Algebra 1 Section 13.8

Definition A rational function is a function expressed as a ratio of polynomials whose denominator is not zero.

Example 1 Graph f(x) = . x ≠ 0; the graph is undefined for x = 0. 2 x x ≠ 0; the graph is undefined for x = 0. Make a table of ordered pairs.

Example 1 x y -4 -½ -2 -1 -1 -2 -½ -4 undefined ½ 4 1 2 2 1 4 ½

Example 1 Graph f(x) = . This graph is called a hyperbola. 2 x This graph is called a hyperbola. The ends of each branch approach but never touch the x- and y-axes.

Example 1 Graph f(x) = . 2 x Therefore, the x-axis is the horizontal asymptote and the y- axis is the vertical asymptote. The center of the hyperbola is at (0, 0).

Example 2 Graph f(x) = + 3. 2 x Since the function is undefined when x = 0, x = 0 is the equation of a vertical asymptote. Make a table of ordered pairs.

Example 2 x y -4 2½ -2 2 -1 1 -½ -1 undefined ½ 7 1 5 2 4 4 3½

Example 2 Graph f(x) = + 3. 2 x This graph is the result of translating the graph of f(x) = three units up. 2 x The center moves up to (0, 3).

Example 2 Graph f(x) = + 3. 2 x The horizontal asymptote moves up to y = 3. The vertical asymptote is still the y-axis.

Graphing Rational Functions x In general, the graph of f(x) = + k is the result of translating the graph of f(x) = k units up if k is positive and k units down if k is negative.

Example 3 Graph g(x) = . -6 x x = 0 is the equation of a vertical asymptote. Make a table of ordered pairs. The graph is in the upper left and lower right quadrants.

Example 3 Graph g(x) = . -6 x The horizontal asymptote is the x-axis, or y = 0. The center is at (0, 0).

Example 4 Graph h(x) = . -6 x – 2 x = 2 is the equation of a vertical asymptote. Make a table of ordered pairs. Graph the function.

Example 4 Graph h(x) = . -6 x – 2 The graph of h(x) is the result of translating the graph of g(x) [Example 3] 2 units right.

Graphing Rational Functions The graph of a function of the form y = + k is a translation of the base function y = . It is a hyperbola with... a x x – h ...a horizontal asymptote of y = k. ...its center at (h, k). ...a vertical asymptote of x = h.

Example 5 Graph f(x) = – 1. (h, k) = (-3, -1) 2 x + 3 (h, k) = (-3, -1) vertical asymptote: x = -3 horizontal asymptote: y = -1 Draw the asymptotes and label the center.

Example 5 Graph f(x) = – 1. 2 x + 3 Make a table of ordered pairs using several convenient points on either side of the hyperbola’s center. Graph the function.

Example 5 The graph of f(x) = – 1 2 The graph of f(x) = – 1 is the result of translating the graph of f(x) = 2/x three units left and one unit down. This is similar to the translations of other types of functions studied earlier. x + 3

Homework: pp. 572-574