Quadratic Functions and Equations

Definition and Properties

Quadratic functions are fundamental in algebra and are defined by the general form , where and , , are real numbers. Quadratic equations are written as and are in standard form when arranged as such.

• Domain: All real numbers ().

• Range: Depends on the direction the parabola opens.

• Extrema: The minimum or maximum value occurs at the vertex.

• Zeros: The solutions to ; these are the x-intercepts of the graph if real.

Graphing Quadratic Functions

The graph of a quadratic function is a parabola. The vertex represents the turning point, and the axis of symmetry passes through the vertex.

• Vertex:

• Axis of Symmetry:

• Opens Up: If

• Opens Down: If

Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

• Factoring: Express as a product of two binomials.

• Completing the Square: Transform the equation into and solve for .

• Quadratic Formula:

Example: Solving by Factoring

Factor :

• Find two numbers that multiply to and add to $2 and ).

• Factored form:

Example: Solving by Completing the Square

Solve :

• Isolate variable terms:

• Add to both sides:

• Rewrite:

• Apply square roots:

Example: Solving by Quadratic Formula

Solve :

• Standard form:

• Apply formula:

• Approximate solutions: and

The Discriminant

The discriminant determines the nature of the solutions:

• : Two distinct real solutions

• : One real solution (repeated root)

• : Two imaginary solutions (complex conjugates)

Complex Numbers

A complex number is any number of the form , where and are real numbers and .

• Add/Subtract: Combine real and imaginary parts.

• Multiply: Use .

• Conjugate: The conjugate of is .

Factoring Quadratic Expressions

Multiplying Binomials (Area Model/Box Method)

The area model helps visualize the multiplication of binomials.

• Example:

• Box entries: , , ,

• Combine:

Factoring Trinomials

To factor , find two numbers and such that and .

• Example: factors to

Graphing Quadratic Functions in Vertex Form

Transformations

The vertex form allows for easy identification of transformations:

• a: Vertical stretch/shrink and reflection

• h: Horizontal translation

• k: Vertical translation

Quadratic Applications

Real-World Problems

• Projectile Motion: models height over time.

• Optimization: Quadratic functions are used to maximize or minimize area, cost, etc.

Rational Equations

Solving Rational Equations

Rational equations contain fractions with variables in the denominator. To solve:

• Multiply both sides by the least common denominator (LCD) to clear fractions.

• Check for extraneous solutions by substituting back into the original equation.

Example:

Solve :

• LCD is 6.

• Multiply both sides:

• Solve:

Radical Equations

Solving Radical Equations

Radical equations have variables under a root. To solve:

• Isolate the radical.

• Apply the principle of powers (raise both sides to the appropriate power).

• Check for extraneous solutions.

Example:

Solve :

• Square both sides:

Absolute Value Equations and Inequalities

Solving Absolute Value Equations

The absolute value is equivalent to or for .

• For , is equivalent to .

• For , has no solution.

Example:

Solve :

• Solutions: or

Solving Absolute Value Inequalities

• is equivalent to

• is equivalent to or

Summary Table: Nature of Quadratic Solutions

Additional info:

• TI-83/84 calculators can be used to graph functions, find zeros, and check solutions.

• Quadratic models are widely used in physics, engineering, and economics for optimization and modeling.