Two things one can say

1) the line cuts y-axis in negative portion, so y-intercept is negative.
Thus put x as 0 and see which all give you the value of y as negative.
(A) $y = 4x + 4........y = 4*0 + 4.....y = 4$....NO
(B) $y = 4x - 4........y = 4*0 - 4.....y = -4$....Yes
(C) $y = x - 6........y = 0 - 6.....y = -6$....Yes
(D) $y = x + \frac{1}{2}........y = 0 + \frac{1}{2}.....y = \frac{1}{2}$....No
(E) $y = -x - 3........y = -0 - 3.....y = -3$....Yes
so Only B, C and E are left

2) the line cuts x-axis in positive portion, so y-intercept is positive.
Thus put y as 0 and see which all give you the value of y as positive.
(B) $y = 4x - 4........0 = 4*x - 4.....4x = 4....x=1$....Yes
(C) $y = x - 6........0 = x - 6.....x = 6$....Yes
(E) $y = -x - 3........0 = -x - 3.....x = -3$....No

So we are left with 2 options B and C.
Now since the figure is drawn to scale x and y are nearly equal without any sign that is |x| ~ |y|
In C, y = -6 and x = 6 so |-6|=|6| .... yes
In B, y = -4 and x = 1 so |-4|is not equal to |1|, so eliminate

C

Since there are no numbers on the graph, the exact equation of the line
cannot be determined, but the line has a positive slope (it points upward when
reading from left to right) and a negative y-intercept (it crosses the y-axis
below the origin). All of the answers are already in slope-intercept form (y =
mx + b, where m = slope and b = y-intercept). Choices (A), (B), (C), and (D)
have positive slope. Of those, only choices (B) and (C) have a negative y-intercept.
Now, is the slope closer to positive 4 or positive 1? A slope of 1 makes 45°
angles when it cuts through the x and y axes, and this figure looks very much
like it represents a slope of 1. A slope of 4 would look much steeper than this
picture. Note that xy-planes are drawn to scale on the GRE, and units on the x-axis and on the y-axis are the same, unless otherwise noted.
The correct answer is (C). Note that the GRE would only give questions in
which the answers are far enough apart that you can determine the intended
answer

Solution:

We can solve these type of questions based on co-ordinate geometry without actually writing anything using the below steps:

1. Find the slope (positive or negative)
Slope is positive when it passes from quadrant 1 & 3 atleast and negative when it passes from quadrant 2 & 4.

In the above it passes from quadrant 1, 3 & 4. Therefore the slop is positive. Eliminate option E as it has a negative slope.

2. Find the y intercept.
Y intercept is nothing but the point which lies on Y axis. In the above diagram although we are not sure what the number is, but we know it is sure negative.

Therefore, the y intercept has to be negative. Eliminate option A & D

Now, I can randomly substitute any negative value for y in B & C

Eg. Let y be -3

(B) y=4x−4 i.e -3=4x-4; 4x=-1; x=-1/4
(C) y=x−6 i.e -3=x-6; x=3

Thus, we know when y has a negative value x is positive and we can eliminate option B for this reason.

Only option C fits.

This might look a bit tedious, but practicing this on a few questions will make this easier.

Hope this helps!