Graph key features of functions, linear equations and linear inequalities - Essay Example

Difference Between a Linear Equation and a Function. A linear equation, y = mx + b, refers to an equation which exhibits a linear relationship between the domain or set of x-values and the corresponding range or set of y-values. Linear equations are normally first degree with respect to both ‘x’ and ‘y’ except in case where x = constant in which the degree of ‘y’ is zero and ‘x’ remains the same at any value of ‘y’ and in case where y = constant in which the slope is zero as well as the degree of ‘x’.

All linear equations are function.Examples of a Linear Equation:(a) y = 2x + 5 (b) y = -3x + 4 On the other hand, a function pertains to a kind of relation with either one-to-one or many-to-one correspondence between the values of ‘x’ in the domain and the matching values of ‘y’ in the range. Given a set of ordered pairs that define a function, each element ‘x’ in the domain is distinct and does not repeat in value when paired with an element ‘y’ in the range.

Through a vertical line test, one may determine whether or not a relation is a function in a graph such that on running down a vertical line, the curve is hit only at a single point everywhere in the curve. In this manner, it may be claimed that a ‘linear equation’ is a function, but not all functions are linear in nature. Examples of a Function:(a) A Set of Ordered Pairs: {(-7, 4) , (-3, 0) , (1, 5) , (2, -2) , (6, 11)}(b) y = 4x – 7 (c) y = x2 - 7x + 6 (d) y = 2x5 - 7x3 + 4x + 15 (e) y = ex + 2 Based on the aforementioned properties and definitions along with the examples shown, linear equation and function share the attribute of having one-to-one correspondence so that the independent variable ‘x’ can assume any value wherein no two or more values of ‘y’ correspond to a common value of ‘x’.

The one-to-one relationship is strict in meaning for linear equations whereas functions take into account correspondence that is many-to-one in type considering equations that represent relations in quadratic and cubic forms. Besides linear equation, a function may also be modelled by nonlinear forms such as rational, polynomial, logarithmic, or exponential. Thus, all linear equations are functions but not all functions are linear equations. What does the equation of a vertical line look like?

Can you give an example of an equation of a vertical line and describe how the graph would look? An equation of a vertical line is given by a constant relation x = c where ‘c’ is a constant value which means that ‘x’ domain stays at a single steady value at any value of ‘y’. An example of a vertical line equation would be x = 7 which is a straight line parallel with the y-axis and whose slope is ‘infinity’. A sketch of its graph would look like: