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, and .

• Adding, Subtracting, and Multiplying Radical Expressions: Combine like terms and use distributive property. For example, .

• Complex Numbers: A complex number is of the form , where .

• Conjugate of a Complex Number: The conjugate of is . Multiplying a complex number by its conjugate yields a real number: .

• Division of Complex Numbers: To write 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 . The solutions are called zeros or roots.

• Factoring: Set the equation to zero and factor. Use the zero product property: if , then or .

• Square Roots: If , then .

• Completing the Square: Transform into and solve for .

• Quadratic Formula: For , the solutions are:

• Equations in Quadratic Form: Some equations, such as , can be solved by substitution (let ).

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

Graphing Quadratic Functions

The graph of a quadratic function is a parabola.

• Vertex: The vertex is the maximum or minimum point. Its coordinates are: ,

• Axis of Symmetry: The vertical line .

• Maximum/Minimum Value: If , the parabola opens upward (minimum); if , 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 or .

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

• Example: Solve . Factor: . Test intervals: , , .

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 , and check for extraneous solutions.

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

• Example: Solve .

Absolute Value Equations and Inequalities

Absolute value equations and inequalities involve expressions like or .

• Equations: has solutions and .

• Inequalities: means ; means or .

• Example: Solve .

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 , if is even and , both ends go up; if , both go down. If is odd, ends go in opposite directions.

Zeros and Multiplicities of Polynomial Functions

Zeros are the values of where . Multiplicity refers to how many times a zero occurs.

• Finding Zeros: Set and solve.

• Multiplicity: If is a factor, is a zero of multiplicity .

• 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 using coefficients.

• Example: Divide by .

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 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:

  • If degrees equal:

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

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

Example: Finding Asymptotes of a Rational Function

Given :

• Vertical Asymptotes: Set denominator to zero: .

• Horizontal Asymptote: Degrees equal, so .

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.