Lecture on Rational Functions

Overview

• Discuss how to deal with rational functions
• Find key features: vertical and horizontal asymptotes, domain, range, x-intercept, y-intercept
• Graph the rational function

Key Concepts

Vertical Asymptote

• Found by setting the denominator equal to zero.
• Example: for (8x - 3), set it to zero: (x = \frac{3}{8}).

Horizontal Asymptote

• Consider (f(x) = \frac{2x + 1}{8x - 3})
• Horizontal asymptote found by comparing the degrees of numerator and denominator.
• If equal degree, ratio of leading coefficients: (y = \frac{1}{4}).

Domain

• Exclude values making the denominator zero.
• Example: (8x - 3 \neq 0) implies (x \neq \frac{3}{8}).
• Domain in interval notation: ((-\infty, \frac{3}{8}) \cup (\frac{3}{8}, \infty)).

Range

• Range discussed later but involves y-values the function cannot take.
• Look at the graph to identify asymptotic behavior and open circles.

X-Intercept

• Set (f(x)) equal to zero and solve for x.
• Example: solve (2x + 1 = 0) gives (x = -\frac{1}{2}).
• Represent as a point: ((-\frac{1}{2}, 0)).

Y-Intercept

• Let (x = 0) and solve for (f(x)).
• Example: (f(0) = \frac{2(0) + 1}{8(0) - 3} = -\frac{1}{3}).
• Represent as a point: ((0, -\frac{1}{3})).

Graphing the Function

• Label asymptotes and intercepts on the graph.
• Check the sign of the function to determine whether the graph is above or below the x-axis.
• Use a graphing calculator to verify.

Additional Example with Reduced Form

Vertical Asymptote

• Reduced form (\frac{3}{x+5}).
• Vertical asymptote at (x = -5).

Horizontal Asymptote

• Reduced form (\frac{3}{x+5}) implies (y = 0).

Domain

• Original form restriction: (x^2 + 7x + 10 \neq 0).
• Exclude (x = -2, -5).
• Domain: ((-\infty, -5) \cup (-5, -2) \cup (-2, \infty)).

X-Intercept

• No x-intercept as numerator doesn't equal zero.

Y-Intercept

• (f(0) = \frac{3}{5}), point ((0, \frac{3}{5})).

Graphing with Open Circles

• Consider domain restrictions and draw open circles where necessary.
• Calculate open circle y-values using reduced form.
• Highlight the domain and range considerations on the graph.

Conclusion

• Understanding and graphing rational functions involves recognizing restrictions, intercepts, and asymptotic behaviors.
• Always verify findings with a graphing calculator.