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## Rational Functions and Their Graphs

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**Rational Functions and Their Graphs**Section 2.6 Page 326**Definitions**• Rational Function- a quotient of two polynomial functions in the form f(x) = p(x) q(x) ≠ 0 q(x) • Domain:**Example 1**• Find the domain of each rational function**Reciprocal Function**Arrow Notation (see page 328)**Use the graph to answer the following questions.**• As x → -2-, f(x) → • As x → -2+, f(x) → • As x → 2-, f(x) → • As x → 2+, f(x) → • As x → -, f(x) → • As x → , f(x) →**Vertical Asymptotes**• Definition: the line x = a is a vertical asymptote of the graph of a function if f(x) increases or decreases (goes to infinity) without bound as x approaches a • Locating Vertical Asymptotes: set the denominator of your rational function equal to zero and solve for x Find the vertical asymptotes of f(x) = x – 1 x2 – 4**Homework**• Page 342 #1 - 28**Holes**• A value where the denominator of a rational function is equal to zero does not necessarily result in a vertical asymptote. • If the numerator and the denominator of the rational function has a common factor (x – c) then the graph will have a hole at x = c • Example: f(x) = (x2 – 4) x – 2**Finding the Horizontal Asymptote**First identify the degree (highest power) of p(x) and q(x). f(x) = p(x) degree n q(x) degree m and identify their leading coefficients.**Review Transformation of Functions**• Describe how the graphs of the following functions are transformed from its parent function.**Homework**• Page 342 #29 - 48**Graphing Rational Functions**• Seven Step Strategy – page 334 • Check for symmetry • Find the intercepts • Find the asymptotes – check for holes • Plot additional points as necessary**Example 6 – Graph**• Symmetry • Intercepts • Asymptotes • Plot points**Example – Graph**Symmetry Intercepts Asymptotes Plot points**Slant Asymptotes**• Slant Asymptotes occur when the degree of the numerator of a rational function is exactly one greater than that of the denominator • Note- when the degrees are the same or the denominator has a greater degree the function has a horizontal asymptote. Line l is a slant asymptote for a function f(x) if the graph of y = f(x) approaches l as x → ∞ or as x → -∞ l**Determine the Slant Asymptote**• Use synthetic division to find the slant asymptote then graph the function**Find the Slant Asymptote**use long division**Partner Work**Check for symmetry then find the intercepts, asymptotes, and holes of each rational function**Homework**• Page 342 #49 – 78 do 2 skip 1