Algebra 2 Semester 1 Final Exam Flashcards

Functions and Zeros

• A function with zeros at x = 2 and x = -1 is represented as f(x) = x^2 - x - 2.
• For the quadratic function f(x) = x^2 - 8x + 12, the zeros are x = 2 and x = 6.
• The zeros for f(x) = x^2 + 2x - 3 are x = -3 and x = 1.
• Zeros of the function f(x) = x^2 + 11x + 24 are x = -8 and x = -3.

Maximum and Minimum Values

• The function g(x) = -2x^2 - 4 has a maximum value of -4.
• For the quadratic function f(x) = 2x^2 - 2x + 5, the minimum occurs at 4.5.
• The minimum of f(x) = x^2 - 6x + 3 is at -6.

Transformations of Functions

• The transformation from the parent function to f(x) - 3 results in a vertical shift of 3 units down.
• Transforming f(x) to f(x - 1) indicates a horizontal shift of 1 unit to the right.
• Increasing a function by 4 units can be expressed as f(x) + 4.

Domain and Range

• For f(x) = x, both the domain and range include all real numbers.
• The domain and range for f(x) = x^2 are: D: all real numbers; R: y ≥ 0.
• For f(x) = √x, the domain is x ≥ 0 and the range is y ≥ 0.
• Domain and range for any cubic function and linear function consist of all real numbers.

Parent Functions

• Parent function for g(x) = √x + 3 is f(x) = √x (square root).
• g(x) = x^2 + 4 corresponds to the parent function f(x) = x^2 (quadratic).
• g(x) = x - √2 represents the linear parent function f(x) = x.

Complex Numbers Simplification

• Simplify (3 + 7i)/(8i) to (7 - 3i)/8.
• The expression (5 + i) / (2 - 4i) simplifies to (3 + 11i)/10.
• Simplifying (3 + 8i)/(-i) yields -8 + 3i.
• The complex expression (3 - i)/(2 - i) simplifies to (7 + i)/5.

Factorization

• The polynomial 2x^4 + 2x^3 - x^2 - 5x - 4 includes (x + 1) as a factor.
• Conversely, (x - 2) is not a factor of 5x^3 + x^2 - 7.
• The polynomial 4x^6 - 12x^5 + 2x^3 - 6x^2 - 5x + 10 does not have (x - 3) as a factor.
• The factorization of x^5 - 8x^4 + 8x - 64 includes (x - 8).

Coordinate Translations

• Translating the point (3, 1) by 5 units left and 4 units down results in (-2, -3).
• The translation of (4, 2) 3 units down gives the new coordinates (4, -1).
• Moving (4, 2) 1 unit right and 6 units up results in (5, 8).
• Translating (3, 1) 4 units up leads to the coordinates (3, 5).

Additional Zeros

• For g(x) = 3x^2 + 12x, the zeros are found at x = 0 and x = -4.
• For f(x) = 2x^2 - x + 2, zeros calculate to 1 ± i√15 / 4.
• The function g(x) = 3x^2 - x + 8 yields zeros of 1 ± i√95 / 6.
• Zeros of f(x) = x^2 + 2x - 3 occur at x = -3 and x = 1.