1.2 Rational functions and graphs

🔑 Key Definitions

Rational Function: Any function that can be defined as an algebraic fraction with polynomials as numerator and denominator.

Asymptote: Generally a line that a curve approaches but does not touch. The curve converges to an asymptote.

Vertical Asymptote: A vertical line $x = k$ that the graph of a function approaches as the function value goes to infinity or negative infinity.

Horizontal Asymptote: A horizontal line $y = k$ that the graph of a function approaches as $x$ goes to infinity or negative infinity.

Oblique (Slant) Asymptote: A non-horizontal, non-vertical line that the curve approaches as $|x| \to \infty$.

Improper Fraction: A fraction where the degree of the polynomial in the numerator is greater than or equal to the degree of the denominator (also called top-heavy fraction).

Range: The set of all possible output values (y-values) of a function.

Turning Point: A point on a curve where the gradient is zero, representing a local maximum or minimum.

Vertical Asymptotes 📐

Finding Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero (and the numerator does not simultaneously equal zero at that point).

For a rational function $y = \frac{p(x)}{q(x)}$, vertical asymptotes are found by solving $q(x) = 0$.

Example: Basic Rational Functions

Consider $y = \frac{1}{x}$:

• Vertical asymptote: $x = 0$
• Horizontal asymptote: $y = 0$
• The curve approaches both coordinate axes but never touches them

For $y = \frac{1}{x - 1}$:

• Vertical asymptote: $x = 1$ (denominator equals zero)
• Horizontal asymptote: $y = 0$
• When $x = 1.0001$, $y$ is large and positive
• When $x = 0.9999$, $y$ is large and negative
• Y-intercept: $(0, -1)$