ALGEBRAIC CURVES Prepared by: Prof. Teresita P. Liwanag – ZapantaB.S.C.E., M.S.C.M., M.Ed. (Math-units), PhD-TM (on-going)
SPECIFIC OBJECTIVES At the end of the lesson, the student isexpected to be able to:• define and describe the properties of algebraiccurves• identify the intercepts of a curve• test the equation of a curve for symmetry• identify the vertical and horizontal asymptotes• sketch algebraic curves
ALGEBRAIC CURVES An equation involving the variables x and yis satisfied by an infinite number of values of xand y, and each pair of values corresponds to apoint. When plotted on the Cartesian plane, thesepoints follow a pattern according to the givenequation and form a definite geometric figurecalled the CURVE or LOCUS OF THE EQUATION.
The method of drawing curves by point-plotting is a tedious process and usually difficult.The general appearance of a curve may bedeveloped by examining some of the properties ofcurves.PROPERTIES OF CURVESThe following are some properties of an algebraiccurve:1. Extent2. Symmetry7.Intercepts8.Asymptotes
1. EXTENT The extent of the graph of an algebraic curveinvolves its domain and range. The domain is theset of permissible values for x and the range is theset of permissible values for y. Regions on which the curve lies and which isbounded by broken or light vertical lines throughthe intersection of the curve with the x-axis. To determine whether the curve lies aboveand/or below the x-axis, solve for the equation of yor y2 and note the changes of the sign of the righthand member of the equation.
2. SYMMETRY Symmetry with respect to the coordinate axesexists on one side of the axis if for every point of thecurve on one side of the axis, there is acorresponding image on the opposite side of the axis. Symmetry with respect to the origin exists ifevery point on the curve, there is a correspondingimage point directly opposite to and at equaldistance from the origin.
Symmetry with respect to the origin exists ifevery point on the curve, there is a correspondingimage point directly opposite to and at equal distancefrom the origin.
Test for Symmetry1. Substitute –y for y, if the equation is unchangedthen the curve is symmetrical with respect to thex-axis.2. Substitute –x for x, if the equation is unchangedthe curve is symmetrical with respect to the y- axis.3. Substitute – x for x and –y for y, if the equation isunchanged then the curve is symmetrical withrespect to the origin.
Simplified Test for Symmetry1. If all y terms have even exponents therefore thecurve is symmetrical with respect to the x-axis.2. If all x terms have even exponents therefore thecurve is symmetrical with respect to the y-axis.3. If all terms have even exponents therefore thecurve is symmetrical with respect to the origin.
3. INTERCEPTS These are the points which the curve crossesthe coordinate axes.a. x-intercepts – abscissa of the points at which thecurve crosses the x-axis.b. y-intercepts – ordinate of the points at which thecurve crosses the y-axis.
Determination of the InterceptsFor the x-intercept For the y-intercepta. Set y = 0 a. Set x = 0b. Factor the equation. b. Solve for the valuesc. Solve for the values of x. of y.
4. Asymptotes A straight line is said to be an asymptote of acurve if the curve approaches such a line more andmore closely but never really touches it except as alimiting position at infinity. Not all curves haveasymptotes.Types of Asymptotes6.Vertical Asymptote7.Horizontal Asymptote8.Slant/Diagonal Asymptote
Steps in Curve Tracing1. If the equation is given in the form of f( x, y) = 0,solve for y (or y2) to express the equation in a formidentical with the one of the four general types ofthe equation.2. Subject the equation to the test of symmetry.3. Determine the x and y intercepts.4. Determine the asymptotes if any. Also determinethe intersection of the curve with the horizontalasymptotes.Note: The curve may intercept the horizontalasymptotes but not the vertical asymptotes.
5. Divide the plane into regions by drawing lightvertical lines through the intersection on the x-axis.Note: All vertical asymptotes must be considered asdividing lines.6. Find the sign of y on each region using thefactored form of the equation to determine whetherthe curve lies above and/or below the x-axis.7. Trace the curve. Plot a few points if necessary.