Unit 46: Functions with Coordinate Box

Some of the odder question types the SAT chooses to include are the ones with function boxes.  You may have never seen these before prior to an SAT test, and at first, they are rather confusing.  Fortunately, once you figure out what’s going on they are very straightforward.

When you see a box like this on a problem:

(x)(f(x))
03
21
40
5-2

You will likely be asked to do one of three things.

  • Determine which answer could be the actual function
  • Determine which answer could be a factor of an unknown function
  • Determine what the function returns when a specified value is inputted

Note the box can also be horizontal, or contain different variable letters:

(n)1234
(f(n))-2147

Or contain two functions

(x)(w(x))(t(x))
1-1-3
23-1
341
433
5-15

Strategy:

  1. Identify the input (usually x), and the output (usually f(x) )
  2. If helpful, you can write each pair in coordinate form (x, y) , with the f(x) as the value. 

Let’s take a look at an example.

(n)1234
(f(n))-2147

The table above shows some values of the linear function f. Which of the following defines 

A) (f(n)=n−3 )
B) (f(n)=2n-4)
C) (f(n)=3n-5)
D) (f(n)=4n-6)

What this box is telling us, is that when we plug 1 into the function, it will return -2.

To find our function, we start plugging 1 into the answer choices.  A) returns -2.  Great!  But wait.  So do B), C), and D)!  How can we find out what the real function is?  We can do so by plugging in a second value.  Let’s use 2, which should return 1.  Now A) fails the test, since it returns -1.  B) does too.  C) works with a second point, so we know we have our right answer.  Remember that these are lines (linear equations), and we can define a line with two points.

In another variation of the “find the function” problem, the digital SAT might use variables for constants. For example, they may ask a question like:

Function (f) is defined by (f(x)=ax+b), what is the value of (a-b)?

For these type of questions, you again use the same strategy above to find the function, and then plug in those numbers to solve for what they are asking for. Make sure to read the questions carefully!

Here’s another one that might be confusing at first.

(x)(f(x))
03
21
40
5-2

The function is defined by a polynomial. Some values of (x) and (f (x)) are shown in the table above. Which of the following must be a factor of (f (x))?

A) (x-2)
B) (x-3)
C) (x-4)
D) (x-5)

If we don’t even have a function, how can we factor it?  Remember that when you factor a polynomial, you put everything on one side and set it equal to 0.  Then, you find which x values will make your function return 0.  It is the same thing here.  We want to see where the function returns 0.  We simply find the 0 in the f (x) column, and look at which x gives us that.

Notice that they slyly included a 0 in the x column.  Don’t fall for it!  That gives us our intercept, not our zeros for the equation.

So we know when we plug in 4 into our mystery equation, it spits out 0.  Therefore, (x – 4) MUST be a factor.  None of the other answers give us 0 when we plug in 4 for x.