1) Functions and their Graphs
A function an equation that each domain has only one range.
· Properties of Lines:
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· Basic Functions, Functions and Graphs: Known as the twelve basic functions :
1) Identity: f(x) = x 7) Natural Logarithmic: f(x) = ln x
2) Squaring: f(x) = x² 8) Sine: f(x) = sin x
3) Cubing: f(x) = x^3 9) Cosine: f(x) = cos x
4) Reciprocal: f(x) = 1/x 10) Absolute Value: f(x) = |x| = abs (x)
5) Square Root:f(x) = √x 11) Greatest Integer: f(x) = int (x)
6) Exponential: f(x) = e^x 12) Logistic: f(x) = 1/1+e^-x
(Ex) Solve for x: 4=2^x Answer is 2 (2*2=4)
Transformations (Shifts, Stretches, and Reflections):
Shifts: These are vertical and horizontal translations
Stretches:
Reflections: A reflection is a mirror image. |
Translations: graph of y = f (x) Horizontal Translations y = f (x - c) a translation to the right by c units y = f (x + c) a translation to the left by c units Vertical Translations y = f (x) + c a translation up by c units y =f (x) - c a translation down by c units Stretches and Shrinks:graph of y = f (x) Horizontal Stretches and Shrinks y = f (x/c) a stretch by a factor of c if c > 1 a stretch 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 stretch by a factor of c if c < 1 Reflections : graph of y = f (x) Across the x-axis y = -f (x) Across the y-axis y = f (-x) |
Examples: 1) Horizontal and Vertical Translations: Describe how the graph of y = x ^2 can be transformed to the graph of the given equation y = (x-1)^2 + 3 A horizontal translation to the right 1 unit, and vertical translation up by 3 units 2) Vertical Stretch : Describe how the graph of y = x^3 can be transformed to the graph of the given equation : y = 2x ^ 3 A vertical stretch by 2 3) Describe how the graph of y = √x can be transformed of the given equation y= √-x A reflection across the y-axis |
Combination of Functions
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Sum: (f + g)(x) = f (x) + g (x)
Difference: (f - g)(x) = f (x) - g (x) Product: (f g)(x) = f (x)g(x) Quotient: (f/g)(x) = f(x)/ g (x), provided g is not equal to zero |
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 degree n has n complex zeroes (real and non real) Some of the zeroes may be repeated. |
synthetic division
Divide:
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3) Exponential and logarithmic functions
Exponential Functions and their Graphs
Logarithmic Functions and their Graphs: The inverse of the exponential function Y= (b)x, denoted by Y = log b( x ) Properties of Logarithms : Basic Properties of Logarithms Basic Properties of Common Basic Properties of Natural Logarithms Solving Exponential and Logarithmic Equations : Ex. a) log [3] √3 = log [3] 3 ^1/2 =1/2 b) log 100= log [10] 100=2 because 10^2 =100 c) 10 ^log6 - 6 |
f(x) = e ^x Bounded Below, but not above
Domain: All reals No local extrema Range: (0, ∞) Horizontal asymptote: y = 0 Continuous No vertical asymptotes Increasing for all x End behavior: lim e^x= 0 and lim e^x= ∞ x --> ∞ x --> ∞ log2 (8)=3 because 2^3 =8 ) |
4) Analytical geometry
Parabolas
Hyperbola
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