With that in mind, let’s just work our way around the figure, starting with C and going clockwise.
CD is 10 units.
DE is 10 units.
EF is 10 units.
FA is 20 units.
From A to the indented corner on the left is 10 units. (Remember, the diagram in the original question isn’t drawn to scale.)
From the indented corner on the left to B is 10 units.
BC is 20 units.
So if we add that all up, we get 10 + 10 + 10 + 20 + 10 + 10 +20 = 90 units.
So the answer is 90.
Notice, once more, that none of the arithmetic in this question was difficult. The difficulty came from figuring out what was being asked, and we did that by reading carefully and knowing to pay attention to the way the College Board draws diagrams.
Page 418, Question 18
The diagram in this question is drawn to scale, so we want to take note of that. Unfortunately, it doesn’t have any kind of scale labeled on it, but we may be able to work around that if it turns out we need to.
The question is also talking about 3 different variables: x, k, and a. This will give a lot of people trouble, because most test-takers who treat SAT Math like school math will think they need to figure out the values of those variables.
But if we look carefully, we see that there’s not enough information to figure out k. Instead of worrying about that, we just realize that k must not matter in the question. Easy enough.
Let’s focus on the end of the question. It asks for the value of a, but all we know about a from the rest of the question is that it’s positive and that g(a – 1.2) = 0.
A lot of people will want to plug a – 1.2 in for x in the original function equation, because that’s the typical knee-jerk reaction that would be appropriate in a school situation. But if we consider that move for a moment, I think we’ll see that it’s not likely to help us very much. We’d end up with this expression, which would be very ugly:
g(x) = k((a – 1.2) + 3)((a – 1.2) – 3)
This doesn’t really look promising, from an SAT Math standpoint. (Now, it’s true that the last question on the grid-ins is often more complicated than other SAT Math questions, but I’d still be reluctant to go about expanding and condensing that hideous expression unless absolutely necessary.)
So let’s try a different tack.
One thing that’s kind of interesting is the fact that we were told the value of g(a – 1.2) is zero. Zero is a unique number with unique properties, particularly when it comes to functions and graphs. On a graph, the zeros of a function are the points where the function crosses the x-axis; in a function equation, we find the roots of an expression by setting the factors of the expression equal to zero.
So let’s think about that for a second. In the equation, the ways to make g(x) come out to zero are to set either (x + 3) or (x – 3) to zero. (k can’t be zero, if it were, every single g(x) value would also be zero, and that isn’t what the graph shows.)
So if either (x + 3) or (x – 3) is zero, that means the g(x) value would be zero when x is either -3 or 3. And that fits with the graph, because the places where the function equals zero seem to be at x = -3 and x = 3. So we know that g(-3) = 0, and g(3) = 0. And that should be kind of our eureka moment: if g(-3) and g(3) are both zero, and g(a – 1.2) is zero as well, than that must mean that (a – 1.2) is the same thing as either (-3) or (3)!
That means that a is either 4.2, or -1.8. But the question tells us a must be positive, so that means it’s 4.2.
Once again, the College Board has created a misleading question out of very basic ideas. In this question, we needed to know that the x-intercepts of a graph are the places where its y-value is zero; we needed to know that 4.2 is bigger than zero but -1.8 is not; we needed to know that 4.2 – 1.2 is 3. That’s about it, from a math standpoint, and none of those ideas is very complicated on its own. I would bet that over 90% of the people who took this test knew every single one of those facts, but they missed the question because they didn’t realize it was asking about those facts in the first place.
So the challenge came from the sheer weirdness of the question, and we had to use our reading skills to understand what was being asked, and our thinking skills to realize that the idea of the zeroes of the function was very important.
Conclusion
By now, you’re probably beginning to develop a strong appreciation for the strange way the College Board designs SAT Math questions. In the next section, we’ll continue to develop your understanding by exploring a large selection of questions from the Blue Book that most people have some trouble with.
Video Demonstrations
If you’d like to see videos of some sample solutions like the ones in this book, please visit http://www.SATprepVideos.com. A selection of free videos is available for readers of this book.