Relations And Functions Algebra Flash Cards Pt.3

Relations refer to the way in which entities or concepts are connected or related to each other. It is about depicting the connections between different elements or entities.

A function is defined as a specific type of relationship between two quantities where each input maps to a unique output. This definition distinguishes a function from other types of mathematical relationships.

The range of a function refers to the set of possible output values that the function can produce, which is distinct from the set of input values (domain) or the gradient (slope) of the function.

Function notation allows us to represent functions using symbols such as f(x), where f is the name of the function and x is the input. It is a standardized way to define and refer to functions in mathematics.

Parent functions are basic functions from which other functions are derived. Linear and quadratic functions are common parent functions that serve as building blocks for more complex functions.

Transformations of parent functions refer to shifting, stretching, compressing or reflecting the graph to create variations of the original function.

Transformations of parent functions involve changing the graph through translations, reflections, stretches, and compressions without altering the general shape of the original parent function.

Dilations in transformations refer to scaling a parent function either vertically or horizontally to create new graphs.

Transformational graphing typically involves linear functions for simplicity and ease of visualization. Linear functions undergo transformations like vertical translations, which allows for shifts in the graph without changing its overall shape.

In transformational graphing of linear functions, the correct form for vertical dilation involves the function g(x) = mx, where m is greater than 0. This means that the function is stretched or compressed vertically based on the value of m.

When graphing linear functions with a negative slope, the transformation involves a vertical dilation (stretch or compression) with a reflection of the function f(x) = x. This transformation results in the linear function G(x) = mx where m is less than 0, indicating a negative slope. The incorrect answers provided suggest transformations related to other types of functions such as quadratic, exponential, and absolute value, each of which involves different operations on the graph.

Quadratic functions can be transformed through various operations such as vertical/horizontal translations, vertical stretches/shrinks, and reflections. The correct answer represents a vertical translation of the original function f(x) = x^2 by adding a constant term c.

The key factor in transformational graphing of quadratic functions involves vertical dilation (stretch or compression) of the function by a factor of 'a' when represented in the form H(x) = ax^2, where 'a' is greater than 0. Other transformations like horizontal translation or reflections over the x-axis involve different aspects of the function's graph.

The correct answer involves both a vertical dilation (stretch or compression) and a reflection of the original function f(x) = x^2 across the x-axis. This results in a change in the shape and orientation of the graph, which is different from horizontal or vertical translations alone.

In this question, the correct answer refers to a horizontal translation of the quadratic function f(x)= x^2 by a distance of c units either to the right or left, depending on the sign of c. The incorrect answers show variations of vertical and horizontal reflections, which do not align with the type of transformation described in the question.

In direct variation, the constant of variation 'k' must not be equal to 0 for the relationship to hold true. Additionally, the graph of all points describing a direct variation is a line passing through the origin.

Inverse variation is a relationship where the product of X and Y remains constant, represented by the equation Y = k/X. The graph of points in an inverse variation relationship forms two curves that are reflections of each other across the line Y=X.