Algebra II Honors Semester 2 Exam Comprehensive Study Guide

Unit 7: Inverses, Exponential, and Logarithmic Functions

• Composition of Functions

  • Function composition involves nested functions where the output of one function becomes the input of another.
  • Notation: $(f \circ g)(x) = f(g(x))$ and $(g \circ f)(x) = g(f(x))$.
  • To evaluate, substitute the entire expression of the inner function into every instance of the variable in the outer function.

• Inverse Functions

  • Definition: A function that "reverses" the action of the original function. If a point $(x, y)$ exists on function $f$, then the point $(y, x)$ exists on the inverse function $f^{-1}$.
  • Finding Inverses Algebraically:
    1. Replace $f(x)$ with $y$.
    2. Swap the roles of $x$ and $y$.
    3. Solve the new equation for $y$.
    4. Replace $y$ with $f^{-1}(x)$.
  • Determining Inverses via Composition: Two functions $f(x)$ and $g(x)$ are inverses of each other if and only if $f(g(x)) = x$ AND $g(f(x)) = x$.
  • Domain and Range: The domain of $f$ becomes the range of $f^{-1}$, and the range of $f$ becomes the domain of $f^{-1}$.

• Exponential Equations and Functions

  • Solving via Same Bases: If the bases can be made equal ($b^x = b^y$), then the exponents must be equal ($x = y$). Example: $3^{x+1} = 27$ becomes $3^{x+1} = 3^3$, so $x+1 = 3$.
  • Solving via Logarithms: If bases cannot be made the same, use the property that $b^x = a$ is equivalent to $x = \log_b(a)$, or take the natural log ($\ln$) or common log ($\log$) of both sides.

• Logarithmic Equations and Functions

  • Evaluating Logarithms: The expression $\log_b(x) = y$ answers the question: "To what power must base $b$ be raised to get $x$?" ($b^y = x$).
  • Properties of Logarithms:
    • Product Property: $\log_b(m \times n) = \log_b(m) + \log_b(n)$
    • Quotient Property: $\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)$
    • Power Property: $\log_b(m^p) = p \times \log_b(m)$
  • Solving Logarithmic Equations: Isolate the log and convert to exponential form, or use the property that if $\log_b(x) = \log_b(y)$, then $x = y$. Always check for extraneous solutions (the argument of a log must be positive).
  • Graphing: The parent function $y = \log_b(x)$ has a vertical asymptote at $x = 0$, an x-intercept at $(1, 0)$, and a domain of $(0, \infty)$.

Unit 8: Rational Functions and Relations

• Simplifying Rational Expressions

  • Factor the numerator and the denominator completely.
  • Divide out common factors to simplify.
  • Stating Restrictions: Restrictions (excluded values) are values of $x$ that make the denominator zero. These must be identified from the original factored denominator before any factors are canceled.

• Graphing Rational Functions

  • Vertical Asymptotes (VA): Occur at the x-values that make the simplified denominator equal to zero.
  • Holes (Removable Discontinuities): Occur if a factor is canceled from both the numerator and the denominator. The x-coordinate of the hole is the value that makes the canceled factor zero.
  • Horizontal Asymptotes (HA): Determined by comparing the degree of the numerator ($n$) and the degree of the denominator ($m$):
    • If $n < m$, the HA is at $y = 0$.
    • If $n = m$, the HA is at $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients.
    • If $n > m$, there is no horizontal asymptote (there may be a slant asymptote).
  • Domain and Range: The domain includes all real numbers except the x-values of vertical asymptotes and holes. The range is generally restricted by horizontal asymptotes and the y-values of holes.

Unit 9: Conic Sections

• Identification of Conic Sections

  • General Form: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$
  • Circle: $A = C$ (both squared terms have the same coefficient and sign).
  • Parabola: Either $A = 0$ or $C = 0$ (only one variable is squared).
  • Ellipse: $A$ and $C$ have the same sign but $A \neq C$.
  • Hyperbola: $A$ and $C$ have opposite signs.

• Writing Equations and Completing the Square

  • To convert general form to standard form, group the $x$ terms and $y$ terms, move the constant to the other side, and complete the square for each variable.
  • Standard Forms:
    • Circle: $(x-h)^2 + (y-k)^2 = r^2$
    • Ellipse (horizontal): $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
    • Hyperbola (horizontal): $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

• Graphing Key Parts

  • Center: Always indicated by $(h, k)$.
  • Vertices and Co-vertices: Distances are determined by the square roots of the denominators ($a$ and $b$) in ellipses and hyperbolas.
  • Foci: Points located along the major axis (ellipse) or inside the curves (hyperbola). For an ellipse, $c^2 = a^2 - b^2$. For a hyperbola, $c^2 = a^2 + b^2$.
  • Asymptotes (Hyperbolas): Diagonal lines that guide the shape of the hyperbola branches, found using $y - k = \pm \frac{b}{a}(x - h)$ or $y - k = \pm \frac{a}{b}(x - h)$.
  • Domain and Range: Must be stated in interval notation, e.g., $[x_1, x_2]$ or $(-\infty, y_1] \cup [y_2, \infty)$.

Unit 10: Sequences and Series

• Arithmetic Sequences and Series

  • Sequence: Defined by a common difference ($d$). Formula: $a_n = a_1 + (n-1)d$.
  • Series: The sum of the terms. Finite sum: $S_n = \frac{n(a_1 + a_n)}{2}$.

• Geometric Sequences and Series

  • Sequence: Defined by a common ratio ($r$). Formula: $a_n = a_1 \times r^{n-1}$.
  • Series: The sum of terms. Finite sum: $S_n = \frac{a_1(1-r^n)}{1-r}$.

• Pascal’s Triangle and Binomial Expansion

  • Pascal's Triangle: A triangular array of numbers where each number is the sum of the two directly above it. The rows provide the coefficients for binomial expansion.
  • Expanding a Binomial: Using $(a + b)^n$, the expansion involves terms with decreasing powers of $a$ and increasing powers of $b$.
  • Finding a Specific Term: Use the formula for the $k+1$ term: $\binom{n}{k} a^{n-k}b^k$, where $\binom{n}{k}$ is the combination formula $\frac{n!}{k!(n-k)!}$.

Unit 11: Trigonometry

• Degree and Radian Conversions

  • To convert degrees to radians: Multiply by $\frac{\pi}{180}$.
  • To convert radians to degrees: Multiply by $\frac{180}{\pi}$.

• The Unit Circle

  • Radius = $1$. Coordinates at any angle $\theta$ are $(\cos(\theta), \sin(\theta))$.
  • Reference Angles: The positive acute angle formed by the terminal side of an angle and the x-axis.
  • Coterminal Angles: Angles that share the same terminal side. Found by adding/subtracting multiples of $360^{\circ}$ or $2\pi$.

• Trigonometric Ratios and Evaluation

  • Six trig ratios: $\sin(\theta)$, $\cos(\theta)$, $\tan(\theta)$, $\csc(\theta) = \frac{1}{\sin(\theta)}$, $\sec(\theta) = \frac{1}{\cos(\theta)}$, and $\cot(\theta) = \frac{1}{\tan(\theta)}$.

• Laws for Non-Right Triangles

  • Law of Sines (Memorize): $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$.
  • Law of Cosines (Formula Given): $a^2 = b^2 + c^2 - 2bc\cos(A)$.
  • Area of Triangles (Formulas Given): $\text{Area} = \frac{1}{2}bc\sin(A)$.

• Graphing Trigonometric Functions

  • General form: $y = a\sin(b(x-c)) + d$ or $y = a\cos(b(x-c)) + d$.
  • Amplitude: $|a|$ (vertical dilation).
  • Period: $\frac{2\pi}{b}$ (for sine/cosine) or $\frac{\pi}{b}$ (for tangent).
  • Phase Shift: $c$ (horizontal translation).
  • Vertical Shift: $d$ (midline translation).