Precalculus Study Guide: Polynomial and Rational Functions (Ch. 3)

Polynomial and Rational Functions

Quadratic Functions

Quadratic functions are polynomials of degree 2 and are fundamental in precalculus. Their graphs are parabolas, and they can be expressed in standard or vertex form.

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Vertex: The point $(h, k)$ is the maximum or minimum of the parabola.

• Axis of Symmetry: $x = h$ (in vertex form), or $x = -\frac{b}{2a}$ (in standard form).

• Direction: If $a > 0$, the parabola opens upward (minimum); if $a < 0$, it opens downward (maximum).

• Domain: $(-\infty, \infty)$

• Range: $[k, \infty)$ if $a > 0$; $(-\infty, k]$ if $a < 0$

• Intervals of Increase/Decrease: Decreasing to the left of the vertex, increasing to the right (if $a > 0$).

Example: For $f(x) = 2x^2 + 3x - 7$, the vertex is at $x = -\frac{3}{4}$.

Transformations of Quadratic Functions

Quadratic functions can be transformed by shifting, stretching, or reflecting their graphs.

• Horizontal Shift: $h$ units right ($h > 0$) or left ($h < 0$)

• Vertical Shift: $k$ units up ($k > 0$) or down ($k < 0$)

• Reflection: If $a < 0$, the graph is reflected over the x-axis.

• Stretch/Compression: $|a| > 1$ stretches vertically; $0 < |a| < 1$ compresses vertically.

Example: $f(x) = -2(x+1)^2 + 3$ is a parabola opening downward, shifted left 1 unit and up 3 units.

Converting Standard Form to Vertex Form

To graph or analyze a quadratic function, it is often useful to rewrite it in vertex form by completing the square.

• Complete the Square: $f(x) = ax^2 + bx + c \rightarrow a(x-h)^2 + k$

• Steps:

  1. Factor $a$ from $x^2$ and $x$ terms.

  2. Add and subtract $(\frac{b}{2a})^2$ inside the bracket.

  3. Simplify to get vertex form.

Example: $f(x) = x^2 + 6x + 7$ Complete the square: $f(x) = (x+3)^2 - 2$

Understanding Polynomial Functions

Definition and Recognition

A polynomial function is an expression 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 and coefficients are real numbers.

• Degree: The highest power of $x$.

• Leading Coefficient: The coefficient of the highest degree term.

• Standard Form: Terms are written in descending order of degree.

Example: $f(x) = 3x^4 - 2x^2 + 7$ has degree 4 and leading coefficient 3.

Graphs of Polynomial Functions

Polynomial functions have smooth, continuous graphs. Their domain is always $(-\infty, \infty)$.

• End Behavior: Determined by degree and leading coefficient.

• Turning Points: Points where the graph changes direction; maximum number is $n-1$ for degree $n$.

End Behavior

The end behavior of a polynomial function describes how the function behaves as $x \to \pm\infty$.

• Even Degree: Both ends go up if leading coefficient $> 0$, both go down if $< 0$.

• Odd Degree: Left and right ends go in opposite directions.

Degree	Leading Coefficient $> 0$	Leading Coefficient $< 0$
Even	Rises left & right	Falls left & right
Odd	Falls left, rises right	Rises left, falls right

Zeros and Multiplicity

Zeros (roots) of a polynomial are values of $x$ where $f(x) = 0$. The multiplicity of a zero is the number of times it occurs as a factor.

• Multiplicity Odd: Graph crosses the x-axis.

• Multiplicity Even: Graph touches but does not cross the x-axis.

Example: $f(x) = (x-2)^2(x+1)$ has a zero at $x=2$ (multiplicity 2, touches) and $x=-1$ (multiplicity 1, crosses).

Turning Points

Turning points are locations where the graph changes direction. The maximum number of turning points for a polynomial of degree $n$ is $n-1$.

• Example: $f(x) = x^3 - 4x^2 + 2x$ (degree 3) has at most 2 turning points.

Graphing Polynomial Functions

Intervals of Unknown Behavior

To graph a polynomial, plot known points (intercepts, turning points, end behavior) and connect with a smooth curve. Break the graph into intervals between these points to analyze behavior.

Graphing Steps

• Determine end behavior.

• Find x-intercepts (zeros) and their multiplicities.

• Find y-intercept ($f(0)$).

• Find turning points.

• Plot points and connect with a smooth curve.

Introduction to Rational Functions

Definition and Domain

A rational function is a function of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Domain: All real numbers except where $Q(x) = 0$.

• Restrictions: Values that make the denominator zero are excluded from the domain.

Example: $f(x) = \frac{x+1}{x-2}$ has domain $x \neq 2$.

Simplifying Rational Functions

• Factor numerator and denominator.

• Cancel common factors.

• State domain after simplification.

Asymptotes

Vertical Asymptotes

Vertical asymptotes occur at values of $x$ where the denominator of a rational function is zero (and not canceled by the numerator).

• Find by: Setting denominator equal to zero and solving for $x$.

• Example: $f(x) = \frac{1}{x-3}$ has a vertical asymptote at $x=3$.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as $x \to \pm\infty$.

• If degree of numerator $<$ degree of denominator: $y=0$

• If degrees are equal: $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$

• If degree of numerator $>$ degree of denominator: No horizontal asymptote

Degree Comparison	Horizontal Asymptote
Num < Den	$y=0$
Num = Den	$y=\frac{a_n}{b_n}$
Num > Den	None

Removable Discontinuities (Holes)

Holes occur when a factor cancels in both numerator and denominator, creating a removable discontinuity at that $x$-value.

• Find by: Factoring numerator and denominator, canceling common factors, and solving for $x$ where the canceled factor equals zero.

• Example: $f(x) = \frac{(x-2)(x+1)}{(x-2)(x-3)}$ has a hole at $x=2$ and a vertical asymptote at $x=3$.

Graphing Rational Functions

Transformations

Rational functions can be graphed by applying transformations to the parent function $f(x) = \frac{1}{x}$.

• Horizontal Shift: $h$ units right/left

• Vertical Shift: $k$ units up/down

• Reflection: Over x-axis or y-axis

Example: $g(x) = \frac{1}{x-2} + 3$ is shifted right 2 units and up 3 units.

Graphing Steps for Rational Functions

• Factor numerator and denominator.

• Find domain and restrictions.

• Identify and plot vertical and horizontal asymptotes.

• Find x- and y-intercepts.

• Identify holes (removable discontinuities).

• Plot points and connect with smooth curves, approaching asymptotes.

Summary Table: Key Properties of Polynomial and Rational Functions

Property	Polynomial Functions	Rational Functions
Domain	$(-\infty, \infty)$	All real numbers except where denominator is zero
End Behavior	Determined by degree and leading coefficient	Determined by degrees of numerator and denominator
Zeros	Set $f(x) = 0$	Set numerator $= 0$ (if not canceled by denominator)
Asymptotes	None	Vertical and horizontal (and possibly oblique)
Holes	None	Where factors cancel in numerator and denominator

Additional info:

• Practice problems and graphing exercises are included throughout the notes to reinforce concepts.

• Tables and diagrams are used to compare properties and illustrate graphical behavior.