[...]... down the y-axis 1-1 6 y=x 2 y=x 2-3 X Fig 1-9 Graphy=x 2-3 Solution: First note that the curve is a parabola with the symmetry attendant to parabolas and it is moved down on the y-axis by the -3 The point x = 0, y = -3 is the key point, being the apex, or lowest point for the curve, and the defining point for the symmetry line, which is the y-axis Now, knowing the general shape of the curve add the point... Because of the symmetry, the parabola must be symmetric about a line halfway between x = 2 and x =-4 , or about the line x = -1 The apex of the parabola is on this x = -1 line so substitute to find the appropriate value of y: f (-1 ) = (-1 + 4)( -1 - 2) = -9 These three points are sufficient to sketch the curve (see Fig 1- 1 1) Y Y=X I Fig 1-1 1 2 +2 ~-8 13 MATHEMATICAL BACKGROUND Before moving on to the graphing... sides: y-3=x2+4x+4 o r y - 3 = ( ~ + 2 ) ~ - ~ = 3 l l l l i x =-2 l X X Fig 1-1 4 2 Now, make the shift in axes with the definitions Y = y - 3 , and X = x + 2 The origin of the "new" coordinate axes is (-2 ,3) Determining the origin from these defining equations helps to prevent scrambling the (-2 ,3) and getting the origin in the wrong place The values (-2 ,3) make X and Y zero and this is the apex of the. .. progress through your calculus course Linear The linear algebraic function (see Fig 1-7 ) is y = mx +b, where m is the slope of the straight line and b is the intercept, the point where the line crosses the y-axis Th~sis not the only form for the linear hction, but it is the one that is used in graphing and is the one most easily visualized X Fig 1-7 1-1 5 Graph the function y = 2x - 3 Solution: This is... about the line x = - 2 If x =-1 or x = - 3 , y = l If x = O or x = - 4 , y = 4 This is sufficient information to sketch the curve Notice, however, in the second solution an even easier means for graphing the bction 2 X x =-2 Fig 1-1 2 Second Solution: The curve can be written i the form y = X 2 if X is defined as n X = x + 2 At x = -2 , X = 0 and the line x = -2 effectively defines a new axis Call it the. .. a completing the square approach Follow M A T H E M A T I C A L BACKGROUND the completing the square approach through the equations below multiplication of the parentheses very carefully 21 Watch the 9x2 - 4y2 - 5 4 - 32y = 19 ~ 9(n2 - 6 ~- 4(y2 + 8 y ) = 19 ) 9 ( - 3)2 - 4(y +4)2 = 19+8 1- 64 = 36 ~ Make the identification X = x - 3 and Y = y + 4 so the function can be written Draw in the new axes... at (3 ,-4 ) When X = O , there are no real Y values When Y = O , X = S Place these points on the graph The asymptotes come out of the y= equation Follow along the rearrangement to find the asymptote lines (See Fig 125.) 9x 2 - 4y2 -5 4x - Y 2 = (9/4)X2 - 9 Y = 4(9/4)X2 -9 For large values of X, Y =(3/2)X Y = +(3/2)X The addition of Fig 1-2 5 these asymptote lines allows completion of the graph Y =-( 3/2)X... is the factorable equation 1-2 Solve x 2 - x - 6 = 0 by factoring Solution: + 2) = 0 The solutions, the values of x that make each parentheses equal to zero, and satis @the factored equation, are x = 3 and x =-2 x2 - x - 6 = 0 4 ( x - 3)(x If the quadratic cannot be solved by factoring, the most convenient solution is by quadratic formula, a general formula for solution of any quadratic equation in the. .. y 2 = 9 - x2 Don't let them 1-2 4 Yl Fig 1-1 7 Graph x 2 - 6 x + 9 + y 2 =16 Solution: At first glance it looks as though a page is missing between problems 1-2 3 and 1-2 4 But if you make the identification that x2 - 6x + 9 is the perfect square of (x - 3) then the equation reads X 2 + Y 2 = 16 if X = x - 3 and Y = y This is the identification 16 CHAPTERI that worked so well for parabolas In the new... is used by some as a convenient construct for drawing the asymptote lines and finding the critical points of the curve Two sides of the rectangle intersect the X-axis at the points where the curve crosses this axis and the diagonals of the rectangle have slopes zf(2/3) 1-3 1 Graph 9x2 -4 y2 - 54x - 3 2 y = 19 Solution1 This is a hyperbola, and the presence of the linear terms indicates it is moved up . Cataloging-in-Publication Data Oman, Robert M. Calculus for the utterly confused / Robert M. Oman, Daniel M. Oman. p. cm. ISBN 0-0 7-0 4826 1-6 1. Calculus- Study and teaching. I. Oman, Daniel. TLFeBOOK Other books in the Utterly Conhrsed Series include: Financial Planning for the Utterly Confrcsed, Fifth Edition Job Hunting for the Utterly Confrcred Physics for the Utterly. the utterly confused calculus student is the inability to set up the problems. In most problems the calculus is easy, the algebra possibly tedious, but writing the problem in mathematical