Chapter 5: Rational Functions, Equations, and Inequalities

1. Chapter 5:Rational Functions, Equations, and Inequalities Danish, Jeffrey, John, Melissa, and Virushan

2. 5.1 – Graphs of Reciprocal Functions • All the y-coordinates of reciprocal functions are the reciprocals of the y-coordinates of the original function • The graph of a reciprocal function has a vertical asymptote at each zero of original function • A reciprocal function will always have y= 0 as a Hortizontal asymptote if the original is linear or quadratic • A reciprocal function has the same positive/negative intervals as original function

3. 5.2 – Exploring Quotients of Polynomial Functions • A rational function, has a hole at x = 0 if . • A rational function has a vertical asymptote at x = a if . • A rational function has a horizontal asymptote only when the degree of p(x) is less than or equal to the degree of q(x). • A rational function has an oblique asymptote only when the degree of p(x) is greater than the degree of q(x) by exactly 1.

4. 5.3 – graphs of rational functions • Rational functions of the form have a vertical asymptote defined by x = —d/c and a horizontal asymptote defined by y = 0. • Most rational functions of the form have a vertical asymptote defined by x = —d/c and a horizontal asymptote defined by y = a/c. • Exception occurs when the numerator and denominator both contain a common linear factor. Ex:

5. 5.4 – Solving rational Equations • Zeros of rational functions are zeros of the function in the numerator. • Reciprocal functions have no zeros. All functions of the form have a horizontal asymptote of x = 0. • Inadmissible solutions that are outside the domain can be determined by the context. • To solve a rational equation algebraically, multiply each term by the lowest common denominator and solve the resulting equation.

6. 5.5 – Solving rational inequalities • Remember to reverse the inequality sign when multiplying or dividing both sides by a negative number. • Can solve by graphing the functions on either side of the inequality sign and identifying the vertical asymptotes and points of intersection. • Can also solve an inequality by creating an equivalent inequality with zero on one side and then identifying the intervals created by the zeros of the new function. Above x-axis where f(x) > 0 and below x-axis where f(x) < 0.

7. 5.6 – Rates of change in rational functions • . Equivalent to slope of secant line that passes though the points (x1,y1) and (x2,y2) on the graph of y = f(x) • . Equivalent to estimating the slope of the tangent line that passes though the point (a,f(a)) on the graph of y = f(x). • The instantaneous rate of change at a vertical asymptote is undefined.