Hoja de repaso: Understanding Affine Functions and Inequalities

📋 Course Outline

1. Definition and properties of affine functions
2. Determining affine function expression from two points
3. Monotonicity and variation of affine functions based on slope
4. Sign analysis of affine functions and solving inequalities

📖 1. Definition and properties of affine functions

🔑 Key Concepts & Definitions

• Coefficient directeur : a real number that indicates the slope of the straight line representing the affine function.
• Ordonnée à l'origine : a real number that corresponds to the y-intercept of the line, where it crosses the y-axis.
• Fonction affine : a function defined on the interval ] -∞ ; +∞ [ that can be expressed in the form f(x) = mx + p, where m and p are real numbers.
• Une fonction affine : an affine function whose graph is a straight line, possibly passing through the origin.

📝 Essential Points

• An affine function is defined on the entire real line, ] -∞ ; +∞ [, and can be written as f(x) = mx + p.
• The number p is called the ordonnée à l'origine, representing the y-intercept of the graph.
• The number m is called the coefficient directeur, representing the slope of the line.
• The graph of an affine function is a straight line.
• If this line passes through the origin (0,0), the function is called a linear function.

💡 Key Takeaway

An affine function is a linear expression with a constant term, whose graph is a straight line characterized by its slope and y-intercept, with the special case of passing through the origin called a linear function.

📖 2. Determining affine function expression from two points

🔑 Key Concepts & Definitions

• Affine function expression : a linear function with a constant term, represented as f(x) = mx + p, where m is the coefficient directeur and p is the y-intercept.

📝 Essential Points

• The coefficient directeur, denoted as m, can be calculated using the coordinates of two points A(xA, yA) and B(xB, yB) with the formula m = (yB - yA) / (xB - xA). This ratio measures the vertical change divided by the horizontal change between the points.
• Once m is determined, the affine function expression f(x) = mx + p can be found by substituting the coordinates of one of the points into the equation to solve for p. This involves replacing x and y with the known point's values and isolating p.

💡 Key Takeaway

Learning how to derive the explicit formula of an affine function from two known points involves calculating the coefficient directeur and then determining the constant term by substitution.

📖 3. Monotonicity and variation of affine functions based on slope

🔑 Key Concepts & Definitions

• Variation : +∞ [ (α IR) m > 0 x | -∞ +∞ Variation de f | -----> m < 0 x | -∞ +∞ Variation de f | <----- m

📝 Essential Points

• Variation de f | -----
• If the slope m > 0, the affine function is strictly increasing on ] -∞ ; +∞ [.

💡 Key Takeaway

The sign of the slope determines whether an affine function increases, decreases, or remains constant over its entire domain.

📖 4. Sign analysis of affine functions and solving inequalities

🔑 Key Concepts & Definitions

📝 Essential Points

• To find where an affine function is positive, solve the inequality mx + p > 0.
• To find where it is negative, solve mx + p < 0.

💡 Key Takeaway

Master the method to analyze the sign of affine functions and solve related inequalities using root and sign tables.

📊 Synthesis Tables

Comparison of Affine Function Properties

Property	Description
Graph	Straight line, possibly passing through origin
Slope (m)	Indicates the steepness of the line
Y-intercept (p)	Point where the line crosses the y-axis
Linear vs Affine	Linear if passing through origin, affine otherwise

⚠️ Common Pitfalls & Confusions

1. Confusing affine functions with linear functions, especially regarding the y-intercept.
2. Incorrectly calculating the coefficient directeur when points have the same x-coordinate.
3. Misinterpreting the sign of the slope as indicating the function's increasing or decreasing nature.
4. Forgetting to substitute a point to find the constant term p after calculating m.
5. Assuming the function is increasing or decreasing without considering the sign of m.
6. Misapplying inequalities by not solving for x properly.

✅ Exam Checklist

1. Identify the slope m from two points.
2. Calculate the y-intercept p using a known point.
3. Determine if the function is increasing or decreasing based on m.
4. Solve inequalities mx + p > 0 and mx + p < 0.
5. Use sign analysis to find intervals where the function is positive or negative.
6. Understand the difference between linear and affine functions.
7. Graph the affine function to visualize properties.
8. Apply the formula for m correctly when points have different x-values.
9. Check the domain of the affine function.
10. Interpret the slope and intercept in real-world contexts.