Medicine.