Algebraic functions

Constant function f(x) = c The constant function f(x) = c: its horizontal-line graph, why its derivative is 0, its integral cx + C, limits, and common mistakes with constants. Linear function f(x) = mx + b The linear function f(x) = mx + b: slope and intercept, its straight-line graph, why its derivative is the slope m, its integral and worked examples. Quadratic function f(x) = ax^2 + bx + c, \quad a \ne 0 The quadratic function f(x) = ax² + bx + c: parabola graph, vertex formula, roots, derivative 2ax + b, integral and worked calculus examples. Power functions f(x) = x^n Power functions f(x) = xⁿ for whole, negative and fractional n: graphs, domains, the derivative nxⁿ⁻¹, the integral and the n = -1 exception. Polynomial functions f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 Polynomial functions explained: degree, leading term, end behavior, roots and turning points, plus how to differentiate and integrate any polynomial. Rational functions f(x) = \frac{P(x)}{Q(x)}, \quad Q(x) \ne 0 Rational functions P(x)/Q(x): domain, vertical and horizontal asymptotes, holes, derivatives with the quotient rule and integrals by partial fractions. Reciprocal function f(x) = \frac{1}{x} The reciprocal function 1/x: its hyperbola graph and asymptotes, the derivative -1/x², why its integral is ln|x| + C, and its one-sided limits. Square root function f(x) = \sqrt{x} = x^{1/2} The square root function f(x) = √x: its domain and graph, the derivative 1/(2√x), the integral (2/3)x^(3/2) + C, and the vertical tangent at 0. Absolute value function f(x) = |x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases} The absolute value function |x|: its V-shaped graph, why it is continuous but not differentiable at 0, its derivative x/|x|, and its integral.