Praxis 5162 Algebra Flashcards

Functions and Sequences

• Recursive Function: A function that calculates successive values step by step starting from an initial condition.
• Explicit Function: A function that defines the dependent variable directly in terms of the independent variable.
• Composite Function: A function formed by applying one function to the results of another, where the output from the first serves as the input for the second.
• Piecewise Function: A function consisting of multiple sub-functions, each applicable to a specific interval of the domain.
• Inverse Function: Achieved by switching the dependent and independent variables and solving for the new dependent variable.

Quadratic Functions

• Vertex Form: Represents a quadratic function as ( y = a(x-h)^2 + k ), where (h, k) is the vertex.
• Vertex: Key point of a quadratic function represented as (h, k).
• Axis of Symmetry: A vertical line given by ( x = -\frac{b}{2a} ) that bisects the parabola.

Sequence Types

• Arithmetic Sequence: Each term is generated by adding a constant to the previous term.
• Geometric Sequence: Each term is created by multiplying the previous term by a constant ratio.
• Geometric Sequence Formula: Defined as ( A(n) = a_1 r^{n-1} ).

Formulas and Calculations

• Distance Formula: Used to calculate the distance between two points: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).
• Midpoint Formula: Provides the midpoint of a line segment as ( \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) ).
• Compound Interest Formula: Given by ( A = P(1+\frac{r}{n})^{nt} ) for calculating compound interest over time.

Discriminants

• Discriminant: Calculated as ( b^2 - 4ac ) to determine the nature of the roots of a quadratic equation.
• Positive Discriminant: Indicates two real solutions.
• Negative Discriminant: Suggests no real solutions exist.
• Zero Discriminant: Indicates exactly one real solution.

Function Types

• Even Function: Exhibits symmetry about the y-axis, characterized by ( f(x) = f(-x) ).
• Odd Function: Exhibits symmetry about the origin, shown by ( f(-x) = -f(x) ).

Statistical Concepts

• Standard Deviation: Measures variability, quantifying average distances of scores from the mean.
• Variance: Defined as the square of the standard deviation.
• Mean Absolute Deviation: The average of the distances between each data point and the mean.

Graphical Analysis

• Domain: Represents the set of all possible x-values of a function.
• Range: Represents the set of all possible y-values of a function.
• Slope-Intercept Form: Written as ( y = mx + b ), where m is the slope and b is the y-intercept.
• Standard Form: Represented as ( Ax + By = C ).

Miscellaneous Concepts

• Exponential Growth: Describes populations reproducing at a constant rate.
• Absolute Value Function: Defined as ( f(x) = |x| ), reflecting distances from zero.
• Imaginary Numbers: Result from the square root of negative numbers.
• Normal Distribution Curve: A bell-shaped curve illustrating the distribution of continuous data.
• Matrices: Organized arrays of numbers used in various calculations, such as linear transformations.
• Scalar Multiplication: Involves multiplying each element in a matrix by a scalar value.