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) ✓
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).
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 )
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 )
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.
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
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.
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.
Mathematics Exam Review