FUNÇÃO DO 1 GRAU | FUNÇÃO AFIM | \Prof. Gis/- AULA 1

Key Concepts

Function of the 1st degree (also known as affine function), general form of the function (f(x) = ax + b), coefficient 'a' (angular coefficient), coefficient 'b' (linear coefficient), graph of the function (a straight line), increasing function (a > 0), decreasing function (a < 0), constant function (a = 0), zero or root of the function (f(x) = 0), finding the root of the function.

Function of the 1st Degree: Definition and General Form

The video focuses on explaining the function of the 1st degree, also known as the affine function. The general form of this function is defined as f(x) = ax + b, where 'a' and 'b' are real numbers. 'a' is called the angular coefficient, and 'b' is called the linear coefficient. The variable 'x' represents the independent variable, and f(x) represents the dependent variable (often denoted as 'y').

Understanding the Coefficients 'a' and 'b'

The coefficient 'a' (angular coefficient) determines the slope and direction of the line. If 'a' is positive (a > 0), the function is increasing, meaning that as 'x' increases, f(x) also increases. If 'a' is negative (a < 0), the function is decreasing, meaning that as 'x' increases, f(x) decreases. If 'a' is zero (a = 0), the function becomes a constant function, f(x) = b, which represents a horizontal line. The coefficient 'b' (linear coefficient) represents the y-intercept, which is the point where the line intersects the y-axis (when x = 0).

Graph of the Function

The graph of a function of the 1st degree is always a straight line. To draw the graph, you need at least two points. These points can be found by assigning arbitrary values to 'x' and calculating the corresponding values of f(x). For example, you can choose x = 0 to find the y-intercept (0, b) and another value for 'x' to find a second point.

Zero or Root of the Function

The zero or root of the function is the value of 'x' for which f(x) = 0. To find the root, you set the function equal to zero (ax + b = 0) and solve for 'x'. This gives you x = -b/a. The root represents the x-intercept, which is the point where the line intersects the x-axis.

Example

To find the root of the function f(x) = 2x + 4, you set 2x + 4 = 0. Solving for 'x', you get 2x = -4, and therefore x = -2. This means that the root of the function is -2, and the graph of the function intersects the x-axis at the point (-2, 0).

Synthesis/Conclusion

The function of the 1st degree is a fundamental concept in algebra. Understanding the general form (f(x) = ax + b), the significance of the coefficients 'a' and 'b', how to graph the function, and how to find the zero or root of the function are essential skills. The video provides a clear and concise explanation of these concepts, making it a valuable resource for students learning about this topic.