Part Two 319Answer 14-3
A horizontal line through point P has the equation y=−3. A horizontal line running through
pointQ has the equation y=−1. A horizontal line through point R has the equation y= 4.
All three of these lines represent functions of x, because they all pass the vertical-line test. A
movable vertical line (that is, a line parallel to the y axis) will never intersect any one of those
three horizontal lines at more than one point.
Question 14-4
Imagine vertical lines running through each of the three points in Fig. 20-3. What are the
equations of these lines? Do any of them represent functions of x?
Answer 14-4
A vertical line through point P has the equation x=−5. A vertical line running through point
Q has the equation x= 0. A vertical line through point R has the equation x= 2. None of these
lines represent functions of x, because they all fail the vertical-line test. A movable vertical line
that intersects any of those three lines at one point will intersect it at infinitely many other
points as well.
Question 14-5
Suppose the coordinates of each of the points in Fig. 20-3 are transposed; that is, the order
of each ordered pair is reversed. Call the new points P,Q, and R*. In which quadrants will
these new points appear?
Answer 14-5
The coordinates of P will be (−3,−5), so P will be in the third quadrant, just as is P. The
coordinates of Q will be (−1, 0), so it will not be in any quadrant, but on the negative x axis.
The coordinates of R will be (4, 2), so it will be in the first quadrant, just as is R.
Question 14-6
Suppose both of the coordinates of each of the points in Fig. 20-3 are multiplied by −1. Call
the new points −P,−Q, and −R. In which quadrants will these new points appear?
Answer 14-6
The coordinates of −P will be (5, 3), so −P will be in the first quadrant. The coordinates of
−Q will be (0, 1), so it will not be in any quadrant, but on the positive y axis. The coordinates
of−R will be (−2,−4), so it will be in the third quadrant.
Question 14-7
Figure 20-4 shows the graphs of four different relations between x and y in Cartesian coordi-
nates. Which of these relations are functions of x? How can we tell?
Answer 14-7
Using the vertical-line test, we can see that curve E and line G both represent functions of x, at
least within the viewing region of this graph. They both pass the test. However, neither curve
F nor curve H are functions of x, because they both fail the vertical-line test.