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# Advanced algebra (some terminologies)

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### Advanced algebra (some terminologies)

1. 1. Rectangular coordinate system- a system that relates the correspondence between the points on a plane to a pair of real numbers. - was called Cartesian coordinate system by some mathematicians in honour of Rene Descartes, the inventor of the system. In some books, it is also called x-y plane. - makes use of two coplanar perpendicular number lines.X-axis - The horizontal number line.Y-axis - The vertical number line.Origin- a point where the axes intersect.Quadrants- four regions formed by the intersection of the axes.Quadrantal points- points that lie exactly on the axis.Ordered pair- is a set of two well-ordered real numbers called coordinates.Coordinates- are the numerical descriptive reference of a point from the two axes.X-coordinate (abscissa) - the first coordinate which corresponds to a real number on the x-axis.Y-coordinate (ordinate) - the second coordinate which corresponds to a real number on the y-axis.Relation- the association of an individual or object to another individual or object. - is any set of one or more ordered pairs. - can be described by a table, a set of ordered pairs, an arrow diagram, an equation or formula or a graph.Ordered pair- consists of two components, the arrangement of which affects the essence of a relation.Domain of a relation- set of all abscissas in a relation.Range of a relation- set of all ordinates in a relation.Function- there corresponds one and only one element of the second set. - A special type of relation. - Actually involves the pairing of elements between two nonempty sets.The Notation f(x) - The functional notation f(x) means the value of the function f at number x.
2. 2. Independent variable- the name of x if y is the value of f at x or y=f(x), since any element of the domain can replace it.Dependent variable- the name of y because its value depends on the value of the independent x.The Vertical Line Test- a graph of a relation is a function if any vertical line drawn passing through the graph intersects it at exactly one point.Constant Function (C) - is a function the range of which consists of a single real number k for all real numbers x in its domain. In symbol C(x) =k. - is a polynomial function of degree 0.Identity Function (I) - defined by I(x) =x.Polynomial Function (P) - defined by P(x) = a1xn+a2xn-1+a3xn-2+...+an-1x+an. where n is a nonnegative integer and a1, a2, a3,...an-1, an are real numbers.Linear Function- a polynomial function to the first degree. It is defined by f(x) =mx+b. Where m and b are real numbers and m≠0.Quadratic Function- a polynomial function is of the second degree.Parabola- the graph of any quadratic function.Cubic Function- a polynomial function is of the third degree.Absolute Value Function- defined by f(x) =|x|.Graph of y=|y±A|- to graph y=|x+A|, simply shift the graph of y=|x|, A units to the left. - To graph y=|x-A|, simply shift the graph of y=|x|, A units to the right.Graph of y=|x±A|- to graph y=|x|+B, simply shift the graph of y=|x|, B units up. - To graph y=|x|-B, simply shift the graph of y=|x|, B units down.Square Root Function- is defined by f(x) =√x.Rational Function- is a function of the form F(x) = N(x) D(x). Where N(x) and D(x) are polynomial functions and D(x) ≠0.Asymptote- is an imaginary line being approached but never touched or intersected by a graph as it goes through infinity.Greatest Integer Function (G) - is defined as G(x) = [[x]]
3. 3. Piece-wise Function- is defined compositely using several expressions and different interval domains.Signum Function- is defined asUnit Step Function- is defined asTranscendental Functions- are non-algebraic functions.The Sum of Functions- If f and g are functions with domains Df and DG, their difference is the function defined as (f+g) (x) =f(x)+g(x). The Domain of (f+g)(x) is Df ∩ DG.The Difference of Functions- If f and g are functions with domains Df and DG their difference is the function defined as (f-g)(x)=f(x)-g(x). The Domain of (f-g)(x) is Df ∩ DG.The Product of Functions- If f and g are functions with domains Df and DG their product is the function defined as (f∙g)(x)=f(x)∙g(x). The domain of (f∙g) is Df ∩ DG.The Quotient of Functions- If f and g are functions with domains Df and DG excluding the values in DG that will make g(x) zero, their quotient is the function defined as The Domain of is Df ∩ DG excluding those values of x that will not defineThe Composite Functions- If f and g are functions with domains Df and DG, the composite fun g) (x) =f[g(x)]. The domainof (f g) (x) consists of all real numbers f in the domain of g for which g(x) is in the domain of f. - new functions that are obtained from existing functions through an operation.
4. 4. Linear Function- Function in the first degree. - is a function of the form f(x)=mx+b. Where m and b are real numbers and m≠0. - The standard form is Ax+By=C and the general form is Ax+By+C=0.Graph of a Linear Function- the graph of a linear function whose domain is a set of all real numbers is a slanting continuous line. - Linear Functions can be graphed by identifying any two points, a slope and a point, y-intercept and a point and through the intercepts.Slope- is a measure of steepness of the line. It is the ration of the "rise" of the line to its "run".The Slope of a Line- If P1(x1,y1 ) and P2(x2,y2) are points of the line representing the linear function f(x)=mx+b, then the slope of the line isThe Slope, The Trend, and the Graph of Linear Function- If the slope is positive, and the graph of a linear function points upward to the right, and the linear function increases all throughout. If the slope is negative, the graph of a linear function points upward to the left, and the linear function decreases all throughout.The y-intercept- The y-intercept is the ordinate of the point of intersection of the graph of a function and the y-axis. A y-intercept of the function f(x) is f (0)The x-intercept- The x-intercept is the abscissa of the point of intersection of the graph of a function and the x-axis. An x-intercept of the function f(x) is the value of x when f(x)=0.The intercepts and the Slope of Linear Function- If the intercepts have the same sign, the slope of the linear function is negative. If the intercepts have different signs, the slope of the linear function is positive.The Slope Intercept form- The linear function has been defined in terms of the equation f(x) =mx+b. - If the graph of a linear function y has a slope m and y-intercept b, then its equation is y=mx+b.The Point Slope form- If the graph of a linear function y has a slope m and passes through the point (x1,y1 ) then its equation is y-y1=m(x-x1).
5. 5. The Two-Point Form- If the graph of a linear function passes through the points (x1,y1) and (x2, y2) then its equation isThe Intercept Form- If the graph of a linear function y has x-intercept a and y-intercept b, then its equation is x/a+y/b=1.The Zero of a linear function- The zero of a linear function f(x) is the real number a such that f(a)=0, a is also called the solution of the equation f(x)=0.The Distance between two points- it is the length of the segment between them. Defined by P1P2=√ (x2-x1)2+ (y2-y1)2.Midpoint of a Segment- is a point that divides it into two congruent segments. It is determined by the formulasDistance between a Point and a Line- is equal to the length of the segment perpendicular to the line whose endpoints are the given point and the point of the perpendicular segment on the line.It is determined by the formulaDistance between two Parallel Lines- is equal to the length of a perpendicular segment whose endpoints join the lines.Quadratic Function- a function is a quadratic function defined by f(x) =ax2+bx+c, where a,b and c are real numbers and a≠0. - A polynomial function in the second degree. - Graph is parabola. - The standard form is f(x) =ax2+bx+c=0 and the vertex form is f(x) =a(x-h) 2+ k.Second Difference Test- A relation f is a quadratic function if equal differences in the independent variable x produce nonzero equal second difference in the function value f(x).Parabolic Function- is another name of Quadratic function because the graph is always a parabola.
6. 6. Vertex of the parabola- the demarcation point between those two parts is a point called the vertex of the parabola.Axis of symmetry- The parabola is symmetric with respect to a line that passes through the vertex this line is the axis of symmetry.The Opening of the Parabola and the Value of a- The graph of f(x)=ax2+bx+c opens upward if a>0. This implies that the parabola has a lowest point at The minimum value of the function is the graph of f(x)=ax2+ bx+c opens downward if a<0. This implies that the parabola has a higest point at if |a|>1, and wider if 0<|a|<1.The Shifting of the Parabola and the Value of h- The graph of a quadratic function f(x)=a(x-h)2+k is shifted c units to the right is f1(x) =a[x-(h+c)]2+k, and is shifted c units to the left if f2(x)=a[x-(h-c)]2+k.The Shifting of the Parabola and the Value of k- The graph of a quadratic function f(x)=a(x-h)2+ k is shifted c units up if f1(x)=a(x-h)2+(k+c), and is shifted c units down if f2(x)=a(x-h)2+(k-c).The Zeroes of a Quadratic Function- The zeroes of a quadratic function f are the x-coordinates of the points where the graph of f intersects the x-axis, if it does.Finding the Zeroes of Quadratic Functions- The zeroes of a quadratic function f(x)=ax2+bx+c are the roots of the quadratic equation ax2+bx+c=0. - The several methods that can be used are Extracting the Square root, Factoring, completing the square and quadratic formula.Imaginary number- a non-real number.Unit Imaginary number- the imaginary i is a number whose square root is -1. In symbols, i=√-1 i2=-1.The Quadratic Formula- The roots of the quadratic equation ax2+bx+c=0 areDiscriminant- used to distinguish the nature of the roots/zeroes of quadratic function.
7. 7. The Discriminant and the Roots of Quadratic Equation- If ax2+bx+c=0, where a, b, and are real numbers, then the discriminant D is D=b2-4ac. If D>0, the two roots are real and unequal. If D=0, th two roots are real and equal. If D<0, the two roots are imaginary and unequal.The Discriminant and the Zeroes of Quadratic Functions- Let f(x)=ax2+bx+c, and the discriminant D=b2-4ac.If D>0, f(x) has two real unequal zeroes and the parabola intersects the x-axis twice.If D=0, f(x) has one real zero and the parabola intersects the x-axis only once.If D<0, f(x) has two imaginary zeroes and the parabola does not intersect the x-axis.Quadratic Inequalities- in one variable are inequalities of the second degree involving the symbols >, <, ≤, ≥ or ≠.Properties of Inequality I. A product is positive when factors are both positive, or both negative. That is, if ab > 0, then a > 0 and b > 0 or a < 0 and b < 0. II. A product is negative when one factor is positive and the other is negative. That is, if ab < 0, then a > 0 and b < 0 or a < 0 and b > 0.Methods in Solving Quadratic Inequalities- The Case Method, The Inspection of Signs Method and The Parabola Method.Graphing Quadratic Inequalities- In order to graph quadratic inequalities, first determine the vertex, then determine the intercepts, graph and perform test (0,0).Circle- is a set of points in a plane, all of which are equidistant from a fixed point called the center of the circle.Radius- the distance from a point of a circle to the center of the circle. The equation of the circle in center-radius form is (x-h)2+(y-k)2=r2. This form is also known as the standard form of the equation of a circle.Circumference- distance around the circle.Diameter- a chord passing through the center of the circle.Chord- distance from any two points in the circumference.Arc- part or portion of the circumference of the circle.Sector- an area in the circle bounded by two radii and arc.
8. 8. Segment- an area in the circle bounded by a chord and an arc.Polynomial- defined by AnXn+An-1Xn-1+A1X+A0n- exponent positive integer.Monomial- a polynomial with one term.Binomial- a polynomial with two terms.Trinomial- a polynomial with three terms.Multinomial- a polynomial with many terms.Degree of a term- in a polynomial in x refers to the exponent of x.Degree of a polynomial- refers to the highest degree among the degrees of the terms in the polynomial.Addition and subtraction of Polynomials- bear in mind that you can only add and subtract similar terms.Multiplication of Polynomials- the distributive property of multiplication and the Laws of exponents are applied. - the methods in multiplying are Horizontal method, FOIL method, Vertical method and Lattice Method.Dividing Polynomials- The laws of exponents and the distributive property are also applied. - there are two ways of dividing polynomials: using the traditional method or the synthetic division method.Synthetic Division- an abbreviated process of dividing. It can be performed by dividing a polynomial in x by a divisor of the form x-c, where c is a nonzero rational number.Polynomial Function- A function is a polynomial function in n defined by p(x)=anxn+an-1xn-1+an-2xn-2+...+a2x2+a1x+a0 where an, an-1, an-2,...,a2,a1 and a0 are real numbers an≠0, and n is a nonnegative integer.Remainder- quantity left after a number or expression can no longer be divided exactly by another number or expression.The Remainder Theorem- If a polynomial function p(x) is divided by x-c, then the remainder is equal to p(c).
9. 9. The Factor Theorem and Its Converse- If p(c) = 0, then x-c is a factor of p(x). Conversely, if x-c is a factor of p(x), then p(c)=0.Fundamental Theorem of Algebra- Every Polynomial equation in one variable has at least one root, real or imaginary.Number of Roots Theorem- Every polynomial equation of a degree n ≥ 1 has exactly n roots.Rational Roots Theorem- If a rational number L/P in lowest terms is a root of the polynomial equation p(x)=anxn+an-1xn-1+an-2xn-2+...+a2x2+a1x+a0 where an, an-1,an-2,...,a2,a1 and a0 are all integers, then L is a factor of a0 and F is a factor of an.Quadratic Surd Roots Theorem- If the quadratic surd a+(square root) b is a root of a polynomial equation, where a and b are rational numbers and √b is an irrational number, then a-√b is also a root of a polynomial equation.Complex Conjugate Roots Theorem- If the complex number a +bi is a root of a polynomial equation with real coefficients, then the complex conjugate a-bi is also a root of the polynomial equation.The zeroes of Polynomial functions- If P(c) =0, then c is a zero of p(x).Corollary- Any rational root of thepolynomial equation equation p(x)=anxn+an-1xn-1+an-2xn-2+...+a2x2+a1x+a0 where an, an-1,an-2,...,a2,a1 and a0 are integers and a factor of A0.Bounds of Zeroes- limit the location of the zeroes of a polynomial function to a certain interval.Upper bound- any number which is greater than or equal to the largest zero of the polynomial function.Lower Bound- any number which is smaller than or equal to the smallest zero of the polynomial function.Bounds of the Zeroes Theorem- suppose p(x), a polynomial function, is divided by x-c using synthetic division, I. if c > 0, and the entries in the third line are positive, some may be zero, then c is an upper bound of the zeroes of p(x). II. If c < 0, and the entries in the third line are alternative in signs, then c is a lower bound of the zeroes of p(x).
10. 10. Descartes Rule of Signs- Let p(x)=0 be a polynomial equation with real coefficients, the leading coefficient an>0, with descending powers of x. 1. The number of positive roots of p(x)=0 is either equal to the number of variations in signs in p(x), or is less than that number by an even counting number. 2. The number of negative roots of p(x)=0 is either equal to the number of variations in signs in p(-x), or is less than that number by an even counting number.Intermediate Value Theorem for Polynomials- If f(x) is a polynomial function with real coefficients, and f(a) and f(b) are opposite in signs, then there exists a value c between a and b such that f(c)=0.Graph of Odd-degree Polynomials- The extreme left and right parts of the graph ofp(x)=an, an-1, an-2,...,a2, a1 and a0 are: I. Increasing, if n is odd and an > 0. II. Decreasing, if n is odd and an< 0.Graph of Even-degree Polynomials- the graph of p(x)=an, an-1, an-2,...,a2,a1 and a0 has: I. decreasing extreme left and increasing extreme right parts, if n is even and an> 0. II.Increasing extreme left and decreasing extreme right parts, if n is even and an<0.Conic section- is a curve formed by the intersection of a plane and a right double circular cone. The equation of a conic section can be written in the form Ax2+Bxy+Cy2+Dx+Ey+F=0.Parabola- is a curve all whose points are equidistant from a fixed point and a fixed line.Focus- the fixed point in a parabola.Directrix- the fixed line in a parabola.Ellipse- is a set of all points in a plane such that the sum of the distances from two fixed points in the plane is constant.Foci- the fixed points in a ellipse.Hyperbola- is the set of all points in a plane such that the absolute value of the difference of the distances of each of these points from two fixed points in the plane is a constant. The fixed points in a hyperbola are also called foci.