Mathematics Exam Review

Equivalent Expressions for Polynomials

• Expression: (3n - 6)(n − 2)
  • Evaluate equivalent expressions:
  • Option a: 3n² - 6n - 12 X
  • Option b (Correct): 3(n-2)² = 3(n² - 4n + 4) ✓
  • Option c: 3n² - 12n + 12 ✓
  • Option d: 6n + 12 X
  • Option e: 3(n² - 4n + 4) ✓

Growth of Functions

• Given Functions:
  • f(x) = 3x² (Quadratic)
  • g(x) = 3^x (Exponential)
• Conclusion:
  • Exponential functions like g(x) will eventually outgrow quadratic functions such as f(x).

Patterns and Function Representation

• S: Squares and Dots Patterns
  • For Squares: Represents number of small squares as a function of step number:
  • Step 1: 2(2)
  • Step 2: 2(3)
  • Expression: ( S = n² + 2n + 2 )
  • For Dots: Represents number of small dots:
  • Step 1: 1
  • Step 2: 2
  • Expression: ( S = n² + n )

Expanding Polynomials

• Example Expansions:
  • (x + 2)(x + 9) = ( x² + 11x + 18 )
  • (2x - 1)(x + 4) = ( 2x² + 7x - 4 )
  • (x - 3)² = ( x² - 6x + 9 )
  • (3x + 2)(3x - 1) = ( 9x² + 3x - 2 )

Revenue Function for Concert Venue

• Function Definition:
  • Revenue function: ( r(x) = x(500 - 10x) )
• Graph Features:
  • Vertex: (25, 6250) → Maximum Revenue
  • x-intercepts (0,0) and (50,0) → No revenue at these ticket prices
• Domain:
  • Appropriate domain: 0 ≤ x ≤ 50
  • Explanation: Represents feasible ticket prices evaluated.

Motion of a Rock

• Height Function:
  • Function: ( h(t) = 80 + 64t - 16t² )
• True/False Statements:
  • Domain includes all real numbers: False; Time can't be negative
  • Initial height: False; height is 80 feet
  • Value at t = 4 in domain: True
  • Landed after 5 seconds: False; calculate height at t=5

Rectangular Exhibit Setup

• Rope Constraint:
  • Total rope: 200 feet for three sides
  • Width w:
  • Length: ( L = 200 - 2w )
  • Area: ( A(w) = w(200-2w) )
• Domain for w:
  • Domain: 0 < w < 100
  • Explanation: The width must be less than half the available rope to form a rectangle.

In-depth Observations:

• The key concepts involve evaluating equivalent expressions, understanding function behavior between quadratics and exponentials, and interpreting the meanings of mathematical results in real-world contexts. Examples given illustrate these relationships clearly, providing a framework for deeper analysis beyond test preparation.