Summary of First Degree Function: Introduction

First Degree Function: Introduction | Traditional Summary

Contextualization

Functions are essential mathematical tools used to describe relationships between variables. A first-degree function, in particular, is a linear relationship represented by the formula y = ax + b, where 'a' and 'b' are real coefficients and 'x' is the independent variable. This formula describes a straight line in the Cartesian plane and is widely used to model situations where there is a direct and constant relationship between two variables.

In everyday life, we can find examples of first-degree functions in various practical situations. For instance, when calculating the total cost of a purchase at a supermarket, where the total price is a function of the number of items purchased. Other applications include forecasting expenses based on the distance traveled or determining profit as a function of the number of units sold. Understanding how these functions operate and how to identify them is essential for solving practical problems and for studying more advanced subjects, such as engineering, economics, and programming.

Definition of First-Degree Function

A first-degree function is a polynomial function of degree 1, represented by the formula y = ax + b, where 'a' and 'b' are real coefficients and 'x' is the independent variable. This means that, for each value of 'x', there is a corresponding value of 'y' obtained by applying the formula. The coefficient 'a' is known as the slope coefficient, which determines the slope of the line on the graph. The coefficient 'b' is called the y-intercept, which indicates the point where the line intersects the y-axis.

The importance of this definition lies in the fact that it allows modeling situations where there is a linear relationship between two variables. This is extremely useful in various fields such as economics, engineering, and social sciences. For example, in economics, the function can be used to represent the relationship between production cost and quantity produced.

Furthermore, the first-degree function is the basis for the study of more complex functions, such as quadratic and exponential functions. Understanding this basic function is essential to advance in more complex mathematical studies and to apply these concepts to real-world problems.

• A first-degree function is represented by y = ax + b.

• 'a' is the slope coefficient and 'b' is the y-intercept.

• Models linear relationships between variables.

Graph of the First-Degree Function

The graph of a first-degree function is always a straight line. The slope of this line is determined by the slope coefficient 'a', while the point where the line intercepts the y-axis is given by the y-intercept 'b'. This means that when altering the values of 'a' and 'b', the slope and position of the line on the graph change.

To plot the graph, it is sufficient to find two points that belong to the line and connect them. For example, for the function y = 2x + 3, we can calculate the values of y for x = 0 and x = 1, obtaining the points (0, 3) and (1, 5). Connecting these points gives us the straight line representing the function.

Visualizing the function on a graph helps to better understand the relationship between variables. For example, a positive slope (a > 0) indicates that y increases as x increases, while a negative slope (a < 0) indicates that y decreases as x increases.

• The graph of a first-degree function is a straight line.

• The slope of the line is determined by 'a' and the intersection with the y-axis is determined by 'b'.

• Two points are sufficient to plot the graph of the function.

Determining Coefficients 'a' and 'b'

To determine the coefficients 'a' and 'b' of a first-degree function from two given points (x1, y1) and (x2, y2), we use specific formulas. The slope coefficient 'a' is calculated using the formula a = (y2 - y1) / (x2 - x1). This coefficient represents the rate of change of y with respect to x.

Once the value of 'a' is known, we can determine the y-intercept 'b' by substituting one of the points into the function equation and solving for 'b'. For example, if the line passes through the points (1, 2) and (3, 6), we first calculate 'a': a = (6 - 2) / (3 - 1) = 4 / 2 = 2. Next, we use one of the points to find 'b': 2 = 2(1) + b, resulting in b = 0.

This process of determining the coefficients is fundamental for constructing the equation of the line from empirical data. It allows modeling real situations and predicting outcomes based on the linear relationship between variables.

• Coefficient 'a' is calculated using the formula a = (y2 - y1) / (x2 - x1).

• Coefficient 'b' is determined by substituting a point into the function equation.

• These coefficients allow constructing the equation of the line from empirical data.

Practical Examples

Applying the first-degree function in everyday situations helps to understand its utility and relevance. For example, consider calculating the total cost of a purchase, where the total price is a function of the number of items purchased. If each item costs R$5 and there is a fixed delivery fee of R$10, the function representing the total cost is y = 5x + 10.

Another practical example is forecasting travel expenses based on the distance traveled. If the fuel cost is R$3 per kilometer and there is a fixed toll fee of R$20, the function representing the total cost of the trip is y = 3x + 20. By altering the value of x (distance), we can predict the total cost of the trip.

These examples show how first-degree functions are used to model and solve real problems. Understanding and applying these functions can facilitate decision-making and forecasting results in various everyday and professional situations.

• First-degree functions are used to calculate total costs.

• They can forecast expenses based on an independent variable.

• Practical applications include purchases, trips, and financial forecasting.

To Remember

• First-Degree Function: A polynomial function of degree 1 in the form y = ax + b.

• Slope Coefficient (a): Determines the slope of the line on the graph.

• Y-Intercept (b): Indicates the point where the line intersects the y-axis.

• Graph: Visual representation of the first-degree function, always a straight line.

• Rate of Change: Represented by the slope coefficient 'a', it is the change in y with respect to x.

Conclusion

In this lesson, we explored the first-degree function, an essential mathematical tool for describing linear relationships between variables. We discussed its representation in the form y = ax + b, where 'a' is the slope coefficient and 'b' is the y-intercept, and how these functions are graphically represented as straight lines. Understanding these concepts is fundamental for modeling and solving problems in various fields of knowledge, such as economics, engineering, and social sciences.

We also addressed the importance of determining the coefficients 'a' and 'b' from given points using specific formulas, and how this information can be applied in everyday situations, such as calculating costs and forecasting expenses. The graphical visualization of functions and practice with real examples help solidify understanding and application of these concepts.

Finally, we discussed the relevance of first-degree functions in various professions and fields of study, highlighting how this knowledge can be useful both in everyday life and in future careers. We encourage students to explore more on the subject to enhance their skills and prepare for more complex challenges.

Study Tips

• Practice solving different types of problems involving first-degree functions to strengthen your understanding and application skills.

• Review and study the practical examples presented in class, trying to create your own examples based on everyday situations.

• Utilize additional resources, such as educational videos, textbooks, and online exercises, to deepen your knowledge and clarify doubts.