Calculus: Analytic Geometry and Calculus, with Vectors

1.5 Equations, statements, and graphs 37

x

Y

0

0

±1T

TW^1

+1

T^1

±1

1

± 43

I

±2 ± ±3

9

we are easily led to the correct conclusion that the graph of y = x2 is the
curve shown in Figure 1.55. It should be noted that the graph contains
no point (x,y) for which y < 0; if y < 0, there is no x for which y = x2.
The y axis is an axis of symmetry of the graph, because if (x,y) is a point
on the graph, then the point (-x,y) is also on the graph.
The graph of the equivalent equations

(1.561) xy = 1,

1

Y x

is more complex. As we shall see later, the graph is a rectangular
hyperbola. It is easy to add more items to the table

x

Y

1

10

1

T^1210

2 1 1

and to sketch the part of the graph to the right of they axis in Figure 1.56.
A similar table in which x and y are both negative enables us to sketch
the part lying to the left of the y axis. The graph contains no point (x,y)
for which x = 0 or y = 0. The x and y axes are not axes of symmetry,
but the origin is a center of symmetry, because if (x,y) is a point on the
graph, then the point (-x,-y) is also on the graph.
The symbol [x] represents, when we are properly warned, the greatest
integer in x, that is, the greatest integer n for which n < x. Thus

[1.99] = 1, [3.14] = 3, [0.25] = 0, [-0.25] = -1, [-3.01] = -4, and

[2] = 2. It is not difficult to show that the graphs of y = [x] and of the
saw-tooth function y = x - [x] - have the forms shown in Figures
1.57 and 1.571.
Trigonometric functions will appear very often in our work, and there
will be very many times when we must know the natures of the graphs of

Figure 1.57

-3-2

y

3

-1

2

1

O -2

0

1 2 3 4 x

Figure 1.571

ra^1

y

-2 -1^01 2 3 x