College Algebra: Quadratic and Polynomial Functions, Equations, and Inequalities Study Guide

Quadratic Functions and Equations

Simplifying Expressions Involving Exponents and Complex Numbers

Understanding how to manipulate exponents and complex numbers is foundational for solving quadratic equations and working with polynomial functions.

• Simplifying Exponents: Apply the laws of exponents to simplify expressions. For example, $a^m \cdot a^n = a^{m+n}$ and $(a^m)^n = a^{mn}$.

• Adding, Subtracting, and Multiplying Radical Expressions: Combine like terms and use distributive property. For example, $\sqrt{a} + \sqrt{a} = 2\sqrt{a}$.

• Complex Numbers: A complex number is of the form $a + bi$, where $i = \sqrt{-1}$.

• Conjugate of a Complex Number: The conjugate of $a + bi$ is $a - bi$. Multiplying a complex number by its conjugate yields a real number: $(a + bi)(a - bi) = a^2 + b^2$.

• Division of Complex Numbers: To write $\frac{a + bi}{c + di}$ in standard form, multiply numerator and denominator by the conjugate of the denominator.

Quadratic Equations: Zeros and Solution Methods

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. The solutions are called zeros or roots.

• Factoring: Set the equation to zero and factor. Use the zero product property: if $ab = 0$, then $a = 0$ or $b = 0$.

• Square Roots: If $x^2 = k$, then $x = \pm \sqrt{k}$.

• Completing the Square: Transform $ax^2 + bx + c = 0$ into $(x + d)^2 = e$ and solve for $x$.

• Quadratic Formula: For $ax^2 + bx + c = 0$, the solutions are: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Equations in Quadratic Form: Some equations, such as $x^4 - 4x^2 + 3 = 0$, can be solved by substitution (let $u = x^2$).

• Applied Problems: Quadratic equations are used to model projectile motion, area problems, and optimization scenarios.

Graphing Quadratic Functions

The graph of a quadratic function $f(x) = ax^2 + bx + c$ is a parabola.

• Vertex: The vertex is the maximum or minimum point. Its coordinates are: $x = -\frac{b}{2a}$, $y = f(-\frac{b}{2a})$

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

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

• Graphing: Plot the vertex, axis of symmetry, and intercepts to sketch the parabola.

• Applied Problems: Use the vertex to solve maximum/minimum value problems, such as maximizing area or profit.

Quadratic Inequalities

Quadratic inequalities involve expressions like $ax^2 + bx + c > 0$ or $< 0$.

• Solving: Find the zeros of the quadratic, plot them on a number line, and test intervals to determine where the inequality holds.

• Example: Solve $x^2 - 4x + 3 > 0$. Factor: $(x - 1)(x - 3) > 0$. Test intervals: $x < 1$, $1 < x < 3$, $x > 3$.

Rational and Radical Equations

Equations involving rational (fractions) and radical (roots) expressions require special techniques.

• Rational Equations: Set the equation to a common denominator, solve for $x$, and check for extraneous solutions.

• Radical Equations: Isolate the radical, square both sides, and solve. Always check for extraneous solutions.

• Example: Solve $\frac{1}{x} + \frac{2}{x+1} = 3$.

Absolute Value Equations and Inequalities

Absolute value equations and inequalities involve expressions like $|x| = a$ or $|x| < a$.

• Equations: $|x| = a$ has solutions $x = a$ and $x = -a$.

• Inequalities: $|x| < a$ means $-a < x < a$; $|x| > a$ means $x < -a$ or $x > a$.

• Example: Solve $|2x - 3| = 5$.

Polynomial Functions and Rational Functions

Matching Polynomial Functions to Graphs

The graph of a polynomial function is determined by its degree and leading term.

• Leading Term: The term with the highest degree determines end behavior.

• End Behavior: For $f(x) = ax^n$, if $n$ is even and $a > 0$, both ends go up; if $a < 0$, both go down. If $n$ is odd, ends go in opposite directions.

Zeros and Multiplicities of Polynomial Functions

Zeros are the values of $x$ where $f(x) = 0$. Multiplicity refers to how many times a zero occurs.

• Finding Zeros: Set $f(x) = 0$ and solve.

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

• Graph Behavior: If multiplicity is odd, the graph crosses the axis; if even, it touches and turns.

Division of Polynomials

Polynomials can be divided using long division or synthetic division.

• Long Division: Divide as with numbers, subtracting multiples of the divisor.

• Synthetic Division: A shortcut for dividing by $x - r$ using coefficients.

• Example: Divide $x^3 - 6x^2 + 11x - 6$ by $x - 1$.

Rational Functions: Domain and Asymptotes

Rational functions are quotients of polynomials. Their domain and asymptotes are key features.

• Domain: All real numbers except where the denominator is zero.

• Vertical Asymptotes: Values of $x$ that make the denominator zero (and not canceled by the numerator).

• Horizontal Asymptotes: Determined by the degrees of numerator and denominator:

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

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

  • If degree numerator > degree denominator: No horizontal asymptote (may be slant asymptote)

• Graphing: Recognize the graph by identifying intercepts, asymptotes, and end behavior.

Feature	Quadratic Function	Polynomial Function	Rational Function
General Form	$ax^2 + bx + c$	$a_nx^n + \ldots + a_0$	$\frac{P(x)}{Q(x)}$
Graph Shape	Parabola	Varies by degree	Hyperbola-like, with asymptotes
Zeros	Up to 2	Up to degree $n$	Set numerator = 0
Asymptotes	None	None	Vertical, horizontal, or slant

Example: Finding Asymptotes of a Rational Function

Given $f(x) = \frac{2x^2 + 3x + 1}{x^2 - 4}$:

• Vertical Asymptotes: Set denominator to zero: $x^2 - 4 = 0 \Rightarrow x = 2, x = -2$.

• Horizontal Asymptote: Degrees equal, so $y = \frac{2}{1} = 2$.

Additional info: These topics are essential for understanding the algebraic and graphical properties of quadratic, polynomial, and rational functions, as well as solving related equations and inequalities.