1)The Locus of an Equation

*locus of an equation -- this represents a graph of a curve where all the points in it will make the equation true. It may be a line, a circle or any other curve. The graph or curve is the geometric representation of an equation while the equation is the analytical representation of a curve.

2)Intercepts on the Axes

*x-intercept -- the point at which a curve meets the x-axis; it is the value of x when y is set to zero. *y-intercept -- the coordinate of a point where the locus of an equation touches the y-axis; it is the value of y when x is equal to zero.

3)Symmetry

*symmetric about the x-axis -- a certain curve is symmetric about the x-axis when the x-axis exactly "mirrors" on both sides (up and down) the locus of the curve; determined when the negative of y (or -y) is replaced for y in the equation; if the equation does not change then we have a symmetry about the x-axis. *symmetric about the y-axis -- x is changed to -x and the equation remains the same. *symmetric about the origin -- simultaneous change of x and y to -x and -y respectively does not change the original equation.

4)Functions

*function -- there exists a unique relationship between the independent and dependent variable. If y is a function of x, then y depends on the value of x; x is the independent variable and y is the dependent variable. There is a unique value for y for every value of x; y maybe the same for different values of x, but y can never take on different values for the same value of x. The independent variable is also called the argument of a function.

5)Graph of a function

graph of a function -- made by plotting the points in a rectangular coordinate where the argument of the function (the independent variable, say x) becomes the abscissa and the function (dependent variable, say y) becomes the ordinate. In this case y = f(x) (read: "y equals f of x"). The plot drawn is also equivalent to a locus of an equation. *vertical line test -- a test made to determine if a graph of a curve is a function; done by passing an imaginary vertical line along the curve. If the vertical line touches the curve two or more times, the graph is not a function.

6)Factorable Equations

*Factorable equations -- means that if an equation can be expressed as factors of two or more expressions and if the right side of the equation is zero, the factors may be equated to zero. As a result the equation is basically branched into several curves each of which is a factor of the original equation and equated to zero.

7)Classification of Curves

*algebraic curve -- the equation is represented by a polynomial in terms of x and y and is set to zero. *transcendental curve -- anything not algebraic; contains transcendental functions such sine, cosine, logarithm and exponential. To classify a curve it has to be first expressed in terms of rectangular coordinates.

8)Degree of an Algebraic Curve

*degree -- the degree of an algebraic is the degree of the polynomial equation representing the curve; the degree of the polynomial is the highest order in a polynomial. The order is found by looking at the exponents. If there is one variable in the term, then the order is the exponent of that variable. If there are more than one variable in the term, then the order is the sum of the exponents of the variables.

9)Points of Intersection of Two Curves

*points of intersection -- this is easily found by solving the equations simultaneously by algebraic means; the points found satisfy the two equations of the curves.

10)Path of a Moving Point

*path of a moving point -- the graph of the equation may be described by a certain condition; the point moves in such a way that it obeys the law stated; the equation for the path is taken from the application of the concepts of distance and other formulas of analytic geometry; the coordinates are represented by either (x,y) or (r, theta).

10)Loci Defined Geometically

*loci defined geometrically -- from a certain curve or figure, we can derive an equation of the locus; this is done by taking note of the properties observed in the curve.