Exam 2

Polynomial and Rational Functions

Graphs of Functions

Understanding the graphical behavior of polynomial and rational functions is essential in precalculus. The graphs provide visual insight into zeros, multiplicity, end behavior, and asymptotes.

• Polynomial Functions: These are functions of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$ where $n$ is a non-negative integer.

• Rational Functions: These are functions of the form $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Graph Features: Key features include intercepts, zeros, turning points, and asymptotes.

Example: The graph of $f(x) = x^2$ is a parabola opening upwards, while $f(x) = \frac{1}{x}$ has vertical and horizontal asymptotes.

Zeros and Multiplicity

The zeros of a polynomial are the values of $x$ for which $f(x) = 0$. The multiplicity of a zero refers to how many times a particular zero occurs.

• Zero: If $f(a) = 0$, then $x = a$ is a zero of $f(x)$.

• Multiplicity: If $(x-a)^k$ is a factor of $f(x)$, then $x=a$ is a zero of multiplicity $k$.

• Graphical Behavior:

  • If the multiplicity is odd, the graph crosses the x-axis at the zero.

  • If the multiplicity is even, the graph touches but does not cross the x-axis at the zero.

Example: For $f(x) = (x-2)^2(x+1)$, $x=2$ is a zero of multiplicity 2 (touches x-axis), $x=-1$ is a zero of multiplicity 1 (crosses x-axis).

Tables: Zeros, Multiplicity, and Graph Behavior

Tables are useful for organizing information about the zeros of a polynomial, their multiplicities, and the corresponding behavior of the graph at each zero.

Zero	Multiplicity	Graph Touch and Turn
$x=a$	Odd	Crosses x-axis
$x=b$	Even	Touches x-axis, turns

Additional info: The original tables were blank; this is a standard academic completion.

End Behavior of Polynomial Functions

The end behavior of a polynomial function describes how the function behaves as $x$ approaches $\infty$ or $-\infty$. It is determined by the leading term $a_nx^n$.

• Even Degree: If $a_n > 0$, both ends go up; if $a_n < 0$, both ends go down.

• Odd Degree: If $a_n > 0$, left end down, right end up; if $a_n < 0$, left end up, right end down.

Example: $f(x) = x^3$ has left end down, right end up.

Graphing Rational Functions: Asymptotes

Rational functions often have vertical and horizontal (or oblique) asymptotes.

• Vertical Asymptotes: Occur at values of $x$ where $Q(x) = 0$ and $P(x) \neq 0$.

• Horizontal Asymptotes: Determined by the degrees of $P(x)$ and $Q(x)$.

  • If degree of $P(x)$ < degree of $Q(x)$, $y=0$ is a horizontal asymptote.

  • If degree of $P(x)$ = degree of $Q(x)$, $y=\frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}$ is a horizontal asymptote.

  • If degree of $P(x)$ > degree of $Q(x)$, there is no horizontal asymptote (may be oblique).

Example: $f(x) = \frac{1}{x-2}$ has a vertical asymptote at $x=2$ and a horizontal asymptote at $y=0$.

Piecewise and Absolute Value Functions

Piecewise functions are defined by different expressions over different intervals. The absolute value function $f(x) = |x|$ is a classic example, with a graph that forms a 'V' shape.

• Piecewise Function: $f(x) = \begin{cases} x^2 & x < 0 \\ x & x \geq 0 \end{cases}$

• Absolute Value Function: $f(x) = |x|$

Example: The graph of $f(x) = |x|$ is symmetric about the y-axis and has a vertex at the origin.

Summary Table: Polynomial Zeros and Graph Behavior

Zero	Multiplicity	Graph Touch and Turn
$x=a$	1	Crosses x-axis
$x=b$	2	Touches x-axis, turns
$x=c$	3	Crosses x-axis, flattens

Additional info: Table entries inferred for academic completeness.