Algebra II Final

Structure of the Honors Algebra 2/Trigonometry Final Exam

Section I: Multiple Choice - Part I

• Properties of Rational Exponents

  • Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$. Move the base to the denominator to make the exponent positive.

  • Zero Exponent Rule: $a^0 = 1$ (where $a \neq 0$).

  • Even Roots: The radicand must be $\geq 0$ to result in a real number. If $n$ is even, $\sqrt[n]{x}$ is undefined for x < 0.

  • Odd Roots: The radicand can be any real number ($x \in \mathbb{R}$). For example, $\sqrt[3]{-8} = -2$.

• Signs of Function Values in Trigonometry

  • Use the ASTC (All Students Take Calculus) mnemonic to determine which functions are positive in each quadrant:

  • Quadrant I (0 to $\frac{\pi}{2}$): All functions (Sine, Cosine, Tangent) are positive.

  • Quadrant II ($\frac{\pi}{2}$ to $\pi$): Only Sine and its reciprocal (Cosecant) are positive.

  • Quadrant III ($\pi$ to $\frac{3\pi}{2}$): Only Tangent and its reciprocal (Cotangent) are positive.

  • Quadrant IV ($\frac{3\pi}{2}$ to $2\pi$): Only Cosine and its reciprocal (Secant) are positive.

• Domain of a Radical Function

  • For the function $f(x) = \sqrt[n]{g(x)}$:

  • If $n$ is even, solve the inequality $g(x) \geq 0$.

  • If $n$ is odd, the domain is $(-\infty, \infty)$.

• Exponential Growth and Decay

  • General Form: $f(x) = a \cdot b^x$, where $a$ is the initial value.

  • Growth: Occurs when b > 1. Often written as $b = 1 + r$, where $r$ is the growth rate.

  • Decay: Occurs when 0 < b < 1. Often written as $b = 1 - r$, where $r$ is the decay rate.

  • Continuous Compounding: $A = Pe^{rt}$, where $e \approx 2.718$.

• Logarithmic and Exponential Conversion

  • Definition: $\log_b(a) = c$ is equivalent to $b^c = a$.

  • Common Logarithm: $\log(x)$ implies base 10.

  • Natural Logarithm: $\ln(x)$ implies base $e$.

Section II: Multiple Choice - Part 2

• End Behavior of Polynomial Functions

  • Determined by the Leading Coefficient Test:

  • Even Degree, Positive Leading Coefficient: Both ends go up ($y \to \infty$ as $x \to \pm\infty$).

  • Even Degree, Negative Leading Coefficient: Both ends go down ($y \to -\infty$ as $x \to \pm\infty$).

  • Odd Degree, Positive Leading Coefficient: Down on the left, up on the right.

  • Odd Degree, Negative Leading Coefficient: Up on the left, down on the right.

• Solving Radical Equations

  • 1. Isolate the radical term.

  • 2. Raise both sides to the power of the index (e.g., square both sides for a square root).

  • 3. Solve the resulting polynomial equation.

  • 4. Check for Extraneous Solutions: Solutions that do not satisfy the original radical constraint.

• Coterminal Angles

  • Angles that share the same terminal side.

  • In Radians: Add or subtract multiples of $2\pi$ (e.g., $\theta \pm 2k\pi$).

  • In Degrees: Add or subtract multiples of $360^\circ$ (e.g., $\theta \pm 360k$).

Section III: Short Answer

• Logarithm Properties

  • Product Rule: $\log<em>b(xy) = \log</em>b(x) + \log_b(y)$.

  • Quotient Rule: $\log<em>b(\frac{x}{y}) = \log</em>b(x) - \log_b(y)$.

  • Power Rule: $\log<em>b(x^p) = p \cdot \log</em>b(x)$.

  • Change of Base: $\log<em>b(a) = \frac{\log</em>k(a)}{\log_k(b)}$.

• Inverse Functions ($f^{-1}(x)$)

  • Algebraic Steps: Replace $f(x)$ with $y$, swap $x$ and $y$, and solve for the new $y$.

  • Existence: A function has an inverse that is also a function if it passes the Horizontal Line Test (it must be one-to-one).

  • Symmetry: The graph of $f(x)$ and $f^{-1}(x)$ are reflections across the line $y = x$.

• Rational Functions: Domain and Asymptotes

  • Domain: All real numbers except where the denominator equals zero.

  • Vertical Asymptotes (VA): Occur at values of $x$ that make the denominator zero (after simplifying).

  • Holes (Removable Discontinuities): Occur if a factor $(x-c)$ cancels out from both numerator and denominator.

Section IV: Problem Solving

• Solving Exponential Equations

  • Method 1 (Same Bases): If $b^x = b^y$, then $x = y$.

  • Method 2 (Different Bases): Take the $\ln$ or $\log$ of both sides and use the Power Rule to bring down the exponent.

• Composition of Functions

  • $(g \circ f)(x) = g(f(x))$. Substitute the entire expression of $f(x)$ into every $x$ in $g(x)$.

  • Domain of Composition: Must consider both the domain of the inner function $f(x)$ and the resulting domain of $g(f(x))$.

Section V: Graphing

• Parent Function Transformations

  • For $y = a \cdot f(x - h) + k$:

  • $a$: Vertical stretch/compression. If negative, reflection over the x-axis.

  • $h$: Horizontal shift (Right if $-h$, Left if $+h$).

  • $k$: Vertical shift (Up if $+k$, Down if $-k$).

Section VI: Trigonometry

• Unit Circle Essentials

  • Coordinates: $(\cos \theta, \sin \theta)$.

  • Important values: $\sin(30^\circ) = \frac{1}{2}$, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, $\tan(45^\circ) = 1$.

• Graphing Sine and Cosine

  • Equation: $y = A \sin(B(x - C)) + D$

  • Amplitude: $|A|$ (half the distance between max and min).

  • Period: $P = \frac{2\pi}{B}$.

  • Phase Shift: $C$.

  • Midline (Vertical Shift): $y = D$.

1. nth Roots and Rational Exponents

• Definition: The expression $a^{m/n}$ can be written as $\sqrt[n]{a^m}$ or $(\sqrt[n]{a})^m$. Here, $n$ is the index and $m$ is the power.

• Conversions:

  • $x^{1/2} = \sqrt{x}$

  • $x^{2/3} = \sqrt[3]{x^2}$

2 & 3. Exponential Growth and Decay Functions

• Standard Function: $y = a(1 \pm r)^t$

  • $a$: Initial amount (y-intercept).

  • $r$: Growth or decay rate (expressed as a decimal).

  • $(1 + r)$: Growth factor (b > 1).

  • $(1 - r)$: Decay factor (0 < b < 1).

• Asymptote: For the basic function $y = ab^x$, there is a horizontal asymptote at $y = 0$.

4. End Behavior of a Function

Determined by the degree and the leading coefficient ($a_n$):

• Even Degree: Ends move in the same direction.

  • a_n > 0: \text{As } x \to \pm\infty, y \to \infty

  • a_n < 0: \text{As } x \to \pm\infty, y \to -\infty

• Odd Degree: Ends move in opposite directions.

  • a_n > 0: \text{As } x \to \infty, y \to \infty; \text{As } x \to -\infty, y \to -\infty

  • a_n < 0: \text{As } x \to \infty, y \to -\infty; \text{As } x \to -\infty, y \to \infty

5. The y-intercept

• Definition: The point where the graph crosses the y-axis.

• Finding the Value: Evaluate the function at $x = 0$ ($f(0)$). In the polynomial $y = ax^n + … + c$, the y-intercept is always $(0, c)$.

6 & 7. Properties of Radicals and Rational Exponents

• Product Property: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$ and $a^m \cdot a^n = a^{m+n}$.

• Quotient Property: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$ and $\frac{a^m}{a^n} = a^{m-n}$.

• Power of a Power: $(a^m)^n = a^{mn}$.

8 & 9. Simplifying Expressions with Positive Exponents

• Simplifying Radicals: Factor out the largest perfect $n$-th power. For example, $\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$.

• Positive Exponents Only: Use the rule $a^{-n} = \frac{1}{a^n}$ to ensure no negative exponents remain. For example, $\frac{x^2y^{-3}}{z} = \frac{x^2}{y^3z}$.

10. Subtracting Polynomials

• Procedure: Distribute the negative sign to every term in the second polynomial, then combine like terms.

• Example: $(5x^2 - 3x) - (2x^2 - 8) = 5x^2 - 3x - 2x^2 + 8 = 3x^2 - 3x + 8$.

11. Common Logarithms

• Definition: A logarithm with base 10, written as $\log x$.

• Evaluation: $\log 100 = 2$ because $10^2 = 100$. $\log 1 = 0$ because $10^0 = 1$.

12. Inverse Functions

• Process: Swap $x$ and $y$ in the equation and solve for the new $y$.

• Notation: $f^{-1}(x)$.

• Domain/Range: The domain of $f(x)$ is the range of $f^{-1}(x)$, and vice-versa.

13. Simplifying Rational Expressions

• Step 1: Factor both the numerator and the denominator completely.

• Step 2: Cancel common factors.

• Step 3: Identify restrictions (values where the original denominator was zero).

14 & 15. Condensing and Expanding Logarithms

• Expanding: Use properties to break down a single log into multiple logs.

  • $\log(\frac{x^3}{y}) = 3\log x - \log y$

• Condensing: Use properties to combine multiple logs into one.

  • $2\log a + \log b = \log(a^2b)$

16. Solving Radical Equations

1. Isolate the radical on one side.

2. Raise both sides to the index power (square both sides for $\sqrt{x}$).

3. Solve the resulting equation.

4. Check for extraneous solutions by plugging answers back into the original radical.

17. Solving Logarithmic Equations

• Case 1: $\log_b(x) = y \implies$ convert to exponential form $b^y = x$.

• Case 2: $\log<em>b(x) = \log</em>b(y) \implies x = y$.

• Note: Always check that the argument of the log is greater than zero.