Functions & Graphs: Types, Rules, Plotting & Business Uses

TAKEAWAYS:

• A function assigns exactly one output (y) to each input (x), written as f(x) or y = f(x).
• Linear functions (y = mx + c) form straight lines; quadratic functions (y = ax² + bx + c) form parabolas.
• Piecewise functions have different rules for different x-intervals (e.g., tax brackets).
• Graphs visually show relationships between variables (e.g., profit vs. sales).
• Domain = all possible x-values; range = all possible y-values.
• Applications include break-even analysis, cost/revenue modeling, and supply-demand curves.

1. What is a Function?

A function is a rule that connects inputs (x) to outputs (y) such that each input has exactly one output. Think of it like a machine:

• You put in a number (input), and the machine gives you one answer (output).
• Example: A vending machine takes money (input) and gives one snack (output).

Notation

• f(x) or y = f(x) means "y is a function of x."
• Example: If f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.

Vertical Line Test

A graph represents a function only if a vertical line crosses it at most once.

2. Types of Functions

(A) Linear Functions (y = mx + c)

• Graph: Straight line.
• Slope (m): Steepness of the line (rise/run).
• Y-intercept (c): Where the line crosses the y-axis.

Example: y = 3x + 2

• Slope (m) = 3 (line rises 3 units for every 1 unit right).
• Y-intercept (c) = 2 (line crosses y-axis at (0,2)).

(B) Quadratic Functions (y = ax² + bx + c)

• Graph: Parabola (U-shaped or upside-down U).
• Vertex: Highest/lowest point of the parabola.
• Axis of symmetry: Vertical line through the vertex (x = -b/(2a)).

Example: y = x² - 4x + 3

1. Find vertex: x = -b/(2a) = 4/(21) = 2* y = (2)² - 4(2) + 3 = -1 Vertex = (2, -1).
2. Find y-intercept (x = 0): y = 3 → (0,3).
3. Find x-intercepts (y = 0): x² - 4x + 3 = 0 → x = 1, 3 → (1,0) and (3,0).

(C) Piecewise Functions

• Different rules for different intervals of x.
• Example: Tax Calculation
  • If income (x) ≤ 200,000: Tax = 0.1x
  • If 200,000 < x ≤ 500,000: Tax = 20,000 + 0.2(x - 200,000)

flowchart TD
    A["x ≤ 200,000"] -->|"Tax = 0.1x"| B["Tax = 0.1x"]
    C["200,000 < x ≤ 500,000"] -->|"Tax = 20,000 + 0.2(x - 200,000)"| D["Tax = 20,000 + 0.2(x - 200,000)"]
    E["x > 500,000"] -->|"Tax = 80,000 + 0.3(x - 500,000)"| F["Tax = 80,000 + 0.3(x - 500,000)"]

3. Domain and Range

• Domain: All possible x-values (inputs).
• Range: All possible y-values (outputs).

Examples:

4. Graphing Functions

Step-by-Step: Graph y = -x² + 4x - 3

1. Find the vertex: x = -b/(2a) = -4/(2-1) = 2* y = -(2)² + 4(2) - 3 = 1 Vertex = (2, 1).
2. Find y-intercept (x = 0): y = -3 → (0, -3).
3. Find x-intercepts (y = 0): -x² + 4x - 3 = 0 → x = 1, 3 → (1,0) and (3,0).
4. Plot points and draw parabola (opens downward because a = -1).

5. Applications in Business

(A) Break-Even Analysis

• Cost Function (C): C = Fixed Cost + Variable Cost Example: C = 500 + 2x (500 = fixed cost, 2x = variable cost per unit).
• Revenue Function (R): R = Price × Quantity Example: R = 5x (price = 5 per unit).
• Break-even point: Where C = R. 500 + 2x = 5x → 3x = 500 → x = 166.67 units.

(B) Profit Function

• Profit (P) = Revenue (R) - Cost (C) Example: P = 5x - (500 + 2x) = 3x - 500
  • If x = 200: P = 3(200) - 500 = 100 (profit).
  • If x = 100: P = 3(100) - 500 = -200 (loss).

6. Comparing Functions

Exam Tip

1. Always check the domain before graphing (e.g., √x requires x ≥ 0).
2. Label axes clearly in graphs (e.g., "Quantity Sold" vs. "Profit").
3. Show all steps in break-even problems:
   • Write C and R functions.
   • Set C = R and solve for x.
   • Interpret the answer (e.g., "The company breaks even at 167 units").
4. For piecewise functions, test points in each interval to avoid mistakes.
5. NEB loves word problems—translate sentences into equations first!

NEB-Style Questions

Short Answer (5 marks)

1. Define a function. Give an example of a linear function and its graph.
2. Find the vertex of y = -2x² + 8x - 5. Is the parabola U-shaped or upside-down?
3. A company’s cost function is C = 1000 + 5x and revenue is R = 10x. Find the break-even point.

Long Answer (10 marks)

4. The profit function of a business is P(x) = -2x² + 50x - 100, where x is the number of units sold.
   • Find the maximum profit and the number of units sold at this profit.
   • How many units must be sold to break even? Show your work with a graph.
5. Define a piecewise function for the following tax rules:
   • 10% tax if income ≤ 250,000.
   • 20% tax if 250,000 < income ≤ 500,000.
   • 30% tax if income > 500,000. Graph the function for income from 0 to 600,000.

Key Formulae to Memorize:

• Slope of a line: m = (y₂ - y₁)/(x₂ - x₁)
• Vertex of parabola: x = -b/(2a)
• Break-even point: Solve C(x) = R(x)
• Profit function: P(x) = R(x) - C(x)