A linear function appertains to those mathematical conceptions that are studied by students during the first lessons of algebra in school. It is one of the fundamental mathematical concepts, the understanding of which is necessary for the study of more complex mathematical terms and concepts. Undoubtedly, a student who does not understand what a linear function is or how to solve elementary linear equations just cannot count on successful mastery over the conceptual material of elementary algebra, not to mention the more sophisticated mathematical disciplines, such as trigonometry or group theory. Therefore, it is recommended to refresh one's memory about the main characteristics of a linear function and the specific methods, which are used to represent it graphically in a Cartesian coordinate system.

In calculus, a linear function is a polynomial function in which the variable (x) has a degree at most one. Therefore, a linear function is a function of the form: f(x) = kx + b, where x is the variable. A graph of a linear function is a set of all points with coordinates of a form (x, f(x)). According to its statement, a geometric representation of a linear function is a straight line on the Cartesian plane (if over real numbers). In fact, that is why this specific type of linear functions is called linear. In other words, a linear function is one of the simplest forms of linear functions because it can be completely described just by one straight line, which is called a linear graph. A linear graph is a line that demonstrates a linear mathematical function or equation in a Cartesian coordinate system.

A linear function has the same fundamental properties as the whole group of linear functions. The basic property of linear functions: increment of the function is proportional to the increment of the argument. That is, the function is a generalization of direct proportionality. A linear function is a function of the form: y = kx + b (for functions of one variable). K (slope of the line) is the tangent of the angle α (a ∈ [0; 휋/2) U (휋/2; 휋), which forms a straight line with the positive direction of the x-axis. If k > 0, a straight line forms an acute angle with the positive direction of the x-axis. If k < 0, a straight line forms an obtuse angle with the positive direction of the x-axis. If k = 0, a line is parallel to the x-axis. A linear function of n variables x = (x1, x2,…xn) is a function of the form: f(x) = a0 + a1x1 + a2x2 + …+ anxn, where a0,a1, a2 – some fixed numbers. The domain of definition of the linear function is all n-dimensional space of the variables x1, x2, …, xn, real or complex. If a = 0, a linear function is called homogeneous or linear form. If all the variables x1, x2, …, xn and the coefficients a0,a1, a2 are real numbers, then the graph of a linear function in the (n + 1) dimensional space of the variables x1, x2, …, xn, y is an n-dimensional hyperplane: y = a0 + a1x1 + a2x2 + …+ anxn. In particular, when n = 1, then a linear function is represented in a Cartesian coordinate system as a straight line in the plane. Therefore, it is obvious that any linear equation with two variables can be represented in a graphical form as a linear graph.

The fundamental properties of a linear function are quite comprehensive, thereby one can easily understand them just by examining the graphical representation of a linear function in a Cartesian coordinate system. In fact, the domain of definition of a linear function consists of all numbers: D: x∈ (-∞; ∞). In accordance with this statement, we can postulate that the range of values of a linear function includes all numbers: E: y∈ (-∞; ∞). In addition, a graph of a linear function demonstrates that a linear graph of the function of the form: y = kx + b may cross the axis of the coordinate system at different angles. Thereby, it is quite obvious that a linear function increases if k> 0 and decreases if k <0.

The graph of a linear function y = kx + b is a straight line parallel to the graph of the function of the form: y = kx, which cuts the intercept b on the y-axis. Let us prove this statement by performing a virtual experiment with the graphs in a Cartesian coordinate system. The line OM is a graph of the function y = kx and b> 0. Let us add to the ordinate (LM) of the point, which belongs to the line OM, the intercept MN, having a length b. Then OL = x, LM = kx and LN = kx + b. Therefore, the point N with abscissa (x) and ordinate (kx + b) belongs to the graph of the function: y = kx + b. The straight line NN', parallel to the line OM, can be drawn through a point N. Thereby, the straight line NN' is a graph of the function: y = kx + b. In fact, M'N'= MN and M'N = b, therefore, the ordinate of any point (for example, N'), which belongs to the line NN', is equal to the corresponding ordinate (L'M') of the point, which belongs to the line OM plus the intercept b. Consequently, the coordinates of all points on the line NN' satisfy the equation y = kx + b. Obviously, the coordinates of any point, which is not lying on the line NN', do not satisfy this formula because the ordinate of this point is obtained from the L'M' by adding a segment greater or less than the intercept b.

According to the properties of a linear function, we can state that if b <0, then the graph of a linear function intersects the negative half ordinates. Eventually, if b is equal to zero (b = 0), then the function: y = kx + b must be referred as the function: y = kx. In fact, the linear function: y = kx is a special case of a linear function if b = 0. If b ≠ 0, and k = 0, then the linear function takes the form: y = b (y = 0*x + b, that is, for any value of the variable (x) the function is equal to b). The linear graph of this function is a straight line parallel to the x-axis that intersects the y-axis at the ordinate b. If k = 0 and b = 0, y = 0*x + 0 = 0. In other words, for all values of the variable (x) the function is equal to zero. The graph of the function: y = 0 is a straight line, which coincides with the x-axis. In order to graph a linear function, such as the function: y = 9x - 3, we have to find to different points of the linear graph: the point with the abscissa x = 0 and the point with the ordinate y = 0. The point with the abscissa x = 0 can be obtained by solving the linear equation: y = 9*0 – 3 = -3. The point with the ordinate y = 0 can be obtained by solving the linear equation: 0 = 9x - 3; x = 1/3. Therefore, we can draw a straight line through the points with coordinates (0, -3) and (1/3, 0). This line is a linear graph of the function: y = 9x – 3.

In fact, the coordinates of the points on a line can be considered as the solutions of linear equations in two variables that define linear functions. In truth, this connection between linear functions and linear equations provides the most common way to produce linear functions. Let us consider this point in more detail. For example, the equation of the form: y = kx + b is a slope-intercept form of a standard linear equation, where y is the value of the function, a, b are the coefficients and x is the variable. As we can see, this linear equation expresses the same relationship between x and y as other linear functions: the values of y depend on the values of the variable (x). Therefore, the linear function, which describes this linear equation, can be referred as f(x) = kx + b.

In fact, one can easily graph a linear function considering the fact, that its graph is a straight line. Thereby, it is a quite simple assignment to create a graph of an elementary linear function, for example, the linear function of the form: y = 2x + 5. In order to demonstrate more complicated assignments as well as practical methods of their accomplishing, let us examine a more complicated function of the form: y = kx + b, considering the fact it passes through the point A (-3, 2) and parallel to the line y = - 4x. Here is a concise practical instruction, which describes the crucial points necessary to complete this objective:

- Firstly, we have to find the value of ‘k’. The fact that the graph of the linear function of the form: y = kx + b is parallel to the line y = - 4x, leads us to the conclusion that k = -4. Therefore, the equation of this function is of the form y = -4x + b.
- Secondly, we have to find the value of ‘b’. It is known that the graph of the function y = -4x + b passes through the point A (-3, 2). If the point belongs to the graph function, we can obtain a true equality: -4 = 2 * (- 3) + b by substituting its coordinates into the equation of a function; hence, b = -10.
- Thirdly, we need to build a graph of the function y = - 4x – 10. According to the conditions pointed out in our assignment, the point A (-3, 2) is already known to us. Thus, we put the points A (-3, 2) and B (0, -10) in the coordinate plane and join them by a straight line.
- Finally, we have to prove that the obtained graph satisfies all the conditions of our initial assignment. We can perform this operation by recalculating the coordinates of the point B. Nevertheless, in our case the coordinates of the point B are correct according to the initial conditions, Thereby, our graph is a graph of the linear function of the form: y = kx + b, which is parallel to the line that satisfies the linear function of the form: y = -4x.

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