Algebra 2 Essential Formulas

Algebra 2 Formulas: Comprehensive Study Notes

Laws of Exponents

• Multiply Powers of the Same Base:

  • Formula: $(a^m)(a^n) = a^{m+n}$

• Divide Powers of the Same Base:

  • Formula: $rac{a^n}{a^m} = a^{m-n}$

• Power Rule:

  • Formula: $(a^m)^n = a^{m \cdot n}$

• Zero Exponent:

  • Definition: $a^0 = 1$

• Distribution of Exponent with Multiple Bases:

  • Formula: $(ab)^n = a^n b^n$

  • Formula: $rac{b^n}{a^n} = rac{b}{a}^n$

• Negative Exponent:

  • Definition: $a^{-n} = \frac{1}{a^n}$

  • Alternative: $\frac{a^{-m}}{b^{-n}} = \frac{m}{n} a^m b^n$

• Distribution of Negative Exponent with Multiple Bases:

  • Formula: $(ab)^{-n} = a^{-n}b^{-n}$

  • Alternative: $\frac{1}{(ab)^n} = \frac{1}{a^n b^n}$

Properties of Radicals

• Distribution of Radicals of the Same Index (for $a \geq 0$ and $b \geq 0$ if $n$ is even):

  • Formula: $\sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}$

  • Alternative: $\sqrt[n]{b} \sqrt[n]{a} = \sqrt[n]{a} \sqrt[n]{b}$

• Power Rule of Radicals:

  • Formula: $\sqrt[n]{a^m} = (\sqrt[n]{a})^m$

• Reverse Operations of Radicals and Exponents:

  • For odd $n$: $\sqrt[n]{a} = a$

  • For even $n$: $\sqrt[n]{a} = | a |$

Special Products

• Difference of Squares:

  • Formula: $(A + B)(A - B) = A^2 - B^2$

• Square of a Binomial:

  • Formula: $(A + B)^2 = A^2 + 2AB + B^2$

  • Formula: $(A - B)^2 = A^2 - 2AB + B^2$

• Cube of a Binomial:

  • Formula: $(A + B)^3 = A^3 + 3A^2B + 3AB^2 + B^3$

  • Formula: $(A - B)^3 = A^3 - 3A^2B + 3AB^2 - B^3$

Special Expressions

• Sum of Cubes:

  • Formula: $A^3 + B^3 = (A + B)(A^2 - AB + B^2)$

• Difference of Cubes:

  • Formula: $A^3 - B^3 = (A - B)(A^2 + AB + B^2)$

Quadratic Formula

• Formula: $x =\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

• Discriminant:

  • Formula: $D = b^2 - 4ac$

  • When $D > 0$: Two Distinct Real Roots

  • When $D = 0$: One Distinct Real Root (or Two Equal Real Roots)

  • When $D < 0$: No Real Roots

Patterns of Imaginary Unit

• Pattern:

  • $i^1 = i$

  • $i^2 = -1$

  • $i^3 = -i$

  • $i^4 = 1$

  • The pattern repeats every 4 terms.

Product of Conjugate Complex Numbers

• Formula:

  • $(a + bi)(a - bi) = a^2 - b^2i^2$

  • Simplified: $a^2 + b^2$

Geometry and Coordinate Geometry

• Midpoint of a Line Segment:

  • Formula: $M = \left(\frac{x<em>1 + x</em>2}{2}, \frac{y<em>1 + y</em>2}{2}\right)$

• Distance of a Line Segment:

  • Formula: $d = \sqrt{(x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2}$

• Slope:

  • Formula: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

Standard Equation for Circles

• Formula:

  • $(x - h)^2 + (y - k)^2 = r^2$

• Definitions:

  • $P(x, y)$ = Any point on the path of the circle

  • $C(h, k)$ = Centre of the circle

  • $r$ = Length of the radius

Linear Equations

• Point-Slope Form:

  • Formula: $y - y<em>1 = m(x - x</em>1)$

• Standard Form:

  • Formula: $Ax + By + C = 0$

  • Condition: $A \geq 0$ (the leading coefficient for the x term must be positive)

• Slope-Intercept Form:

  • Formula: $y = mx + b$

  • Where $m$ = slope and $b$ = y-intercept

Parallel and Perpendicular Lines

• Parallel Lines:

  • Condition: $m<em>1 = m</em>2$

• Perpendicular Lines:

  • Condition: $m<em>1 = -\frac{1}{m</em>2}$

Functions Overview

• Types of Functions:

  • Linear Functions: $f(x) = mx + b$

  • Power Functions: $f(x) = x^n$ for $n > 1$

  • Root Functions: $f(x) = \sqrt[n]{x}$ for $n \geq 2$

  • Reciprocal Functions: $f(x) = \frac{1}{n}x$

  • Absolute Value Functions: $f(x) = |x|$

  • Greatest Integer Functions: $f(x) = [[x]]$ or $int(x)$

Transformation of Functions

• General Transformation:

  • Formula: $g(x) = f(x + h) + k$

  • Where:

  • $h$ = amount of horizontal movement

    • $h > 0$: move left

    • $h < 0$: move right

  • $k$ = amount of vertical movement

    • $k > 0$: move up

    • $k < 0$: move down

• Reflection:

  • Off x-axis: $g(x) = -f(x)$

  • Off y-axis: $g(x) = f(-x)$

• Vertical Stretching and Shrinking:

  • Formula: $g(x) = af(x)$

  • Where:

  • $a > 1$: stretches vertically

  • $0 < a < 1$: shrinks vertically

• Horizontal Stretching and Shrinking:

  • Formula: $g(x) = f(bx)$

  • Where:

  • $0 < b < 1$: stretches horizontally

  • $b > 1$: shrinks horizontally

Quadratic Functions

• Standard Form: $f(x) = a(x - h)^2 + k$

  • Vertex: $(h, k)$

  • Axis of Symmetry: $x = h$

  • Vertical Stretch Factor:

  • $a > 0$: vertex at minimum (opens up)

  • $a < 0$: vertex at maximum (opens down)

• General Form:

  • Formula: $f(x) = ax^2 + bx + c$

  • Y-intercept: $(0, c)$

  • X-intercepts at: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (if $b^2 - 4ac \geq 0$)

  • Vertex at: $x = -\frac{b}{2a}, \quad y = f \left(-\frac{b}{2a}\right)$

Functions and Their Inverses

• One-to-One Function:

  • Inverse Function: $f^{-1}(x)$

  • Inverse condition: $(x, y) \to (y, x)$

  • Domain of $f(x)$ → Range of $f^{-1}(x)$

  • Range of $f(x)$ → Domain of $f^{-1}(x)$

• Note:

  • $f^{-1}(x) \neq \frac{1}{f(x)}$ (Inverse is different from reciprocal)

End Behaviors and Polynomial Functions

• Multiplicity:

  • Definition: When a factored polynomial expression has exponents on the factor greater than 1.

• Odd Degree Polynomial Functions:

  • $a > 0$: Left is downward, right is upward.

  • Behavior: $y \to -\infty$ as $x \to -\infty$; $y \to \infty$ as $x \to \infty$.

  • $a < 0$: Left is upward, right is downward.

  • Behavior: $y \to \infty$ as $x \to -\infty$; $y \to -\infty$ as $x \to \infty$.

• Even Degree Polynomial Functions:

  • $a > 0$: Both ends upward.

  • Behavior: $y \to \infty$ as $x \to -\infty$; $y \to \infty$ as $x \to \infty$.

  • $a < 0$: Both ends downward.

Polynomial Division and Remainders**

• If $R = 0$ when evaluating $P(b)$, then $(x - b)$ is a factor of $P(x)$ and $P(b) = 0$.

• Remainder Theorem: To find the remainder of $P(x)$ when divided by $(x - b)$, substitute $b$ into $P(x)$.

• Factor Theorem:

  • A statement on the relationship between the factor and the polynomial.

Graphs of Exponential Functions

• Exponential Function:

  • Formula: $y = a^x$, with key points and intercepts determined by the value of $a$.

• Natural Exponential Functions:

  • Formula: $y = e^x$

  • Graph features: $y$-intercept at 1, no $x$-intercept.

Logarithmic Functions

• Logarithmic Definitions:

  • For base $a$:

  • $log_a(1) = 0$ because $a^0 = 1$

  • $log_a(a) = 1$ because $a^1 = a$

  • $a^{log_a(x)} = x$

• Common and Natural Logarithm:

  • Common Logarithm: $log_{10}(x)$

  • Natural Logarithm: $ln(x)$

Exponential and Logarithmic Relationships

• Exponent Laws:

  • For multiplication of bases: $(a^m)(a^n) = a^{m+n}$

  • For division of bases: $\frac{a^m}{a^n} = a^{m-n}$

• Logarithmic Laws:

  • For multiplication: $log<em>a(xy) = log</em>a(x) + log_a(y)$

  • For division: $log<em>a(\frac{x}{y}) = log</em>a(x) - log_a(y)$

  • For exponents: $log<em>a(x^n) = n \cdot log</em>a(x)$

Common Logarithm Mistakes

• Key point: Logarithm properties do not hold for addition or subtraction directly:

  • $log<em>a(x+y) \neq log</em>a(x) + log_a(y)$

  • $log<em>a(x-y) \neq log</em>a(x) - log_a(y)$

Inverse of Exponential Function

• To find inverse: Start from $y = a^x$, then switch $x$ and $y$: $x = a^y$ and rearrange to $y = log_a(x)$.

Graphing

• Exponential Functions:

  • $y$-intercept = 1, no $x$-intercept, Domain = $(-\infty, \infty)$, Range = $(0, \infty)$.

• Logarithmic Functions:

  • $x$-intercept = 1, no $y$-intercept, Domain = $(0, \infty)$, Range = $(-\infty, \infty)$.

Basic Trigonometric Definitions and Identities

• $\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}$

• $\csc{\theta} = \frac{1}{\sin{\theta}}$

• $\sec{\theta} = \frac{1}{\cos{\theta}}$

• $\cot{\theta} = \frac{1}{\tan{\theta}}$

• Fundamental Identity:

  • $\sin^2{\theta} + \cos^2{\theta} = 1$