Advanced Math: Quadratics, Functions, and Exponential Models
Core ideas
Advanced Math emphasizes nonlinear relationships. For quadratics, know factoring, completing the square, the quadratic formula, and the connection between roots, x-intercepts, and factors. For functions, understand notation such as f(a), domain restrictions, composition, and transformations like shifts and stretches. For exponentials, distinguish growth from decay and recognize models of the form a(b)^t, where b > 1 means growth and 0 < b < 1 means decay.
Traps
When solving radical or rational equations, check for extraneous solutions caused by squaring or multiplying by expressions that can equal zero. Do not cancel terms across addition; cancel factors only. In exponential word problems, identify whether the rate is applied per year, per month, or continuously. Also watch for questions asking for an equivalent expression rather than a numerical solution.
Worked example
Solve x^2 - 5x + 6 = 0. Factor as (x - 2)(x - 3) = 0, so x = 2 or x = 3. If the question asks for the sum of the solutions, compute 2 + 3 = 5. For ax^2 + bx + c = 0, the sum of roots is -b/a and the product is c/a.
Rule of thumb
Before choosing a method, inspect the structure: a perfect square, a common factor, or a simple substitution often saves time. Always state domain restrictions for rational and radical expressions.