SAT Mathematics Practice Test 4

Question : 28 of 58

Marks: +1, -0

Line

l

in the xy-plane contains points from each of Quadrants II, III, and IV, but no points from Quadrant I. Which of the following must be true?

The slope of line $l$ is undefined.
The slope of line $l$ is zero.
The slope of line $l$ is positive.
The slope of line $l$ is negative.

Validate

Solution:

The quadrants of the

xy-

plane are defined as follows: Quadrant I is above the

x-

axis and to the right of the

y-

axis; Quadrant II is above the x-axis and to the left of the y-axis; Quadrant III is below the

x-

axis and to the left of the y-axis; and Quadrant IV is below the x-axis and to the right of the

y-

axis. It is possible for line

l

to pass through Quadrants II, III, and IV, but not Quadrant I, only if line

l

has negative

x-

and

y-

intercepts. This implies that line l has a negative slope, since between the negative x-intercept and the negative

y-

intercept the value of

x

increases (from negative to zero) and the value of

y

decreases (from zero to negative); so the quotient of the change in y over the change in

x

, that is, the slope of line

l

, must be negative.
Choice A is incorrect because a line with an undefined slope is a vertical line, and if a vertical line passes through Quadrant IV, it must pass through Quadrant I as well. Choice B is incorrect because a line with a slope of zero is a horizontal line and, if a horizontal line passes through Quadrant II, it must pass through Quadrant I as well. Choice C is incorrect because if a line with a positive slope passes through Quadrant IV, it must pass through Quadrant I as well.

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