Classe de Mathématiques - Fonction Affine

Fonctions Affines

• A function f defined on R by f(x) = ax + b is called an affine function.
• Its graph is a straight line with equation y = ax + b.
• 'a' is the slope (coefficient directeur).
• 'b' is the y-intercept (ordonnée à l'origine), the point where the line crosses the y-axis (0, b).

Examples

• f(x) = 2x - 5
• If x = 0, f(0) = -5, the line passes through point A(0, -5).
• If x = 2, f(2) = 2(2) - 5 = -1, the line passes through point B(2, -1).

Special Cases

• Linear function: If b = 0, the function passes through the origin (0, 0).
• Constant function: If a = 0, the function is a horizontal line parallel to the x-axis.

Characteristic

• The increments of the images are proportional to the increments of the x values.

Variation

• Increasing: If a > 0, the function is increasing for all real numbers.
• Constant: If a = 0, the function is constant.
• Decreasing: If a < 0, the function is decreasing for all real numbers.

Demonstration

• For two real numbers u and v with u < v:
• f(u) - f(v) = a(u - v)
• If a > 0, f(u) < f(v) , the function is increasing.
• If a = 0, f(u) = f(v) , the function is constant.
• If a < 0, f(u) > f(v) , the function is decreasing.

Examples of Variation

• f(x) = 2x - 5 is increasing.
• g(x) = -3x + 2 is decreasing.
• h(x) = 2 is constant.

Sign of ax + b (a ≠ 0)

• The sign of ax + b depends on the value of x relative to -b/a.
• Table showing signs for a > 0 and a < 0.

Solving Inequalities

• Solving inequalities of the form (ax + b) > 0 or (ax + b) < 0 involves finding the intervals where the expression is positive or negative.
• Create a sign chart to determine the intervals.
• A product of factors is 0 if at least one factor is 0.

Solving Equations

• The solution to ax + b = 0 is x = -b/a.
• Consider the sign of a when determining the solution intervals.