dgboy765 wrote:
In the xy-plane shown, the shaded region consists of all points that lie above the graph of y=x^2 - 4x and below the the x-axis. Does the point (a,b) (not shown) lie in the shaded region if b<0?
(1) 0 < a < 4
(2) a^2 - 4a < b
Source: Official GMAT Quantitative Review 2016
P. 162 DS #124
Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.)
Dear
dgboy765,
I'm happy to help.
Here are the basic ideas. Consider the graphs of the form y = [some expression of x]. These can include
a) oblique lines,
y = mx + b (e.g.
y = (3/7)x + (5/7))
b) parabolae: e.g.
y = x^2, or y =
x^2 - 4c) higher powers of x: e.g.
y = x^5 - x^3d) square roots:
y = sqrt(x - 3) e) all kinds of other exotic curves:
y = 2^x, or
y = (1 + x)/(1 - x), or etc.
For the purpose of this discussion, the nature of that variety doesn't matter. What I am going to say applies to all graphs of the form y = [expression of x]. I will use the parabola y = x^2 - 4 as my example graph, but what I am saying about this graph applies to any graph of the form y = [expression of x]
The first may be obvious: all the values (x, y) that satisfy the equation y = x^2 - 4 must lie exactly on the graph of y = x^2 - 4. More generally, the graph of any equation is the set of all ordered pairs that satisfy this equation.
Now, think about any point on that line. Imagine we take an (x, y) that live on the graph, and we change the y-coordinate to make it bigger. That would result in a new point that is not on the graph but above the graph. This corresponds to the inequality y > x^2 - 4. Any ordered pair that satisfies this inequality is somewhere above the graph.
Similarly, any ordered pair that satisfies the inequality y < x^2 - 4 lies somewhere below the graph.
In this question, notice that for point (a, b), a is the x-coordinate and b is the y-coordinate. The shaded region shown in the picture is below the x-axis, so y < 0, and it is above the parabola y = x^2 - 4, so it is entirely defined in terms of these two inequalities:
y < 0 and y > x^2 - 4
The prompt already specified that b < 0, and that takes care of the first inequality. Notice that statement #2 is just the second inequality written in terms of a & b instead of x & y. That's why it's sufficient. Once we specify both inequalities, we have specified the shaded region entirely.
On a side note, notice that y = x,
the line through the origin with a slope of 1, is the line that contains all points with equal x- and y-coordinates. The inequality y > x are all the points above y = x, and the inequality x > y are all the points below y = x.
Does all this make sense?
Mike