Algebra Midterm #3 Review
Algebra Midterm #3 Review
Complex Numbers
- §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!
Quadratics
- §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.
Exponents and Logarithms
- §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 = Bx, x = By, and y = Bx−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 Bx = A, then logB(A) = x.
- logB(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."
- logB(B) = 1, logB(1) = 0, logB(B)x = x, BlogB(x) = x
- §12.4: Properties of Logarithms
- Product Rule: logB(x) + logB(y) = logB(x · y)
- Quotient Rule: logB(x) − logB(y) = logB(x / y)
- Power Rule: logB(xr) = r · logB(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 loge(x).
- Common logarithms: log(x) with no base listed means log10(x).
- Know where to find the buttons "ln," "log," "10x," and "ex" on your calculator.
- Change-of-base formula: logA(x) = logB(x) / logB(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·ert, 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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