Equations, Inequalities And Graphs (Copy)
EQUATIONS, INEQUALITIES AND GRAPHS — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Linear Equation | y = mx + c | Mixing gradient and intercept |
| Gradient | m = (y₂ − y₁)/(x₂ − x₁) | Reversing subtraction |
| Equation of Line | y − y₁ = m(x − x₁) | Using wrong coordinates |
| Parallel Lines | Same gradients | Thinking intercepts matter |
| Perpendicular Lines | m₁m₂ = −1 | Forgetting negative reciprocal |
| Quadratic Equation | ax² + bx + c = 0 | Forgetting a ≠ 0 |
| Quadratic Formula | x = (−b ± √(b² − 4ac))/2a | Missing brackets |
| Discriminant | Δ = b² − 4ac | Forgetting square on b |
| Nature of Roots | Δ > 0 → 2 roots | Mixing conditions |
| Nature of Roots | Δ = 0 → repeated root | Forgetting tangent meaning |
| Nature of Roots | Δ < 0 → no real roots | Trying to force real answers |
| Completing Square | (x + b/2)² − (b/2)² | Forgetting subtraction term |
| Perfect Square | (x ± a)² = x² ± 2ax + a² | Wrong middle term |
| Difference of Squares | a² − b² = (a − b)(a + b) | Wrong signs |
| Cubic Equation | ax³ + bx² + cx + d = 0 | Forgetting factor theorem |
| Factor Theorem | f(a) = 0 → (x − a) factor | Substituting incorrectly |
| Remainder Theorem | f(a) gives remainder | Mixing with factor theorem |
| Simultaneous Equations | Solve together | Solving separately |
| Substitution Method | Replace one variable | Partial substitution |
| Elimination Method | Eliminate variable | Wrong multiplication |
| Linear Inequality | Solve like equations | Forgetting sign change |
| Inequality Rule | Multiply/divide by negative → reverse sign | Most common mistake |
| Quadratic Inequality | Use roots + intervals | Solving directly |
| Interval Testing | Test values between roots | Testing roots themselves |
| Modulus Equation | |x| = a → x = ±a | Giving one answer only |
| Modulus Inequality | |x| < a → −a < x < a | Wrong interval direction |
| Exponential Equation | Same bases → equate powers | Ignoring base equality |
| Log Equation | logₐx + logₐy = logₐ(xy) | Adding logs incorrectly |
| Log Equation | logₐx − logₐy = logₐ(x/y) | Dividing logs directly |
| Log Restriction | logₐ(x) only if x > 0 | Allowing negatives |
| Graph of Line | Straight line | Wrong gradient direction |
| Graph of Quadratic | Parabola | Drawing straight line |
| Vertex Formula | x = −b/2a | Missing negative sign |
| Axis of Symmetry | x = −b/2a | Writing y instead |
| y-intercept | Put x = 0 | Confusing with x-intercept |
| x-intercept | Put y = 0 | Wrong substitution |
| Circle Equation | (x − a)² + (y − b)² = r² | Forgetting centre signs |
| Radius | √r² = r | Using r² as radius |
| Tangent to Circle | Radius ⟂ tangent | Forgetting 90° |
| Asymptote | Line graph approaches infinitely | Thinking graph touches |
| Exponential Graph | y = aˣ | Writing xᵃ |
| Log Graph | y = logₐx | Forgetting vertical asymptote |
| Reciprocal Graph | y = 1/x | Forgetting asymptotes |
| Transformation Up | y = f(x) + a | Moving graph downward |
| Transformation Right | y = f(x − a) | Moving graph left |
| Reflection in x-axis | y = −f(x) | Reflecting in y-axis |
| Reflection in y-axis | y = f(−x) | Reflecting in x-axis |
| Stretch | y = af(x) | Confusing with translation |
| Compression | y = f(bx) | Wrong direction |
| Intersection Points | Solve simultaneously | Giving only x-values |
| Tangent Condition | Δ = 0 | Thinking 2 intersections exist |
| Domain Restriction | Denominator ≠ 0 | Ignoring undefined values |
| Root Restriction | Inside √ ≥ 0 | Allowing negatives |
| Sketching Graphs | Show intercepts + turning points | Random sketch |
| Gradient of Curve | dy/dx | Using average gradient |
| Stationary Point | dy/dx = 0 | Forgetting second step |
| Increasing Function | dy/dx > 0 | Mixing signs |
| Decreasing Function | dy/dx < 0 | Mixing signs |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK GRAPH FEATURES TABLE
| Graph | Shape | Important Features |
|---|---|---|
| y = mx + c | Straight line | Gradient m |
| y = x² | Parabola | Vertex at origin |
| y = x³ | Cubic | S-shaped |
| y = |x| | V-shape | Vertex at origin |
| y = √x | Half curve | Starts at origin |
| y = 1/x | Hyperbola | Two asymptotes |
| y = aˣ | Exponential | Horizontal asymptote |
| y = logₐx | Log curve | Vertical asymptote |
| Circle | Closed curve | Fixed radius |
QUICK QUADRATIC RULES
| Feature | Formula |
|---|---|
| General Form | ax² + bx + c = 0 |
| Discriminant | Δ = b² − 4ac |
| Vertex x-coordinate | x = −b/2a |
| Axis of Symmetry | x = −b/2a |
| Quadratic Formula | x = (−b ± √Δ)/2a |
| Two Roots | Δ > 0 |
| Repeated Root | Δ = 0 |
| No Real Roots | Δ < 0 |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK INEQUALITY RULES
| Situation | Result |
|---|---|
| Multiply/divide by negative | Reverse inequality sign |
| Quadratic > 0 | Above x-axis |
| Quadratic < 0 | Below x-axis |
| |x| < a | −a < x < a |
| |x| > a | x < −a or x > a |
QUICK TRANSFORMATION RULES
| Transformation | Effect |
|---|---|
| y = f(x) + a | Up by a |
| y = f(x) − a | Down by a |
| y = f(x + a) | Left by a |
| y = f(x − a) | Right by a |
| y = −f(x) | Reflect in x-axis |
| y = f(−x) | Reflect in y-axis |
| y = af(x) | Vertical stretch |
| y = f(bx) | Horizontal stretch/compression |
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Tangent question | Use Δ = 0 |
| Vertex question | Use x = −b/2a immediately |
| Graph sketch | Find intercepts first |
| Modulus | Split into cases |
| Simultaneous equations | Substitute line into curve |
| Circle questions | Radius to tangent is 90° |
| Transformations | Inside opposite, outside normal |
| Inequalities | Draw sign chart |
| Logs | Convert to same base if stuck |
| Exponentials | Make bases equal |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Missing brackets | Entire expression becomes wrong |
| Wrong sign reversal in inequalities | Full answer incorrect |
| Forgetting ± | Missing one root |
| Wrong discriminant sign | Wrong nature of roots |
| Poor graph sketch | Method marks lost |
| Not labeling asymptotes | Incomplete graph |
| Wrong transformation direction | Incorrect graph |
| Ignoring domain restrictions | Undefined answers |
| Incorrect expansion | Entire algebra becomes incorrect |
| Forgetting to substitute back | Missing coordinates |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.