W4_Functions and Graphs (continued)

Fundamentals of Mathematics 1

Session 4: Functions and Graphs (continued)

Presented by: Erwan Lamy

Institution: ESCP BUSINESS SCHOOL

Locations: Berlin, London, Madrid, Paris, Turin, Warsaw

Objectives

  • Understanding Slope: Learn about the slope and different forms of line equations.

  • Solving Linear Equations: Use elimination by addition or substitution for systems of linear equations in 2 and 3 variables.

  • Working with Nonlinear Systems: Utilize substitution to solve nonlinear systems.

  • Demand and Supply Curves: Understand and solve systems related to equilibrium and break-even points.

Outline

  • Equation of a Line (Chapter 3.1)

  • Systems of Equations (Chapters 3.4 and 3.5)

  • Applications of Systems of Equations (Chapter 3.6)

Equation of a Line (3.1)

  • Slope Definition: Measures the steepness of a line.

    • For points (1, 1) and (2, 2):

      • Slope m = (y2 - y1) / (x2 - x1) = (2 - 1) / (2 - 1) = 1

Methods to Determine Slope:

  1. Two-Point Formula: m = (y2 - y1) / (x2 - x1) using two points on the line.

  2. Graphical Interpretation: Counting the rise over the run on a graph.

  3. Slope from an Equation: Identify the slope directly from the slope-intercept form y = mx + b.

  • Types of Lines:

    • Vertical Lines: Represented as x = a.

    • Horizontal Lines: Represented as y = b.

  • Equations of a Line:

  1. Point-Slope Form: For a line passing through (1, 1) with slope m:

    • Point-Slope: y - y1 = m(x - x1)

  2. Slope-Intercept Form: y = mx + b where m is the slope and b is the y-intercept.

  3. General Linear Form: Ax + By + C = 0 where A, B, and C are constants.

  • Parallel and Perpendicular Lines:

  1. Parallel Lines: Have the same slope.

  2. Perpendicular Lines: Slopes are opposite reciprocals.

    • If the slope of one line is m1, then for it to be perpendicular to another line with m2, it must hold true that m1 * m2 = -1.

Example of Slope-Intercept and Point-Slope Form

  • Example: Given the line y = 3x + 1,

    • For the line passing through (3, -2):

      • Parallel Line: y + 2 = 3(x - 3)

      • Perpendicular Line: y + 2 = - (1/3)(x - 3)

Systems of Equations (3.4 and 3.5)

  • Definition: Linear System: a1x + b1y = c1

  • Goal: Find values for which all equations are simultaneously true.

Methods to Solve Linear Equations:

  1. Elimination-by-Substitution: Solve one equation for a variable and substitute into the other.

  2. Elimination-by-Addition: Adjust equations so that adding them eliminates one variable.

  3. Graphical Representation: Plotting each equation and finding intersections.

Example to Solve System:

  • {2 + 3 = 5 (1)}

  • {4 - 2 = 7 (2)}

  • Graphical Representations:

    • Unique Solutions: Intersection points of lines.

    • No Solutions: Parallel lines.

    • Infinitely Many Solutions: Identical lines.

Nonlinear Systems

  • Definition: At least one equation is nonlinear.

  • Example: Solve the following system:

    • {2 - 2 + y - 7 = 0}

    • {3 - y + 1 = 0}

Applications: Demand and Supply Curves

  • Equilibrium: Intersection of demand and supply curves.

    • Methods: Set demand equals supply to find equilibrium price and quantity.

  • Equilibrium Price and Quantity: Where the market balances.

  • Break-Even Points:

    • Profit/Loss Formula: Profit = Total Revenue - Total Cost.

    • Example: Selling price, fixed and variable costs to determine break-even points.