IndietroCollege Algebra Comprehensive Study Guide
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Rational numbers can be expressed as p/q (q ≠ 0) with decimals that terminate or repeat. Irrational numbers have decimals that never terminate or repeat. Real numbers include both.
Use parentheses ( ) for excluded endpoints and brackets [ ] for included endpoints. Infinity always uses parentheses.
Commutative, associative, identity, inverse, and distributive properties govern addition and multiplication of real numbers.
\(x^0=1\), \(x^{-n}=\frac{1}{x^n}\), \(x^m \cdot x^n = x^{m+n}\), \(\frac{x^m}{x^n} = x^{m-n}\), and power of a product/quotient rules.
Express numbers as \(N \times 10^m\) where 1 ≤ N < 10. Negative exponents for numbers < 1, positive for numbers ≥ 10.
The degree is the highest sum of exponents in any term of the polynomial.
Start with GCF, then use grouping, trinomials, ac method, or special patterns like difference of squares and sum/difference of cubes.
If \(ab=0\), then \(a=0\) or \(b=0\).
Exclude values that make the denominator zero by factoring and setting factors equal to zero.
Pull out perfect squares (or cubes), combine like radicals, and rationalize denominators by multiplying by conjugates if needed.
\(d=\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) calculates the distance between two points.
A graph is a function if no vertical line intersects it more than once.
\(m=\frac{y_2 - y_1}{x_2 - x_1}\) measures the steepness or rate of change of a line.
\(y - y_1 = m(x - x_1)\) is used when you know a point and the slope.
Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: \(m_1 \cdot m_2 = -1\).
\(\frac{f(x+h) - f(x)}{h}\) represents the average rate of change of a function over an interval.
\((f \circ g)(x) = f(g(x))\). The inner function g(x) is evaluated first.
Even: \(f(-x) = f(x)\) (symmetry about y-axis). Odd: \(f(-x) = -f(x)\) (symmetry about origin).
\(x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) solves any quadratic equation \(ax^2 + bx + c = 0\).
\(b^2 - 4ac > 0\): two real solutions; \(= 0\): one real solution; \(< 0\): two complex solutions.
\(f(x) = a(x - h)^2 + k\) with vertex at \((h, k)\). Minimum if \(a > 0\), maximum if \(a < 0\).
Remainder theorem: remainder of \(f(x)\div (x-c)\) is \(f(c)\). Factor theorem: \((x-c)\) is a factor if \(f(c)=0\).
Possible rational zeros are ± (factors of constant) / (factors of leading coefficient).
Vertical asymptotes where denominator = 0 (after simplification). Horizontal asymptotes depend on degrees: top < bottom → y=0; equal degrees → ratio of leading coefficients; top = bottom + 1 → oblique asymptote.
Replace f(x) with y, swap x and y, solve for y, then write f⁻¹(x). Check by composing functions.
\(y = \log_a x\) means \(a^y = x\).
\(\log(MN) = \log M + \log N\), \(\log(M/N) = \log M - \log N\), \(\log(M^p) = p \log M\).
\(\log_a x = \frac{\ln x}{\ln a}\) or \(\log_a x = \frac{\log x}{\log a}\) to compute logs with any base.
Periodic: \(A = P(1 + \frac{r}{n})^{nt}\). Continuous: \(A = Pe^{rt}\).
Growth: \(P(t) = P_0 e^{kt}, k > 0\). Decay: \(P(t) = P_0 e^{-kt}, k > 0\). Doubling/half-life: \(T = \frac{\ln 2}{k}\).