precalculus:
1) FUNCTIONS AND THEIR GRAPHS
A function is an equation where to every domain there is one range.
Properties of Lines:
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Basic Functions, Functions and Graphs: (Known as the twelve basic functions):
1) Identity function: f(x) = x 2) Squaring: f(x) = x² 3) Cubing: f(x) = x^3 4) Reciprocal: f(x) = 1/x 5) Square Root:f(x) = √x 6) Exponential: f(x) = e^x 7) Natural Logarithmic: f(x) = ln x 8) Sine: f(x) = sin x 9) Cosine: f(x) = cos x 10) Absolute Value: f(x) = |x| = abs (x) 11) Greatest Integer: f(x) = int (x) 12) Logistic: f(x) = 1/1+e^-x |
Transformations (Shifts, Stretches, and Reflections):
Shifts: vertical and horizontal translations
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Translations: graph of y = f (x)
Horizontal Translations y = f (x - c) translation to the right by c units y = f (x + c) translation to the left by c units Vertical Translations y = f (x) + c translation up by c units y =f (x) - c translation down by c units Horizontal Stretches and Shrinks y = f (x/c) a stretch by a factor of c if c > 1 a shrink by a factor of c if c < 1 Vertical Stretches or Shrinks y = c * f (x) a stretch by a factor of c if c > 1 a shrink by a factor of c if c < 1 Across the x-axis y = -f (x) Across the y-axis y = f (-x) |
2) POLYNOMIALS AND RATIONAL FUNCTIONS
Quadratic Function
Synthetic Division: A shortcut to dividing two polynomials only by their coefficients of the several powers of the variable. Rational Functions : are ratios of polynomial functions The Fundamental Theorem of Algebra : A function of a degree (n) has n complex zeroes (real and non real) Some of the zeroes can be repeated. |
3) EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Exponential Functions and their Graphs
Y= (b)x, denoted by Y = log b( x ) Properties of Logarithms : Basic Properties of Logarithms:
Basic Properties of Common logarithms:
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4) ANALYTICAL GEOMETRY
Parabolas:
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