Calculus: Analytic Geometry and Calculus, with Vectors

1.2 Slopes and equations of lines 13

2 Plot the points 1I(-7,-1), B(-5,O), C(-3,1), D(-1,2), and E(S,5).
(These points all lie on a line, and the figure should not contradict this fact.)
3 Plot the points (6,2), (2,6), (-6,2), (-6,-2), (-2,-6), (2,-6), and
(6,-2). (These points all lie on the circle with center at the origin and radius
N/:F01 and the figure should not contradict this fact.)
4 Three vertices of a rectangle are (4,-1), (-6,-1), and (4,5). Sketch the
rectangle and find the coordinates of the fourth vertex.
5 For each of several values of x, plot the point P(x, 2 - x). What can be
said about the resulting set of points?
(
6 Plot the points Pl(x1,Y1), Pz(xz,Yz), Q(xl + xz, Yl + yz),

xi ± xz

R ` 2

Y1 + Y2) and makean observation about the figure obtained by drawing the

2

line segments from these points to each other and to the origin when

(a) xi=6,Y1=0,xz=0,yz=4 (b) xi=2,y1=5,xz=6,yz=3

(c) xi=-2, Y1=-4,xz=7,Y2=1 (d) xi= -1, Y1= 1,X =1,Yz=0

.Ins.: The figure is a parallelogram together with its diagonals. Remark: Invest-

ing time in a good problem can produce dividends. Observe and remember that

the mid-point of the line segment joining (xi,yi) and (xz,yz) is (x1

2

xz, Y1

More information about such matters will appear in the next chapter.^2 Yz

7 Draw the triangle having vertices at the points P1(-3,1),P2(7,-1),P3(1,5).

For each k = 1, 2, 3, let mk be the slope of the

side opposite the vertex Pk. Work out the

slopes shown in Figure 1.291 and observe that

=

the answers look right. jI 1\ m1=-1

8 Show that the equation of the line P1P3 P,

of the preceding problem is y = x + 4. Find n x

the x coordinate of the point on this line for

which y = 0. 11ns.: The answer is -4, and

inspection of Figure 1.291 shows that this answer

looks right. Figure 1.291

9 When numerical values are assigned to the coordinates (xi,yi) of a point P1

and to m, it is possible to plot the point P1, to sketch the line L through P1 having

slope m, and (provided m 5,5 0) to estimate the x coordinate xo of the point

(xo,0) on L for which y = 0. It is then possible to find the equation of L and

determine xo algebraically. Do all this and make the results agree when

(a) (xi,yi) _ (1,2), in = 1 (b) (xi,yi) (1,2), m = -1

(c) (x1,Y1) _ (-2,1), m = 1 (d) (xi,yi) (-2,1), m = -1

(e) (xi,yi) _ (-2,-3), m = (f) (xi,yi) _ (4,-2), in = 2

(g) (xi,yi) _ (-1,4), m = (h) (x1,Y1) _ (1,1), in = I

10 When numerical values are assigned to the coordinates (x1,y1) and (x2,Y2)

of two points P1 and F2, it is possible to plot these points, to sketch the line L

through them, and (except when L is parallel to the x axis) to estimate the x

coordinate of the point (xo,0) on L for which y = 0. It is then possible to find the