Description

comment trouver la règle d’une fonction linéaire ?

To determine the rule of a linear function, follow these organized steps:

### 1. Understand the Linear Function Form
A linear function is typically expressed as ( y = mx + b ), where:
– ( m ) is the slope (rate of change).
– ( b ) is the y-intercept (where the line crosses the y-axis).

### 2. Identify Two Points
You need at least two points on the line. These can be obtained from a graph, table, or word problem. If given a word problem, translate the information into two points.

### 3. Calculate the Slope (( m ))
Use the formula:
[ m = frac{y_2 – y_1}{x_2 – x_1} ]
where ( (x_1, y_1) ) and ( (x_2, y_2) ) are your two points.

### 4. Determine the Y-Intercept (( b ))
Once you have ( m ), substitute one of the points into the equation ( y = mx + b ) to solve for ( b ).

### 5. Write the Equation
Combine ( m ) and ( b ) into the equation ( y = mx + b ).

### 6. Verify with Additional Points
If you have more points, plug them into your equation to ensure they satisfy it. If they don’t, check your calculations for errors.

### Special Cases:
– **Horizontal Line**: The slope ( m = 0 ), so the equation is ( y = b ).
– **Vertical Line**: Not a function; disregard.
– **Origin Crossing**: If the line passes through (0,0), then ( b = 0 ), so the equation is ( y = mx ).

### Example:
Given points (1, 4) and (3, 8):
– Slope ( m = (8 – 4)/(3 – 1) = 4/2 = 2 ).
– Using point (1,4): ( 4 = 2(1) + b ) → ( b = 2 ).
– Equation: ( y = 2x + 2 ).

### Conclusion:
By systematically finding the slope and y-intercept, you can derive the rule of any linear function. Practice with various examples to reinforce your understanding.