Polynomial and rational functions Knowledge-Based Study

Study notes

1. Defining Polynomial Functions

   A polynomial function in one variable is a function of the form:

   $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$

   where $n$ is a non-negative integer, and the coefficients $a_n, a_{n-1}, \dots, a_0$ are real numbers with $a_n \neq 0$. The highest exponent $n$ is the degree of the polynomial, and $a_n$ is the leading coefficient.

   Key Properties of Polynomial Graphs

   • Domain: The domain of any polynomial function is all real numbers, $(-\infty, \infty)$.
   • Continuity: Graphs of polynomials are smooth and continuous with no breaks, holes, or sharp corners.
   • Turning Points: A polynomial of degree $n$ has at most $n - 1$ local extrema (turning points).
   • Zeros: A polynomial of degree $n$ has at most $n$ real zeros (x-intercepts).

   End Behavior: The Leading Coefficient Test

   As $x \to \infty$ or $x \to -\infty$, the graph of a polynomial behaves like its leading term, $a_n x^n$:

   • Even degree ($n$ is even): Both ends point in the same direction. If $a_n > 0$, both ends rise ($y \to \infty$). If $a_n < 0$, both ends fall ($y \to -\infty$).
   • Odd degree ($n$ is odd): The ends point in opposite directions. If $a_n > 0$, the graph falls to the left and rises to the right. If $a_n < 0$, the graph rises to the left and falls to the right.

   Multiplicity of Zeros

   If a polynomial has a factor $(x - c)^k$, then $c$ is a zero of multiplicity $k$:

   • If $k$ is odd, the graph crosses the x-axis at $x = c$.
   • If $k$ is even, the graph touches the x-axis and turns around at $x = c$.

   Defining Rational Functions

   A rational function is the quotient of two polynomial functions:

   $R(x) = \frac{P(x)}{Q(x)}$

   where $P(x)$ and $Q(x)$ are polynomials, and $Q(x) \neq 0$.

   Domain and Vertical Asymptotes

   The domain of $R(x)$ consists of all real numbers except values where $Q(x) = 0$. These excluded values produce either vertical asymptotes or holes (removable discontinuities):

   • Vertical Asymptote: If $x = c$ is a zero of $Q(x)$ but not of $P(x)$ (after simplifying common factors), the line $x = c$ is a vertical asymptote. As $x \to c$, $R(x) \to \pm\infty$.
   • Hole (Removable Discontinuity): If a factor $(x - c)$ cancels completely from both the numerator and the denominator, a hole occurs at the point $\left(c, \lim_{x \to c} R(x)\right)$.

   Determining End Behavior of Rational Functions

   Let $R(x) = \frac{a_n x^n + \dots}{b_m x^m + \dots}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator:

   • Case 1 ($n < m$): The x-axis, $y = 0$, is a horizontal asymptote.
   • Case 2 ($n = m$): The line $y = \frac{a_n}{b_m}$ is a horizontal asymptote.
   • Case 3 ($n = m + 1$): There is no horizontal asymptote, but there is a slant (oblique) asymptote. We find its equation $y = mx + b$ using polynomial long division: $R(x) = (mx + b) + \frac{r(x)}{Q(x)}$.
   • Case 4 ($n > m + 1$): There are neither horizontal nor slant asymptotes; the end behavior resembles $\frac{a_n}{b_m} x^{n-m}$.

   Step-by-Step Example

   Analyze the function $f(x) = \frac{2x^2 - 8}{x^2 - x - 2}$.

   1. Factor Numerator and Denominator

   $$f(x) = \frac{2(x - 2)(x + 2)}{(x - 2)(x + 1)}$$

   2. Identify Holes and Vertical Asymptotes

   The factor $(x - 2)$ is common to both numerator and denominator:

   • A hole occurs at $x = 2$. For $x \neq 2$, the simplified form is $g(x) = \frac{2(x + 2)}{x + 1}$. Evaluating at $x = 2$: $g(2) = \frac{2(4)}{3} = \frac{8}{3}$. The hole is at $\left(2, \frac{8}{3}\right)$.
   • A vertical asymptote occurs at $x = -1$, where the denominator of the simplified form equals zero.

   3. Identify Horizontal Asymptote

   The degrees of the numerator and denominator are both 2 ($n = m = 2$).

   $$y = \frac{2}{1} = 2$$

   4. Intercepts

   • x-intercept: Set the simplified numerator to zero: $2(x + 2) = 0 \implies x = -2$, giving $(-2, 0)$.
   • y-intercept: Evaluate $f(0)$: $f(0) = \frac{-8}{-2} = 4$, giving $(0, 4)$.