Polynomial and Rational Functions: Key Concepts and Problem-Solving Strategies

Polynomial and Rational Functions

Identifying the Vertex and Axis of Symmetry of a Parabola

Quadratic functions, or parabolas, are fundamental in algebra. The vertex is the highest or lowest point on the graph, and the axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror images.

• Standard Form: $y = ax^2 + bx + c$

• Vertex Formula: The vertex $(h, k)$ is found using $h = -\frac{b}{2a}$ and $k = f(h)$.

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Example: For $y = 3x^2 + 18x + 22$, $a = 3$, $b = 18$, so $x = -\frac{18}{2 \times 3} = -3$.

Domain and Range of Functions

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values).

• Polynomial Functions: Domain is usually $(-\infty, \infty)$.

• Quadratic Example: $f(x) = x^2 - 2$ has domain $(-\infty, \infty)$ and range $[-2, \infty)$.

• Rational Functions: Exclude values that make the denominator zero.

End Behavior of Polynomial Functions

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

• Even Degree, Positive Leading Coefficient: Both ends up.

• Even Degree, Negative Leading Coefficient: Both ends down.

• Odd Degree, Positive Leading Coefficient: Left down, right up.

• Odd Degree, Negative Leading Coefficient: Left up, right down.

• Example: $f(x) = 2x^6 + 5x^2 - 3x + 3$ has even degree and positive leading coefficient, so both ends up.

Symmetry of Functions

Functions can be classified by their symmetry:

• Even Function: $f(-x) = f(x)$ (symmetric about the y-axis)

• Odd Function: $f(-x) = -f(x)$ (symmetric about the origin)

• Neither: If neither condition holds.

• Example: $f(x) = x^4 - 2x^2 - 3$ is even.

Leading Term, Leading Coefficient, and Degree

For a polynomial $f(x) = a_nx^n + ... + a_0$:

• Leading Term: The term with the highest power of $x$.

• Leading Coefficient: The coefficient of the leading term.

• Degree: The highest exponent of $x$.

• Example: $f(x) = 7x^8 - 10 + 0.12x - x^3$ has leading term $7x^8$, leading coefficient $7$, and degree $8$.

Polynomial Division

Polynomials can be divided using synthetic division (for linear divisors) or long division (for higher-degree divisors).

• Synthetic Division: Used for divisors of the form $x - c$.

• Long Division: Used for more complex divisors.

• Example: Divide $3x^2 - 2x - 6$ by $x + 2$ using synthetic division.

Zeros of Polynomial Functions

The zeros (or roots) of a polynomial are the values of $x$ for which $f(x) = 0$. The multiplicity of a zero is the number of times it appears as a factor.

• Example: $f(x) = 2x^3 + 5x^2 + 12x - 8$

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

Descartes' Rule of Signs

Descartes' Rule of Signs helps determine the possible number of positive and negative real zeros of a polynomial.

• Count sign changes in $f(x)$ for positive zeros.

• Count sign changes in $f(-x)$ for negative zeros.

• The number of positive/negative real zeros is equal to the number of sign changes or less by an even number.

Rational Functions: Domain and Asymptotes

Rational functions are quotients of polynomials. Their domain excludes values that make the denominator zero. Asymptotes describe the behavior of the function as $x$ approaches certain values.

• Vertical Asymptotes: Set denominator equal to zero and solve for $x$.

• Horizontal Asymptotes: Compare degrees of numerator and denominator.

• Slant (Oblique) Asymptotes: Occur when the degree of the numerator is one more than the denominator.

• Example Table:

Type	How to Find	Example
Vertical	Set denominator = 0	$x^2 - 4 = 0 \implies x = \pm2$
Horizontal	Compare degrees	Degree numerator < denominator: $y=0$
Slant	Numerator degree = denominator degree + 1	Divide numerator by denominator

Solving Polynomial and Rational Inequalities

To solve inequalities, find the zeros, test intervals, and graph the solution set on a number line.

• Example: Solve $3x^2 - 11x + 4 < 0$.

• Find zeros, test intervals, and shade the solution set.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a polynomial function changes sign over an interval, then it has at least one real zero in that interval.

• Application: If $f(a)$ and $f(b)$ have opposite signs, there is a zero between $a$ and $b$.

Summary Table: Key Properties of Polynomial and Rational Functions

Property	Polynomial	Rational Function
Domain	$(-\infty, \infty)$	All real numbers except where denominator = 0
Zeros	Set $f(x) = 0$	Set numerator = 0 (if denominator ≠ 0)
Asymptotes	None	Vertical, horizontal, or slant
End Behavior	Determined by leading term	Determined by degrees of numerator and denominator

Additional info: These notes expand on the problem set by providing definitions, formulas, and examples for each concept, making them suitable for exam preparation in College Algebra.