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Introduction

This tutorial is designed to help you understand the fundamentals of first-degree functions, also known as linear functions. Using the Curió method, we will break down key concepts and solve exercises that frequently appear in exams. By the end of this guide, you will have a solid grasp of how to work with first-degree functions.

Step 1: Understanding the Definition of First-Degree Functions

First-degree functions are mathematical expressions of the form: [ f(x) = ax + b ] where:

• ( a ) is the slope of the line (how steep it is).
• ( b ) is the y-intercept (the point where the line crosses the y-axis).

Practical Tips

• The slope ( a ) determines the direction of the line:
  • If ( a > 0 ), the line rises from left to right.
  • If ( a < 0 ), the line falls from left to right.
• The y-intercept ( b ) indicates where the line intersects the y-axis.

Step 2: Graphing First-Degree Functions

To graph a first-degree function, follow these steps:

1. Identify the slope ( a ) and the y-intercept ( b ) from the function.
2. Plot the y-intercept (0, b) on the graph.
3. Use the slope to determine another point:
   • For example, if the slope ( a = 2 ), from the y-intercept, move up 2 units and to the right 1 unit to find the next point.
4. Draw a straight line through the points.

Common Pitfalls

• Ensure you understand how to interpret the slope as a ratio (rise/run).
• Double-check your plotted points for accuracy.

Step 3: Solving First-Degree Equations

To solve equations of the form ( ax + b = c ), follow these steps:

1. Isolate the variable ( x ):
   • Subtract ( b ) from both sides: ( ax = c - b ).
2. Divide by ( a ):
   • ( x = \frac{c - b}{a} ).

Example

For the equation ( 3x + 4 = 10 ):

1. Subtract 4: ( 3x = 6 ).
2. Divide by 3: ( x = 2 ).

Step 4: Applications in Real-World Problems

First-degree functions can represent real-world scenarios, such as:

• Calculating costs based on a fixed fee and a variable rate.
• Predicting profits based on the number of items sold.

Example Scenario

If a company has a fixed cost of $50 and makes $10 for each product sold, the function can be expressed as: [ f(x) = 10x + 50 ] where ( x ) is the number of products sold.

Conclusion

In this tutorial, you learned about the basic concepts of first-degree functions, how to graph them, solve related equations, and apply them to real-world situations. Practice these techniques with various exercises to strengthen your understanding. For further study, consider exploring more advanced topics in linear functions or taking a course that utilizes the Curió method for deeper learning.