Rational graphs

Vertical asymptote: condition

x = a is a vertical asymptote when Q(a) = 0 AND P(a) ≠ 0 (cancel common factors first)

Horizontal asymptote: deg(P) < deg(Q)

y = 0 (the x-axis is the horizontal asymptote)

Horizontal asymptote: deg(P) = deg(Q)

y = a/b where a and b are the leading coefficients of P and Q

Horizontal asymptote: deg(P) > deg(Q)

No horizontal asymptote

Oblique/slant asymptote: condition

Exists when deg(P) = deg(Q) + 1

Oblique/slant asymptote: how to find

Perform polynomial long division: P(x)/Q(x) = mx + b + R(x)/Q(x) — the asymptote is y = mx + b

Can a rational function have both a horizontal AND oblique asymptote?

No — it can only have one or the other, never both

Can a rational function have two horizontal asymptotes?

No — maximum of one horizontal asymptote

Crossing a horizontal asymptote y = L: how to test

Solve f(x) = L — if a solution exists, the graph crosses it

Crossing an oblique asymptote y = mx + b: how to test

Solve R(x) = 0 where R(x) is the remainder from long division

y-intercept: how to find

Evaluate f(0)

x-intercept: how to find

Set P(x) = 0 and solve (provided Q(x) ≠ 0 at those values)

Sign chart: what does f(x) > 0 mean?

Graph is above the x-axis

Sign chart: what does f(x) < 0 mean?

Graph is below the x-axis

Sign chart: how to construct

Mark all x-intercepts and vertical asymptotes on a number line, then test the sign of f(x) in each interval

Stationary points: how to find

Solve f'(x) = 0

Local minimum: second derivative test

f''(x) > 0 at the stationary point

Local maximum: second derivative test

f''(x) < 0 at the stationary point

Point of inflection: condition

f''(x) = 0 AND f''(x) changes sign at that point

Symmetry about the y-axis: condition

f(-x) = f(x) — even function

Symmetry about the origin: condition

f(-x) = -f(x) — odd function

Steps to sketch a polynomial function

1) Stationary points: f'(x) = 0 2) Classify with f''(x) 3) Test concavity: sign change of f''(x) 4) y-intercept: f(0) 5) Sketch 6) x-intercepts if asked: f(x) = 0 via Newton-Raphson

Steps to sketch a rational function

1) Domain and range 2) Stationary points 3) Classify and test concavity 4) y-intercept 5) x-intercept 6) All asymptotes (vertical, horizontal, oblique) 7) Symmetry 8) Sign chart

Removable discontinuity (hole): when does it occur in a rational function?

When a factor cancels from both P(x) and Q(x) — the function is undefined at that point but has no asymptote there