Concepts & Formulas to Master
Slope: \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\) | Slope-intercept: \(y = mx + b\) | Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.
Point-slope: y − y₁ = m(x − x₁)
Number of solutions: 1 (intersect), 0 (parallel), ∞ (same line)
The system: \(2x + 3y = 12\) and \(4x + 6y = k\) has infinitely many solutions. Find \(k\).
Quadratic formula: \(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\) | Discriminant \(\Delta = b^2-4ac\): positive → 2 real roots; zero → 1 real root; negative → no real roots.
Standard: y = ax² + bx + c → vertex x = −b/(2a)
Factored: y = a(x − r₁)(x − r₂), roots = r₁, r₂
For \(f(x) = 2x^2 - 8x + 5\), find the vertex.
\(a^m \cdot a^n = a^{m+n}\) | \(\dfrac{a^m}{a^n} = a^{m-n}\) | \((a^m)^n = a^{mn}\) | \(a^{-n} = \dfrac{1}{a^n}\) | \(a^{1/n} = \sqrt[n]{a}\)
Half-life: A = A₀(1/2)^(t/T½)
Rational exponent: x^(m/n) = (ⁿ√x)^m
Simplify: \(\dfrac{8^{2/3} \cdot 2^3}{4^2}\)
\(f(x)+k\): shift up \(k\) | \(f(x+k)\): shift left \(k\) | \(-f(x)\): reflect over x-axis | \(f(-x)\): reflect over y-axis | \(af(x)\): vertical stretch/compress
Inverse: swap x and y, then solve for y
Domain restriction: denominator ≠ 0; radicand ≥ 0
If \(f(x) = x^2+1\) and \(g(x) = 2x-3\), find \(f(g(2))\).
Circle: \((x-h)^2+(y-k)^2=r^2\) | Arc length: \(s = r\theta\) (radians) | SOH-CAH-TOA | Special triangles: 30-60-90 (1:\(\sqrt{3}\):2) and 45-45-90 (1:1:\(\sqrt{2}\))
Pythagorean: a²+b²=c²
sin²θ + cos²θ = 1 | sin(90°−θ) = cos θ
Circle: \(x^2+y^2-6x+4y-12=0\). Find center and radius.
Mean = sum/count | Median = middle value (sorted) | Mode = most frequent | Range = max−min | P(A∪B) = P(A)+P(B)−P(A∩B)
Conditional probability: P(A|B) = P(A∩B) / P(B)
Margin of error: the ± value around a sample estimate
Data set: {3, 7, 7, 9, 14}. Find mean and median.