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.


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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.