Summary of First Degree Function: Graph and Table

First Degree Function: Graph and Table | Traditional Summary

Contextualization

Linear functions are essential mathematical tools that describe linear relationships between variables. In the form f(x) = ax + b, where 'a' and 'b' are constants and 'a' ≠ 0, these functions are graphically represented by lines on the Cartesian plane. They have wide applicability in various fields such as economics, engineering, and social sciences, allowing for practical and efficient modeling and prediction of linear behaviors.

In everyday life, linear functions are used to calculate the average speed of a vehicle, predict monthly expenses based on a fixed budget, and even estimate population growth. Understanding how these functions operate and how to graphically represent them is fundamental for solving real problems and critically interpreting data.

Definition of Linear Function

A linear function is a polynomial function of degree 1, which can be expressed in the form f(x) = ax + b, where 'a' and 'b' are constants and 'a' is not zero. This function is called 'linear' because the highest exponent of x is 1. The constant 'a' is known as the slope, and the constant 'b' is known as the y-intercept.

The slope 'a' determines the inclination of the line in the graph. If 'a' is positive, the line rises from left to right; if 'a' is negative, the line falls from left to right. This means that the slope indicates the rate of change of the function.

The y-intercept 'b' indicates the point where the line crosses the y-axis, that is, the value of f(x) when x is zero. This point is known as the y-intercept. Together, the coefficients 'a' and 'b' determine the position and inclination of the line on the Cartesian plane.

• A linear function is expressed as f(x) = ax + b.

• The slope 'a' determines the inclination of the line.

• The y-intercept 'b' determines the intersection point with the y-axis.

Graph of a Linear Function

The graph of a linear function is always a straight line on the Cartesian plane. To draw this line, it is necessary to identify at least two points that belong to the function. These points can be found by substituting values of x into the equation f(x) = ax + b and calculating the corresponding values of f(x).

A practical method to draw the graph is to identify the points of intersection of the line with the x and y axes. The point of intersection with the y-axis occurs when x is zero, that is, f(0) = b. The point of intersection with the x-axis occurs when f(x) is zero, that is, when ax + b = 0, resulting in x = -b/a.

Once these points are identified, simply draw a line that passes through them. The slope of the line, determined by the slope 'a', will indicate whether the line rises or falls. The line is continuous and extends infinitely in both directions.

• The graph of a linear function is a straight line.

• Identifying intersection points with the x and y axes makes drawing the graph easier.

• The slope of the line is determined by the slope 'a'.

Table of Values

A table of values is a useful tool for visualizing and analyzing the relationship between x and f(x) in a linear function. To build a table of values, we choose a series of values for x and calculate the corresponding values of f(x) using the function's equation.

For example, for the function f(x) = 2x + 3, we can choose values for x such as -2, -1, 0, 1, and 2. By substituting these values in the equation, we obtain the corresponding values of f(x), creating a table that clearly shows the linear relationship between x and f(x).

The table of values can be used to plot the graph of the function, as each pair (x, f(x)) represents a point on the Cartesian plane. Additionally, the table allows for identifying patterns and predicting values of f(x) for other values of x.

• A table of values helps visualize the relationship between x and f(x).

• We choose values for x and calculate the corresponding f(x).

• The table can be used to plot the graph of the function.

Data Interpretation in Tables

Data interpretation in tables involves analyzing the relationship between the variables x and f(x) represented in table form. This allows for identifying the behavior of the function and predicting future values. For a linear function, the relationship between x and f(x) is linear, meaning that the difference between successive values of f(x) is constant.

For example, if we have the table:

We can observe that the difference between consecutive values of f(x) is always 2, indicating a linear relationship. The corresponding function can be determined by observing the initial values and the constant difference, resulting in f(x) = 2x + 1.

Interpreting tables allows not only for plotting graphs but also for understanding how the variables relate and predicting values of f(x) for new values of x. This is especially useful in practical situations where data is presented in tabular form.

• Interpreting tables helps understand the linear relationship between x and f(x).

• The constant difference between values of f(x) indicates a linear function.

• It allows predicting future values and plotting graphs.

To Remember

• Linear Function: A polynomial function of degree 1 in the form f(x) = ax + b.

• Slope: The constant 'a' that determines the inclination of the line in the graph.

• Y-Intercept: The constant 'b' that determines the intersection point of the line with the y-axis.

• Cartesian Plane: A two-dimensional coordinate system used to graph functions.

• Intersection Point: The point where the line crosses the x or y axes.

• Table of Values: A table that shows the relationship between values of x and the corresponding values of f(x).

• Linear Relationship: A relationship where the difference between successive values is constant.

Conclusion

The linear function is a fundamental mathematical tool that describes linear relationships between variables. During the lesson, we understood its definition and characteristics, such as the slope 'a' and the y-intercept 'b', which determine the inclination and the intersection of the line on the graph, respectively. We also learned to represent these functions graphically on the Cartesian plane and interpret tables of values that show the relationship between x and f(x).

The ability to represent and interpret linear functions is essential for solving practical problems in various fields, such as economics, engineering, and social sciences. These functions allow for modeling everyday situations, predicting behaviors, and making data-driven decisions. Understanding how to build and analyze graphs and tables gives us a powerful tool for understanding and predicting linear relationships.

We encourage students to further explore the topic, using the knowledge acquired to solve real problems and deepen their understanding of the applicability of linear functions. Continuous practice and application in different contexts will strengthen their ability to interpret and use these functions effectively.

Study Tips

• Practice building tables of values and graphs for different linear functions, varying the coefficients 'a' and 'b' to observe how they affect the slope and position of the line.

• Use software or dynamic geometry apps to visualize and manipulate graphs of linear functions, facilitating an understanding of their properties.

• Solve practical problems involving linear relationships, such as budget calculations and data analysis, to apply the knowledge gained in real situations, reinforcing learning.