Algebra Midterm #3 Review

• §10.8: Complex Numbers
  • Definition: i is a number such that i² = -1, or i = √(-1).
  • Anything that can be written as a+bi is a complex number.
  • Complex conjugate: same real part; opposite imaginary part (a+bi and a−bi).
  • Complex arithmetic: know how to add, subtract, multiply, and divide complex numbers.
  • Know how to raise i to any power. This cycles every four steps!

• §11.1: Solving Quadratic Equations
  • ...by factoring or by completing the square


• §11.2: The Quadratic Formula
  • If ax² + bx + c = 0, then x = ( -b ± √(b²−4ac) ) / (2a).
  • Know how this formula is derived (completing the square) and how to use it.


• §11.3: Analyzing Solutions of Quadratics
  • The discriminant is D = b²−4ac.
  • Trichotomy: check if D<0 (complex roots), D>0 (real roots), or D=0 (one root).
  • Complex roots always come in conjugate pairs (a+bi and a−bi).
  • Given the roots of a quadratic, find the quadratic itself.


• §11.4: Applications: speed problems, falling objects, solving formulas


• §11.6: Graphs of Quadratics
  • Standard form: y = ax² + bx + c
  • Vertex form: y = a·(x−h)² + k, with vertex at (h, k)
  • Understand how a, h, and k in the equation affect the shape of the graph.
  • Given the equation, sketch the graph; given the graph, find the equation.


• §11.7: More Graphs of Quadratics
  • Completing the square can also be used to change standard form to vertex form!
  • From standard form, the vertex can also be found with the formula x = -b/(2a).
  • The x-intercepts can be found by setting y = 0 and solving for x by any method.


• §11.8: More Applications
  • Optimization (of profit, area, etc.): write an equation and use it to find
    the maximum or minimum value, which will occur at the vertex.
  • Finding a parabola's equation from 3 points: use standard form, plug in
    the 3 (x,y) values to get a system of 3 equations, and solve for a, b, and c.

• §12.1: Function Theory
  • Function composition and f∘g(x) notation. Note that f∘g ≠ g∘f!
  • Inverse functions and f ^-1(x) notation. Note that this does not mean 1/f(x)!
  • Finding inverse of a function, verifying if functions are inverses, graphs of inverses.


• §12.2: Exponential Functions
  • Graphing y = B^x,   x = B^y,   and y = B^x−h+k for various B-values.
  • Know what "asymptote" means, where to expect one, and how to write its equation.


• §12.3: Introduction to Logarithms
  • Definition of logarithm as inverse exponential function: if B^x = A, then log_B(A) = x.
  • log_B(A) can be thought of as asking "To what power must B be raised to equal A?
  • Convert an equation from exponential form to logarithmic form or vice-versa.
  • Evaluate simple logarithms by hand: consider what it is "asking."
  • log_B(B) = 1,   log_B(1) = 0,   log_B(B)^x = x,   B^log_B(x) = x


• §12.4: Properties of Logarithms
  • Product Rule: log_B(x) + log_B(y) = log_B(x · y)
  • Quotient Rule: log_B(x) − log_B(y) = log_B(x / y)
  • Power Rule: log_B(x^r) = r · log_B(x)
  • Simplifying or expanding expressions: consider "What is the last operation being done here?


• §12.5: Important Bases of Logarithms
  • The number e is what (1 + 1/n)ⁿ becomes as n gets very large. e ≈ 2.71828183.
  • Natural logarithms: ln(x) is shorthand for log_e(x).
  • Common logarithms: log(x) with no base listed means log₁₀(x).
  • Know where to find the buttons "ln," "log," "10^x," and "e^x" on your calculator.
  • Change-of-base formula: log_A(x) = log_B(x) / log_B(A). Useful for converting to base 10 or e.


• §12.6: Solving Exponential and Logarithmic Equations
  • If the variable is in an exponent...
    • Try isolating the thing being raised to a power.
    • If possible, rewrite everything in terms of a common base, and set exponents equal.
    • Otherwise, either rewrite in log form and use change-of-base formula
      OR take log or ln of both sides and use the Power Rule.
  • If the variable is in a logarithm...
    • Try isolating the logarithm (if there's only one), then rewrite in exponential form.
    • If there are several logarithms, combine using properties from §12.4.
    • Check results in original equation; some may be extraneous!


• §12.7: Applications of Exponents and Logarithms
  • Logarithmic scales (Richter, pH, decibels)
  • Population growth: P(t) = P₀·e^rt, where P₀ = starting population
  • Radioactive decay: same equation, but r will be negative.
    • "Half-life" = amount of time required for half of it to decay.
  • To find r-value, plug in a known value of t and P(t) and solve for r.

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