Discussion X-intercepts

X-intercepts of Rational Function Notes X-intercepts of Rational Function To find the x-int of Rational Functions, set the numerator equal to zero and solve for x.

Find all x-intercepts of each function. Practice Find all x-intercepts of each function.

Discussion y-intercepts

y-intercepts of Rational Function Notes y-intercepts of Rational Function To find the y-int of Rational Functions, substitute 0 for x.

Find all y-intercepts of each function. Practice Find all y-intercepts of each function.

Discussion Domain and Range What is the domain of this function? Are there any numbers that x is not allowed to equal?

The domain is “split” by the vertical asymptotes! Practice Find the Domain. The domain is “split” by the vertical asymptotes!

Notes Asymptote An asymptote is a line that a function approaches but never actually reaches. Vertical Asymptote Horizontal Asymptote

Notes Vertical Asymptotes A rational function has a vertical asymptote at each value of x that makes only the denominator equal zero. Example: Vertical Asymptotes:

Find the Vertical Asymptotes: Practice Find the Vertical Asymptotes:

Horizontal Asymptotes Notes Horizontal Asymptotes a: leading coefficient of the numerator b: leading coefficient of the denominator n: degree of the numerator m: degree of the denominator If m = n, then y = a/b is an asymptote If n < m, then y = 0 is an asymptote If n > m, then there is no horizontal asymptote

Find the Horizontal Asymptotes: Practice Find the Horizontal Asymptotes:

Discussion End Behavior

Notes End Behavior The shape of a fucntion when x approaches positive or negative infinity. Where does the graph “level out?” The horizontal asymptote is sometimes called the “end-behavior asymptote.” Why?

State the end behavior of each function. Practice State the end behavior of each function.

Notes Holes If a value of x makes both the numerator and denominator of the fraction equal zero, then the function has a hole at that point. Example: Hole:

Practice Find the Holes:

Notes Slant Asymptote A rational Function will have a slant asymptote if the degree of the top is exactly one more than the degree of the bottom. Example:

Determine if the function has a slant asymptote. Practice Determine if the function has a slant asymptote.

Find the equation of the slant asymptote: Discussion Find the equation of the slant asymptote:

Notes Slant Asymptote To find the slant asymptote of a rational function, divide the top by the bottom. The equation of the asymptote is the quotient (ignore the remainder).

Find the slant asymptote. Practice Find the slant asymptote.

Discussion Graph the function. x-int: V.A.: H.A.: S.A.: Holes:

Practice Graph each function.

Class Work (30 minutes) Find the HOLES: Factor both numerator and denominator and look for common factors. If there are any, set them equal to zero and solve. Those are your holes. If no factors repeat, there are no holes. Find the X-INTERCEPT(S): Set your numerator equal to zero and solve Find the Y-INTERCEPT: Plug in zero for all x-values Find the VERTICAL ASYMPTOTE: Set your denominator equal to zero and solve. If there is a hole, there is no V.A. Find your HORIZONTAL ASYMPTOTE: If the degree of the numerator is equal to the degree of the denominator, take the coefficient of the terms. If the degree of the numerator is larger than the denominator, there is no H.A, there is a SLANT ASYMPTOTE. If the degree of the numerator is less than the denominator then the H.A is the line y=0.

Inverses of Rational Functions Cross multiply Solve for x (factor) Switch x and y Rewrite as an inverse Example: Class Work (30 minutes)