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1. From Wikipedia, the free encyclopedia2. Lexicographical order

Transcript of Elementary Algebra Pq

  • Elementary algebra pqFrom Wikipedia, the free encyclopedia

  • Contents

    1 Parent function 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    2 Pointwise product 22.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Algebraic application of pointwise products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    3 Quadratic equation 43.1 Examples and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Solving the quadratic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    3.2.1 Factoring by inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2.2 Completing the square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.3 Quadratic formula and its derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.4 Reduced quadratic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.5 Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.6 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.7 Quadratic factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.8 Graphing for real roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.9 Avoiding loss of signicance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    3.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Advanced topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    3.4.1 Alternative methods of root calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4.2 Generalization of quadratic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    3.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    4 Quadratic formula 214.1 Derivation of the formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    i

  • ii CONTENTS

    4.2 Geometrical Signicance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Historical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Other derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    4.4.1 Alternate method of completing the square . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4.2 By substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4.3 By using algebraic identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4.4 By Lagrange resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    4.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    5 Quartic function 305.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4 Inection points and golden ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.5 Solving a quartic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    5.5.1 Nature of the roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.5.2 General formula for roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.5.3 Simpler cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.5.4 Converting to a depressed quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.5.5 Ferraris solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.5.6 Solving by factoring into quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.5.7 Solving by Lagrange resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.5.8 Solving with algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    5.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.10 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    5.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

  • Chapter 1

    Parent function

    In mathematics, a parent function is the simplest function of a family of functions that preserves the denition (orshape) of the entire family. For example, for the family of quadratic functions having the general form

    y = ax2 + bx+ c ;

    the simplest function is

    y = x2

    This is therefore the parent function of the family of quadratic equations.For linear and quadratic functions, the graph of any function can be obtained from the graph of the parent function bysimple translations and stretches parallel to the axes. For example, the graph of y = x2 4x + 7 can be obtained fromthe graph of y = x2 by translating +2 units along the X axis and +3 units along Y axis. This is because the equationcan also be written as y 3 = (x 2)2.For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). For example, thegraph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle along the positive X axis (where tan() = A B), then stretching it parallel to the Y axis using a stretch factor R, whereR2 = A2 + B2. This is because A sin(x) + B cos(x) can be written as R sin(x) (see List of trigonometric identities).The concept of parent function is less clear for polynomials of higher power because of the extra turning points, butfor the family of n-degree polynomial functions for any given n, the parent function is sometimes taken as xn, or, tosimplify further, x2 when n is even and x3 for odd n. Turning points may be established by dierentiation to providemore detail of the graph.

    1.1 See also Curve sketching

    1.2 External links Video explanation at VirtualNerd.com

    1

  • Chapter 2

    Pointwise product

    For entrywise product, see Matrix multiplication#Hadamard product.

    The pointwise product of two functions is another function, obtained by multiplying the image of the two functionsat each value in the domain. If f and g are both functions with domain X and codomain Y, and elements of Y can bemultiplied (for instance, Y could be some set of numbers), then the pointwise product of f and g is another functionfrom X to Y which maps x X to f(x)g(x).

    2.1 Formal denitionLet X and Y be sets, and let multiplication be dened in Ythat is, for each y and z in Y let the product

    : Y Y ! Y given by y z = yz

    be well-dened. Let f and g be functions f, g : X Y. Then the pointwise product (f g) : X Y is dened by

    (f g)(x) = f(x) g(x)

    for each x in X. In the same manner in which the binary operator is omitted from products, we say that f g = fg.

    2.2 ExamplesThe most common case of the pointwise product of two functions is when the codomain is a ring (or eld), in whichmultiplication is well-dened.

    If Y is the set of real numbers R, then the pointwise product of f, g : X R is just normal multiplication ofthe images. For example, if we have f(x) = 2x and g(x) = x + 1 then

    (fg)(x) = f(x)g(x) = 2x(x+ 1) = 2x2 + 2x

    for every real number x in R.

    The convolution theorem states that the Fourier transform of a convolution is the pointwise product of Fouriertransforms:

    Fff gg = Fffg Ffgg

    2

  • 2.3. ALGEBRAIC APPLICATION OF POINTWISE PRODUCTS 3

    2.3 Algebraic application of pointwise productsLet X be a set and let R be a ring. Since addition and multiplication are dened in R, we can construct an alge-braic structure known as an algebra out of the