Rational Function Graphing Analysis

Domain:

• Determine the domain by identifying all real values of x that make the denominator zero (after cancellation).
• Express the domain in set notation (e.g., {x | x ≠ a, b}).

Intercepts:

• Find x-intercepts by setting the numerator equal to zero (excluding holes).
• Find the y-intercept by evaluating f(0), if defined.
• State intercepts as ordered pairs.

Behavior at X-Intercepts:

• For each x-intercept, determine if the graph crosses the x-axis or touches but does not cross it (based on the multiplicity of the zero).

Vertical Asymptotes:

• Identify vertical asymptotes from the remaining denominator factors.
• Determine the behavior of the graph on either side of each vertical asymptote (approaching +∞ or -∞).

Horizontal Asymptotes:

• Compare the degrees of the numerator and denominator.
• If degree(num) < degree(denom), the horizontal asymptote is y = 0.
• If degree(num) = degree(denom), the horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator).
• If degree(num) > degree(denom), there is no horizontal asymptote.

Oblique Asymptotes:

• If the degree of the numerator is exactly one greater than the degree of the denominator, find the oblique (slant) asymptote using polynomial division.
• Otherwise, state there is no oblique asymptote.

Intersection with Asymptotes:

• Determine if the graph intersects the horizontal or oblique asymptote by solving f(x) = asymptote equation.
• State the point(s) of intersection or confirm there are none.

Interval Analysis:

• Use the real zeros of the numerator and denominator to divide the x-axis into intervals.
• Choose a test value in each interval to determine if the graph is above or below the x-axis.
• Present the results using interval notation.