Comprehensive Review and Exam Preparation: Algebra II and Trigonometry

Writing Equations of Lines: The following problems involve determining the equation of a line based on specific characteristics: - Problem 5: Slope $m = 2$ containing the point $(3, -1)$ . - Problem 6: Slope $m$ through the point $(-5, 4)$ . - Problem 7: Containing the points $(0, -4)$ and $(4, 1)$ . - Problem 8: Containing the points $(4, 2)$ and $(-2, -1)$ .
Function Identification: Determine if the following relations represent a function (each input having exactly one output): - 1. $y = x^2 + 7$ : Identified as a function. - 2. $y = \frac{3}{x - 6}$ : Identified as a function. - 3. $y = \sqrt{x + 4}$ : Identified as a function. - 4. $2x^2 + 3y^2 = 10$ : Not a function (describes an ellipse). - 5. Set of Ordered Pairs: $\{(-4, 3), (0, 2), (1, -2), (3, 2)\}$ . This is a function. - 6. Set of Ordered Pairs: $\{(-4, 3), (0, 2), (1, -2), (0, 3)\}$ . This is not a function because the input $0$ is mapped to both $2$ and $3$ . - Graphs (7-10): Visual identification of functions using the vertical line test. Results vary (e.g., Problem 7 is not a function, Problem 8 is a function, Problem 9 is not a function).

For any given function (specifically referenced as $f(x)$ , $g(x)$ , and $h(x)$ in problems 18, 19, and 20), the following features must be analyzed:

Domain and Range: The sets of all possible input and output values respectively.
Intercepts: Identify both $X$ -intercepts (points where the graph crosses the horizontal axis) and $Y$ -intercepts (points where the graph crosses the vertical axis).
Local Maxima and Minima: Identify the input values where the function reaches a relative high or low point. Explicitly state whether the point is a minimum or maximum.
Absolute Maxima and Minima: Identify the absolute highest and lowest output values over the entire domain.
Parity (Even, Odd, or Neither): - Even: Symmetry respect to the $y$ -axis ( $f(-x) = f(x)$ ). - Odd: Symmetry respect to the origin ( $f(-x) = -f(x)$ ).
Interval Behaviors: Identify intervals on the $x$ -axis where the function is increasing, decreasing, or constant.

Standard Form of a Circle: $(x - h)^2 + (y - k)^2 = r^2$ , where $(h, k)$ is the center and $r$ is the radius.
Specific Circle Problems: - 13: Center $(0, 0)$ and radius $r = 5$ . - 14: Center $(-2, 3)$ and radius $r = \sqrt{5}$ . - 15: Center $(0, 0)$ and passes through the point $(3, -4)$ . - 16: Center $(-5, -3)$ and passes through the point $(-3, 2)$ . - 19: Endpoints of a diameter are located at $(-1, -7)$ and $(1, -3)$ . - 20: Endpoints of a diameter are located at $(-3, 2)$ and $(5, -6)$ .

Piecewise Function Definition: A function defined by multiple sub-functions, each applying to a certain interval of the domain. - Example 16: $f(x) = \begin{cases} x^2 & \text{if } -2 \le x \le 2 \\ 2x & \text{if } x > 2 \end{cases}$ - Example 17: If $f(x) = \begin{cases} 3x + 2 & \text{if } -2 \le x < 1 \\ 4 & \text{if } 1 \le x \le 4 \end{cases}$ . Calculate specific values: - (a) $f(-1)$ - (b) $f(0)$ - (c) $f(1)$ - (d) $f(3)$ - Example 18: If $f(x) = \begin{cases} -x + 1 & \text{if } x < 0 \\ 2 & \text{if } x = 0 \\ 2x + 1 & \text{if } x > 0 \end{cases}$ . Calculate specific values: - (a) $f(-2)$ - (b) $f(0)$ - (c) $f(2)$
Composite and Inverse Functions: - Tabular Data for $x, g(x), h(x)$ : - For $x = 1$ : $g(1) = 5$ , $h(1) = 5$ . - For $x = 2$ : $g(2) = 3$ , $h(2) = 7$ . - For $x = 3$ : $g(3) = 4$ , $h(3) = -1$ . - For $x = 4$ : $g(4) = 2$ , $h(4) = 1$ . - Linear Function $k(x)$ : $k(x) = 3x - 4$ . - Evaluations: - 1. $h(g(2))$ - 2. $g(h(4))$ - 3. $k(g(2))$ - 4. $f(k(2))$ - 5. $g(f(1))$ - 6. $k(f(4))$ - 7. $h(h(4))$ - 8. $k(k(4))$ - 9. $g^{-1}(5)$ - 10. $h^{-1}(1)$ - 11. $k^{-1}(11)$ - 12. $g^{-1}(h^{-1}(7))$

Factored Form of Polynomials: Given degree and zeros (with multiplicity). Polynomials must have real coefficients. - Degree 3: Zeros: $-2, 1, 3$ . - Degree 3: Zeros: $0, 1$ (multiplicity 2). - Degree 5: Zeros: $2$ (multiplicity 3), $+5i, -5i$ (Implies complex conjugate pairs). - Degree 4: Zeros: $-4, 0, 2+i, 2-i$ . - Degree 3: Zeros: $5, 7i$ (Implies $-7i$ for real coefficients). - Degree 5: Zeros: $-1$ (multiplicity 2), $0, 5+2i$ (Implies $5-2i$ ).
Finding Zeros of Polynomials: - 7. $f(x) = x^3 + 2x^2 - 5x - 6$ - 8. $g(x) = x^3 - 9x^2 + 15x + 25$ - 9. $h(x) = x^3 - 9x + 10$ - 10. $j(x) = x^4 + 7x^2 - 9$ - 11. $k(x) = x^3 + 2x^2 + 9x + 18$ - 12. $m(x) = x^3 - 2x^2 - 3x + 10$
Rational Function Sketches: Create sketches satisfies specific conditions: - EB (End Behavior): Expressed as horizontal asymptotes (e.g., $y = 0, y = 1, y = -2$ ). - VA (Vertical Asymptotes): Identify specific $x$ values and signage (positive/negative). - Holes: Specific points where terms cancel out (e.g., $x = -2$ ). - Intercepts: Identify $X$ -intercept points.
Rational Function Analysis: - 1. $f(x) = \frac{x-4}{x+4}$ - 2. $g(x) = \frac{-2x - 8}{x^2 - 3x - 10}$ - 3. $h(x) = \frac{x^2 - 3x - 10}{x^2 - x - 2}$ - 4. $k(x) = \frac{3x^2 - 12x}{x^2 + x - 20}$ - 5. $m(x) = \frac{x^2 + 2x - 24}{x^2 - 9}$ - 6. $n(x) = \frac{x^2 - 6x + 9}{x^2 - 9}$

Logarithmic Evaluation: - 1. $\log_2(16)$ - 2. $\log(1000)$ - 3. $\log_{1/2}(1/8)$ - 4. $\log_9(3)$
Trigonometry Standards and Skills: - 6.3: Evaluating trig functions. - 6.5: Identifying features of wave functions including Amplitude, Period, Phase Shift, and Vertical Shift (Handwritten note additions). - 7.1: Writing solutions to trig equations. - 7.2/7.3: Recognizing and using trig identities involving algebra. - 8.1: Law of Sines. - 8.2: Law of Cosines.

Structure: - Part 1: Multiple Choice (20 questions). - Part 2: Constructed Response (11 questions). - The exam spans the entire school year, not just the current semester. - Class time will be given for each section on different days.
Permitted Resources: - A calculator is allowed for the entirety of the exam. - A $3\text{ in} \times 5\text{ in}$ cards is allowed for the entirety of the exam.
General Math Skills Covered: - Interpreting Functions/Representations. - Evaluating Functions. - Creating Expressions. - Sketching Graphs. - Multiple Choice Strategy. - Solving Equations. - Manipulating Expressions.
Skills Excluded from Standalone Questions: 1.1, 1.3, 1.4, 2.1, 2.3, 2.4, 2.5, 2.7ab, 4.1, 4.2, 4.3, 4.5, 4.8, 5.3, 5.4, 5.5, 5.6, 5.7, 6.1, 6.2, 6.4, 8.3.