Understanding Rational Functions and Graphs
Learn about rational functions, including how to define, simplify, and graph them. Explore the parts of graphs such as x-intercepts, y-intercepts, vertical asymptotes, horizontal asymptotes, angled asymptotes, and holes. Discover steps to find all information about rational functions, including horizontal asymptotes, y-intercepts, factors, x-intercepts, vertical asymptotes, domain, and range.
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Rational Functions I.. Rational functions. ( ) p x A) Rational function: f(x) = ( ) q x where p(x) & q(x) are polynomials [and q(x) 0] B) Simple rational function: f(x) = 1/x C) Graph of parent rational function: f(x) = 1/x
Rational Functions ( ) p x = ( ) f x II.. Parts of graphs of rational functions: A) x-intercepts (the solutions, roots, etc.) 1) Factor the TOP polynomial (if needed). 2) Set each factor = 0 and solve. 3) The solutions are the x-intercepts. Put dots on graph. B) y-intercept (0, #) 1) Change all the x s to zero. Solve for y. Plot the point. C) Vertical asymptote lines (lines the graph doesn t touch). 1) Factor the BOTTOM polynomial (if needed). 2) Set each factor = 0 and solve. 3) The solutions are the equations of the vertical asymptote lines. Draw vertical dotted lines at x = #. ( ) q x
Rational Functions ( ) p x = ( ) f x II.. Parts of graphs of rational functions: ( ) q x D) Horizontal asymptote lines (cheerleader test). 1) Look at the degree of the top & bottom polynomial. Ask yourself, Could cheerleaders make this pyramid? a) If YES (more on bottom than on top), then horizontal asymptote is y = 0. b) If NO (more on top than on bottom), then there is NO horizontal asymptote. c) If TOP = BOTTOM (special case), then the horizontal asy is y = a/b(leading coeff).
Rational Functions ( ) p x = II.. Parts of graphs of rational functions: E) Angled asymptote lines. 1) If the TOP degree is exactly 1 bigger than the BOTTOM degree, then you have an angled asymptote. a) The angled line has an equation of y = mx + b, 1) To get the equation use long division on f(x) (or synthetic division if the bottom is x1 k). 2) The answer is y = mx + b (ignore remainder). Draw the dotted angled asymptote. F) Holes in the graph (neither a zero nor a vertical asymptote). 1) If you have a common factor in the top and the bottom that cancel out, you will have a hole at its (x,y) solution. ( ) f x ( ) q x
Rational Functions III.. Steps for finding all info on rational functions: A) Use the Cheerleader test to find the horizontal asymptote. 1) Gives a dotted horizontal asymptote line at y = #. B) Find the y-intercept by changing the x s to 0 s and solve. 1) Gives a point on the y-axis (0 , #). C) Factor the TOP and factor the BOTTOM of the function. 1) Cancel out like factors. These cancelled factors will be holes in the graph. Set cancelled factor = 0 and solve. This is the x coordinate of a hole in the graph. 2) x-intercepts (solutions, zeroes, roots) are found by solving the TOP factors. Gives pts on x-axis (# , 0). 3) Vertical asymptotes are found by solving the BOTTOM factors. Gives dotted vertical asymptote line(s) at x = #. D) Domain & Range is all real #s except the asymptote lines & x holes. Domain = {r: x #, x } and Range = {r: y #}
Rational Functions ( ) p x = ( ) f x IV.. Graphing rational functions: ( ) q x A) Factor the TOP & BOTTOM parts. (Cancel down = holes) B) Use the factors to find x-intercepts & vertical asymptotes. C) Use the Cheerleader test to find the horizontal asymptote. D) Plot points on each side of the vertical asy. (y =, table) 1) Choose pts very close to each vertical asy. 2) Choose a point in the middle (between vert asy) 3) Choose a large & + pt for begin / end behavior. E) Connect the points with smooth curved lines. 1) Note: The graph might cross the horizontal asy near the vert asymptotes, but the beginning / end will never cross the horizontal asy (or angled asy).
