Quadratic Functions and Characteristics

Quadratic Functions and Characteristics

• Quadratic function: A mathematical function characterized by the variable raised to the greatest power of 2.
• Vertex: The turning point of a parabola, representing either its maximum or minimum value.
• Axis of Symmetry: A vertical line that divides a parabola into two mirror-image halves.

Maximum and Minimum Values

• Maximum: The highest point on a downward-opening parabola, occurring when the coefficient of x squared is negative.
• Minimum: The lowest point on an upward-opening parabola, occurring when the coefficient of x squared is positive.

Domain and Range

• Domain: All possible values for the variable x in a function.
• Range: All possible values for the variable y generated by a function.

Solutions and Forms of Quadratic Functions

• Solution: The x-coordinate(s) where the graph intersects the x-axis.
• Standard form of quadratic function: Typically represented as y = ax^2 + bx + c, where a, b, and c are constants.

Transformations of Quadratic Functions

• y = x^2 + k: Translates the graph vertically by k units.
• y = (x+h)^2: Translates the graph horizontally by h units.
• y = ax^2 when a is a fraction: Makes the graph wider due to the smaller value of a.
• y = ax^2 when a is a whole number: Makes the graph narrower as a increases.
• y = ax^2 when a is negative: Inverts the graph, flipping it upside down.

Special Techniques and Related Functions

• Completing the square: A method used to transform a quadratic expression into a perfect square trinomial of the form (x+b)^2 = x^2 + 2bx + b^2.
• Quadratic Formula: A tool used for finding roots of quadratic equations; however, no specific definition provided.

Differences in Functions

• First Differences: If the differences of consecutive outputs in a table are consistent, the relationship is linear.
• Second Differences: If the differences of consecutive first differences are consistent, the relationship is quadratic.

Additional Function Types

• Step function: Exhibits discrete jumps and is represented graphically by a series of connected line segments.
• Absolute Value Function: Expressed as y = |x|, produces a V-shaped graph.
• Exponential function: Defined as y = ab^x, where a is a constant and b is the base.
• Linear function: A function that generates a straight line graph.