Fonction linéaire 1ere année

Introduction

This tutorial provides a comprehensive overview of linear functions, specifically tailored for first-year students. Understanding linear functions is crucial in mathematics as they form the foundation for more complex concepts. This guide will walk you through the essential elements of linear functions, including their definition, properties, and how to work with them.

Step 1: Understand the Definition of a Linear Function

A linear function is a mathematical function that creates a straight line when graphed. It can be expressed in the general form:

• Equation: y = mx + b

Where:

• y is the output value.
• x is the input value.
• m is the slope of the line (rate of change).
• b is the y-intercept (the point where the line crosses the y-axis).

Practical Tips

• The slope (m) indicates how steep the line is. A positive slope means the line rises, while a negative slope means it falls.
• The y-intercept (b) helps you identify where the function starts on the graph.

Step 2: Identify Slope and Y-Intercept

To analyze a linear function effectively, you need to identify its slope and y-intercept from its equation or graph.

How to Find Slope and Y-Intercept

• From the equation y = mx + b, the slope is m, and the y-intercept is b.

• From a graph, choose two points on the line:

  • Calculate the slope using the formula:

  [ m = \frac{y_2 - y_1}{x_2 - x_1} ]

  • The y-intercept is where the line crosses the y-axis (x = 0).

Common Pitfalls

• Ensure you are using the correct points when calculating the slope.
• Remember that horizontal lines have a slope of 0, while vertical lines are undefined.

Step 3: Graphing a Linear Function

Graphing a linear function involves plotting points and drawing the line that connects them.

Steps to Graph

1. Identify the y-intercept (b) and plot it on the graph.
2. Use the slope (m) to find another point:
   • From the y-intercept, rise (up or down) according to the slope and run (left or right).
3. Plot the second point.
4. Draw a straight line through the points extending in both directions.

Practical Tips

• Use a ruler for accuracy when drawing the line.
• Label the axes clearly to avoid confusion.

Step 4: Solve Linear Equations

Solving linear equations involves finding the value of x that makes the equation true.

Steps to Solve

1. Isolate the variable (x) on one side of the equation.
2. Perform inverse operations to simplify the equation.
3. Check your solution by substituting it back into the original equation.

Example

For the equation 2x + 3 = 11:

1. Subtract 3 from both sides: 2x = 8
2. Divide by 2: x = 4
3. Check: 2(4) + 3 = 11 (True)

Conclusion

In this tutorial, you learned about linear functions, including their definition, how to identify slope and y-intercept, graphing techniques, and solving linear equations. Mastering these concepts will provide a strong foundation for more advanced topics in mathematics. As a next step, practice graphing various linear equations and solving them to solidify your understanding.