Math Unlocked: The Ultimate Formula Guide
1. Foundational Algebra
A. Operations, Properties & Simplification
| Order of Operations | PEMDAS: Parentheses → Exponents → Multiplication & Division (Left to Right) → Addition & Subtraction (Left to Right). |
|---|---|
| Commutative Property | $$a + b = b + a$$ $$a \cdot b = b \cdot a$$ (Order doesn’t matter) |
| Associative Property | $$(a + b) + c = a + (b + c)$$ $$(a \cdot b) \cdot c = a \cdot (b \cdot c)$$ (Grouping doesn’t matter) |
| Distributive Property | $$a(b + c) = ab + ac$$ $$-(a – b) = -a + b$$ |
| Identity Properties | Additive: $$a + 0 = a$$ Multiplicative: $$a \cdot 1 = a$$ |
| Inverse Properties | Additive: $$a + (-a) = 0$$ Multiplicative: $$a \cdot \frac{1}{a} = 1$$ ($$a \neq 0$$) |
| Combining Like Terms | Only terms with the exact same variable parts can be combined. Ex: $$3x^2 + 5x^2 = 8x^2$$ (Correct) Ex: $$3x^2 + 5x$$ (Cannot combine) |
B. Laws of Equality & Inequalities
| Addition/Subtraction Prop | If $$a = b$$, then $$a + c = b + c$$. (Same for inequalities) |
|---|---|
| Multiplication/Division Prop | If $$a = b$$, then $$ac = bc$$. (Same for inequalities if $$c > 0$$) |
| The Golden Rule of Inequalities | FLIP the inequality symbol when multiplying or dividing by a NEGATIVE number. Ex: $$-2x > 6 \Rightarrow x < -3$$ |
| Transitive Property | If $$a = b$$ and $$b = c$$, then $$a = c$$. If $$a < b$$ and $$b < c$$, then $$a < c$$. |
| Trichotomy Property | For any two real numbers $$a$$ and $$b$$, exactly one is true: $$a < b$$, $$a = b$$, or $$a > b$$. |
C. Absolute Value Equations & Inequalities
| Definition | $$|x|$$ is the distance from zero. $$|x| = x$$ if $$x \ge 0$$ $$|x| = -x$$ if $$x < 0$$ |
|---|---|
| Solving Equations | $$|ax + b| = c$$ Creates two cases: $$ax + b = c$$ OR $$ax + b = -c$$ |
| “Less Than” Inequality ($$<$$) | $$|x| < c$$ means distance is less than $$c$$. Solution: $$-c < x < c$$ (AND / Intersection) |
| “Greater Than” Inequality ($$>$$) | $$|x| > c$$ means distance is more than $$c$$. Solution: $$x > c$$ OR $$x < -c$$ (Union / Disjoint) |
| No Solution Case | $$|x| = -5$$ (Impossible, absolute value cannot be negative). |
D. Linear Equations & Graphs
| Slope Formula ($$m$$) | $$m = \frac{y_2 – y_1}{x_2 – x_1} = \frac{\text{Rise}}{\text{Run}}$$ |
|---|---|
| Slope-Intercept Form | $$y = mx + b$$ ($$m$$ = Slope, $$b$$ = y-intercept) |
| Point-Slope Form | $$y – y_1 = m(x – x_1)$$ (Used when you know a point and slope) |
| Standard Form | $$Ax + By = C$$ Slope = $$-\frac{A}{B}$$, y-intercept = $$\frac{C}{B}$$ |
| Horizontal Line | $$y = k$$ (Slope = 0) |
| Vertical Line | $$x = k$$ (Slope = Undefined) |
| Parallel Lines | Slopes are equal: $$m_1 = m_2$$ |
| Perpendicular Lines | Slopes are negative reciprocals: $$m_1 \cdot m_2 = -1$$ |
E. Systems of Equations & Inequalities
| Consistent Independent | Lines intersect at exactly one point. (Different slopes). |
|---|---|
| Inconsistent System | Lines are Parallel. No solution ($$\emptyset$$). (Same slope, different y-intercepts). |
| Consistent Dependent | Lines are Identical. Infinitely many solutions. (Same slope, same y-intercepts). |
| Solving Methods | 1. Substitution: Solve one eq for x or y, plug into the other. 2. Elimination: Add/Sub equations to cancel a variable. 3. Graphing: Find intersection point. |
| Linear Inequalities (Graphing) | 1. Graph boundary line (Solid for $$\le, \ge$$, Dashed for $$<, >$$). 2. Test a point (0,0). Shade true region. |
| Systems of Inequalities | Solution is the overlapping shaded region of all inequalities. |
2. Advanced Algebra & Functions
A. Laws of Exponents & Radicals
| Product Rule | $$x^a \cdot x^b = x^{a+b}$$ (Add exponents when multiplying like bases) |
|---|---|
| Quotient Rule | $$\frac{x^a}{x^b} = x^{a-b}$$ (Subtract exponents when dividing) |
| Power Rule | $$(x^a)^b = x^{a \cdot b}$$ (Multiply exponents) |
| Negative Exponent | $$x^{-n} = \frac{1}{x^n}$$ (Flip the fraction) |
| Zero Exponent | $$x^0 = 1$$ (For any $$x \neq 0$$) |
| Fractional (Rational) Exponents | $$x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$$ (Top is power, bottom is root) |
| Product/Quotient of Roots | $$\sqrt{ab} = \sqrt{a}\sqrt{b}$$ $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ |
B. Complex Numbers
| Imaginary Unit ($$i$$) | $$i = \sqrt{-1}$$ $$i^2 = -1$$ $$i^3 = -i$$ $$i^4 = 1$$ (Pattern repeats every 4) |
|---|---|
| Standard Form | $$a + bi$$ (Real part: $$a$$, Imaginary part: $$b$$) |
| Addition / Subtraction | Combine real parts and imaginary parts separately. $$(a+bi) + (c+di) = (a+c) + (b+d)i$$ |
| Complex Conjugate | The conjugate of $$a + bi$$ is $$a – bi$$. Product: $$(a+bi)(a-bi) = a^2 + b^2$$ (Result is Real) |
| Division | To divide, multiply numerator and denominator by the conjugate of the denominator. |
C. Quadratic Functions (Parabolas)
| 1. General (Standard) Form | $$f(x) = ax^2 + bx + c$$ y-intercept is $$c$$. Axis of symmetry: $$x = \frac{-b}{2a}$$. |
|---|---|
| 2. Vertex Form | $$f(x) = a(x – h)^2 + k$$ Vertex is at $$(h, k)$$. |
| 3. Factored (Intercept) Form | $$f(x) = a(x – p)(x – q)$$ x-intercepts (roots) are at $$p$$ and $$q$$. |
| The Quadratic Formula | $$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$ |
| The Discriminant ($$\Delta$$) | $$\Delta = b^2 – 4ac$$ $$\Delta > 0$$: 2 Real Roots. $$\Delta = 0$$: 1 Real Root (Double root). $$\Delta < 0$$: 2 Complex/Imaginary Roots. |
| Sum & Product of Roots | Sum ($$x_1 + x_2$$) = $$\frac{-b}{a}$$ Product ($$x_1 \cdot x_2$$) = $$\frac{c}{a}$$ |
D. Polynomials & Higher Order Equations
| Remainder Theorem | If polynomial $$P(x)$$ is divided by $$(x – c)$$, the remainder is $$P(c)$$. |
|---|---|
| Factor Theorem | $$(x – c)$$ is a factor of $$P(x)$$ if and only if $$P(c) = 0$$. |
| Sum/Difference of Cubes | Sum: $$a^3 + b^3 = (a + b)(a^2 – ab + b^2)$$ Diff: $$a^3 – b^3 = (a – b)(a^2 + ab + b^2)$$ |
| Multiplicity of Roots | If factor is $$(x-c)^n$$: $$n$$ is Odd: Graph crosses x-axis. $$n$$ is Even: Graph touches/bounces off x-axis. |
E. Function Operations & Analysis
| Domain Restrictions | 1. Denominator $$\ne 0$$. 2. Value inside even root (e.g., $$\sqrt{x}$$) must be $$\ge 0$$. |
|---|---|
| Composition of Functions | $$(f \circ g)(x) = f(g(x))$$ Plug $$g(x)$$ into $$f(x)$$. |
| Inverse Functions ($$f^{-1}$$) | 1. Replace $$f(x)$$ with $$y$$. 2. Swap $$x$$ and $$y$$. 3. Solve for $$y$$. Graphically: Reflection over line $$y = x$$. |
| Even vs Odd Functions | Even: $$f(-x) = f(x)$$ (Symmetric to y-axis). Odd: $$f(-x) = -f(x)$$ (Symmetric to origin). |
F. Transformations
| Vertical Shifts | $$f(x) + k$$: Shift UP $$k$$ units. $$f(x) – k$$: Shift DOWN $$k$$ units. |
|---|---|
| Horizontal Shifts | $$f(x – h)$$ : Shift RIGHT $$h$$ units. $$f(x + h)$$ : Shift LEFT $$h$$ units. |
| Reflections | $$-f(x)$$: Reflect over x-axis (Vertical flip). $$f(-x)$$: Reflect over y-axis (Horizontal flip). |
| Stretches & Compressions | $$a \cdot f(x)$$ where $$|a| > 1$$: Vertical Stretch. $$a \cdot f(x)$$ where $$0 < |a| < 1$$: Vertical Compression. |
G. Rational Functions
| Definition | $$f(x) = \frac{P(x)}{Q(x)}$$ where $$P$$ and $$Q$$ are polynomials. |
|---|---|
| Vertical Asymptotes | Set denominator $$Q(x) = 0$$ and solve (ensure values don’t make numerator 0). |
| Holes (Removable Discontinuity) | Occur at x-values that make BOTH numerator and denominator zero (common factors cancel out). |
| Horizontal Asymptotes | Compare degree of Top ($$n$$) vs Bottom ($$d$$): 1. $$n < d$$: Asymptote is $$y = 0$$. 2. $$n = d$$: Asymptote is $$y = \frac{\text{Lead Coeff Top}}{\text{Lead Coeff Bottom}}$$. 3. $$n > d$$: No Horizontal Asymptote (Slant). |
3. Data Analysis & Probability
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A. Ratios, Rates & Percentages
| Ratio & Proportion | Ratio: $$\frac{a}{b}$$ or $$a:b$$. Proportion: $$\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$$ (Cross Product). |
|---|---|
| Unit Rate | A rate with a denominator of 1. Example: 60 miles per hour ($$\frac{60}{1}$$). |
| Percent Formula | $$\frac{\text{Part}}{\text{Whole}} = \frac{\%}{100}$$ |
| Percent Change | $$\frac{\text{New Value} – \text{Original Value}}{\text{Original Value}} \times 100\%$$ (Positive = Increase, Negative = Decrease). |
| Scaling | If scale factor is $$k$$: Length ratio = $$k$$ Area ratio = $$k^2$$ Volume ratio = $$k^3$$ |
B. Variations (Direct, Inverse, Joint)
| Direct Variation | $$y = kx$$ ($$y$$ increases as $$x$$ increases. Graph goes through origin). $$k = \frac{y}{x}$$ (Constant of Variation). |
|---|---|
| Inverse Variation | $$y = \frac{k}{x}$$ or $$xy = k$$ ($$y$$ decreases as $$x$$ increases). |
| Joint Variation | $$z = kxy$$ ($$z$$ varies directly with both $$x$$ and $$y$$). |
C. Statistics: Central Tendency & Spread
| Mean (Average) | $$\bar{x} = \frac{\sum x}{n} = \frac{\text{Sum of all values}}{\text{Number of values}}$$ |
|---|---|
| Median | The middle value when data is ordered from least to greatest. If $$n$$ is even, average the two middle numbers. |
| Mode | The value(s) that appear most frequently. |
| Range | $$\text{Range} = \text{Maximum} – \text{Minimum}$$ |
| Standard Deviation ($$\sigma$$) | Measures spread/consistency. Low $$\sigma$$: Data is close to the mean (Consistent). High $$\sigma$$: Data is spread out (Volatile). |
D. Data Visualization (Graphs)
[Image of box and whisker plot diagram labeled]| Box-and-Whisker Plot | Displays 5-Number Summary: Min, Q1 (Lower Quartile), Median, Q3 (Upper Quartile), Max. IQR (Interquartile Range) = $$Q3 – Q1$$ (Represents middle 50%). |
|---|---|
| Histogram vs. Bar Graph | Bar Graph: Categorical data (spaces between bars). Histogram: Continuous numerical data ranges (bars touch). |
| Scatterplots | Shows relationship between two variables. Positive Correlation: Trend goes up. Negative Correlation: Trend goes down. Line of Best Fit: Used for prediction/inference. |
| Nonlinear Relationships | Scatterplots may show curved trends (Quadratic/Exponential) rather than straight lines. |
| Pie Charts | Sector Angle = $$\frac{\text{Part}}{\text{Total}} \times 360^\circ$$ |
| Venn Diagrams | Intersection ($$A \cap B$$): The overlapping region (Both). Union ($$A \cup B$$): Everything in circles A and B combined. |
E. Probability & Counting Principles
| Basic Probability | $$P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$ |
|---|---|
| Complement | $$P(\text{Not } E) = 1 – P(E)$$ |
| Independent Events | $$P(A \text{ and } B) = P(A) \times P(B)$$ (One event does not affect the other). |
| Dependent Events | $$P(A \text{ and } B) = P(A) \times P(B|A)$$ (Outcome of A changes probability of B). |
| Mutually Exclusive | $$P(A \text{ or } B) = P(A) + P(B)$$ (Cannot happen at the same time). |
| Conditional Probability | $$P(A | B) = \frac{P(A \text{ and } B)}{P(B)}$$ (Probability of A given B happened). |
| Fundamental Counting Principle | If task A has $$m$$ ways and task B has $$n$$ ways, total ways = $$m \times n$$. |
| Permutations (Order Matters) | $$_nP_r = \frac{n!}{(n-r)!}$$ (e.g., Arranging people in a line, Officers in a club). |
| Combinations (Order Doesn’t Matter) | $$_nC_r = \frac{n!}{r!(n-r)!}$$ (e.g., Selecting a team, Choosing toppings). |
4. Geometry & Trigonometry
A. Lines, Angles & Polygons
| Angle Pairs | Complementary: Sum = $$90^\circ$$ Supplementary: Sum = $$180^\circ$$ Vertical Angles: Equal measures. |
|---|---|
| Parallel Lines | If two parallel lines are cut by a transversal: – Alternate Interior angles are equal. – Corresponding angles are equal. – Consecutive Interior sum to $$180^\circ$$. |
| Polygon Interior Angles | Sum of interior angles = $$(n – 2) \times 180^\circ$$ Measure of one interior angle (Regular) = $$\frac{(n-2)180}{n}$$ |
| Polygon Exterior Angles | Sum of exterior angles is always $$360^\circ$$ (for any convex polygon). |
B. Triangles (Congruence & Similarity)
[Image of similarity of triangles]| Pythagorean Theorem | $$a^2 + b^2 = c^2$$ (Only for Right Triangles). Converse: If $$a^2+b^2=c^2$$, it’s a right triangle. If $$a^2+b^2 < c^2$$, it's Obtuse. If $$a^2+b^2 > c^2$$, it’s Acute. |
|---|---|
| Triangle Inequality | The sum of any two sides must be greater than the third side. $$a + b > c$$ |
| Area of Triangle | $$A = \frac{1}{2}bh$$ (Height must be perpendicular to base). |
| Congruence ($$\cong$$) | Triangles are identical (Same shape, Same size). Tests: SSS, SAS, ASA, AAS, HL. |
| Similarity ($$\sim$$) | Triangles have same angles but proportional sides. Tests: AA, SAS, SSS. Ratio of Areas = $$(\text{Scale Factor})^2$$ |
| Special Right Triangles | 30-60-90: Sides are $$x : x\sqrt{3} : 2x$$ 45-45-90: Sides are $$x : x : x\sqrt{2}$$ |
C. Circles
[Image of circle anatomy radius diameter chord]| Area & Circumference | $$A = \pi r^2$$ $$C = 2\pi r$$ or $$C = \pi d$$ |
|---|---|
| Arc Length | $$L = 2\pi r \left(\frac{\theta}{360}\right)$$ (Fraction of circumference) |
| Sector Area | $$A = \pi r^2 \left(\frac{\theta}{360}\right)$$ (Fraction of area) |
| Tangents | A line tangent to a circle is perpendicular to the radius at the point of tangency. Two tangents from the same external point are equal in length. |
| Chords | A radius perpendicular to a chord bisects the chord. |
| Radians to Degrees | $$\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$$ |
| Degrees to Radians | $$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$$ |
D. Coordinate Geometry
| Distance Formula | $$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$$ |
|---|---|
| Midpoint Formula | $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ |
| Equation of a Circle | $$(x – h)^2 + (y – k)^2 = r^2$$ Center: $$(h, k)$$, Radius: $$r$$ |
E. Solids: Volume & Surface Area
[Image of volume formulas for cylinder sphere cone]| Rectangular Prism | $$V = l \cdot w \cdot h$$ $$SA = 2(lw + lh + wh)$$ |
|---|---|
| Cylinder | $$V = \pi r^2 h$$ $$SA = 2\pi r^2 + 2\pi r h$$ |
| Cone | $$V = \frac{1}{3}\pi r^2 h$$ |
| Sphere | $$V = \frac{4}{3}\pi r^3$$ $$SA = 4\pi r^2$$ |
| Pyramid | $$V = \frac{1}{3}Bh$$ ($$B$$ = Area of base) |
F. Trigonometry
| SOH CAH TOA | $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$ $$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$ $$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$$ |
|---|---|
| Pythagorean Identity | $$\sin^2 \theta + \cos^2 \theta = 1$$ |
| Tangent Identity | $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ |
| Complementary Angles | $$\sin(x) = \cos(90^\circ – x)$$ $$\cos(x) = \sin(90^\circ – x)$$ |