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ReviewPrecalculusReference : TC7 by Leithold

This presentation is a copyright of Robert T. Nericua 2015 FundamentalsReal Numbers and InequalitiesThe real number system consist of a set R of elements called real numbers and two operations called addition and multiplication, denoted by the symbols + and , respectively.If a and b are elements of a set R, a + b indicates the sum of a and b, and a b (or ab) indicates their product.The operation of subtraction is defined by the equation a b= a + (-b) , where b denotes the negative of b such that b + (-b) =0.The operation of division is defined by the equation a b = a b-1 , b 0, where b-1 denotes the reciprocal of b such that b b-1 = 1.

Real Numbers and inequalitiesAxiom is used to indicate a formal statement assumed to be true without proof.Properties that can be shown to be logical consequences of axioms are theorems.In the statement of most theorems there are two parts: the if part, called the hypothesis, and the then part is called conclusion. The argument verifying a theorem is a proof.A real number is either a positive, negative, or zero, and any real number can be classified as either rational or irrational.A rational number is one that can be form as the ratio of two integers.Real Numbers and inequalitiesThe rational numbers consist of the following:IntegersFractionsTerminating decimalsNonterminating repeating decimals

The real numbers that are not rational are called irrational numbers.

Real Numbers and inequalitiesReal numbers and inequalitiesReal Numbers and InequalitiesFundamentalsCoordinates and graphs of EquationsThe origination of analytic geometry is credited to Ren Descartes (1596-1650).In his book Geometry, published in 1635, Descartes established the union of algebra and geometry by a rectangular Cartesian coordinate system.Any two real numbers form a pair, and when the order of appearance of the numbers is significant, we call it an ordered pair.The set of all ordered pairs of real numbers is called the number plane, denoted by R2, and each ordered pair (x , y) is a point in the number plane.The first number x of the pair is called the abscissa (or x coordinate) of P ,a and the second number y is called ordinate (or y coordinate) of P.Coordinate and graphs of equationsCoordinate and Graphs of equationsCoordinate and Graphs of EquationsFundamentalsLinesLinesTwo vertical lines l1 and l2 having slopes m1 and m2, respectively, are perpendicular if and only if m1m2=-1.

LinesFundamentalsParabolasA parabola is a set of all points in a plane equidistant from a fixed point and a fixed line. The fixed point is called the focus.The fixed line is called the directrix. Theorem

An equation of the parabola having its focus at (0, p) and having as its directrix the line y=-p is x2 =4py.

An equation of the parabola having its focus at (p,0) and its directrix is the line x=-p is y2 =4px.ParabolasThe intersection of the parabola with its axis is called Vertex; latus rectum is the chord perpendicular to the fixed axis (directrix) and passing through the focus.ParabolasFundamentalsCirclesCircleFundamentalsTranslation of AxesEquations for Translating the Axes If (x,y) represents a point P with respect to a given set of axes, and (x,y) is a representative of P after the axes are translated to a new origin having coordinates (h,k) with respect to the given axes, then x = x h and y=y-k

Standard Forms of an Equation of a Parabola If p is the directed distance from the vertex to the focus , an equation of the parabola with its vertex (h, k) and with its axis vertical is (x-h)2 = 4p(y-k)

Translation of AxesA parabola with the same vertex and with axis horizontal has the equation(y-k)2 = 4p(x-h) Translation of AxesFUNDAMENTALSellipsesAn ellipse is the set of points in a plane the sum whose distance from two points is a constant. Each fixed point is called a focus.The line through the foci of an ellipse is called principal axis; The points of intersection of the ellipse and its principal axis are called the vertices; The point on the principal axis that lies between the two vertices is called the center.The segment of the principal axis between the two vertices is called the major axis, and its length is 2a units. The minor axis is the line segment perpendicular to the principal axis and has a length of 2b.

ellipsesEllipseellipseFundamentalsHyperbolasHyperbolashyperbolasFundamentalsTrigonometric functions In Geometry, an Angle is defined as the union of two rays called the sides, having a common endpoint called the vertex.

Definition of Sine and Cosine Function Definition of a Periodic Function

Fundamental Pythagorean Identitysin2 + cos2 =1 Definition of the Tangent, Secant, Cotangent, and CosecantTrigonometric functionsFundamentalsPartial FractionsPartial Fraction is method of expressing a single rational expression as a sum of two or more simpler quotients.The degree of the numerator is less than the degree of the denominator is called a proper fraction. The degree of the numerator is not less than the degree of the denominator is called a improper fraction.

Partial Fraction