Welcome to the last lesson of our function series. Before you start, check out parts I, II, and III. Today, we’re going to discuss table and/or graph-related function problems. These can come in many forms, so we will discuss some of the most common and useful ones, which should serve you well in your SAT studies!
The Table Function Problem
This one is an SAT classic: the type of problem in which you’re given a table of x and f(x) values and have to answer the question based on that. Most often, the question will be something like, “which of the following equations could represent f(x)?” To answer these, you will want to pick a pair of x and f(x) values from the table and plug them into each option until you find one that works.
Which of the following equations could represent f(x)?
(A) f(x) = x +2
(B) f(x) = x-2
(C) f(x) = 3x -2
(D) f(x) = 5x – 2
(E) f(x) = 6x -2
Let’s pick a pair of values – say, (2, 8) – and test it out for each answer choice to see which one works.
(A) 8 = 2 + 2 ? No, 8 does not = 4.
(B) 8 = 2 – 2? No, 8 des not = 0.
(C) 8 = 3(2) – 2? No, 8 does not = 4.
(D) 8 = 5(2) – 2? Yes, 8 = 8.
(E) 8 = 6(2) – 2? No, 8 does not = 10.
It’s possible that one pair of values will work for more than one equation. If that’s the case, then keep those couple of equations around and test out another set of values until you can find which equation works for all of the pairs of values in the table.
The Function Graph Problem
These kinds of problems involve a randomly-shaped graph of a function (i.e. not a function graph shape that you will recognize like a parabola or an absolute value graph – just some curves and/or lines – see below for an example), and will ask you some sort of question or questions related to your ability to read and interpret that graph.
One key thing – perhaps the key thing – to remember here is that f(x) always graphs on the y axis. So, if they ask you something like, what is the value of f(5), then you need to find what the y value of the graph is when x = 5.
Be careful of switching your x’s and y’s – that is a very common mistake on these problems.
Similarly, if the question asks you how many times f(x) equals a given number (e.g. “for how many values of x does f(x) equal 2?” or “for how many values of x does f(x) intercept the x axis?”), then check the graph to see how many times the y value is equal to that number. If it asks you for intervals where f(x) is positive or negative, look for intervals where the graph falls above or below the x axis. And so on. Keep your variable straight and don’t get scared by the strange graph shapes, and these problems are entirely doable!
The Graph – Ordered Pair Problem
These types of problems might come in a variety of forms, but they all involve the crucial idea of the “marriage” or the function and the ordered pair notation.
Sometimes, you’ll be given a problem something like this one:
In the linear function g(x), g(2) = 10 and g(6) = 22. Given that, what is the value of g(-2)?
The key here is to see that you can convert these function-equations into ordered pairs. If g(2) = 10, then you know the line passes through the point (2,10). If g(6) = 22, then you know the line passes through the point (6,22). Once you have these two ordered pairs, you can figure out anything you need by plugging those coordinates into the slope formula and then plugging the slope and one of those ordered pairs into the y = mx + b equation to find the value of b. Finally, you can plug in the x value they’re asking about — -2, in this case – to find what the corresponding g(x) value will be there. Problem solved!
Sometimes, you’ll be given problem involving a function graph and a shape that intersects that function graph at a couple of points. In that kind of a case, the same principle applies, in that you should try to come up with an ordered pair of (x,y) coordinates and then use that to figure out whatever you need to figure out.
In the figure above, square ABCD intersects the graph of y = h(x) at points C and D. The area of square ABCD is 36. h(x) = k x², where k is a constant, then what is the value of k?
Fist thing: label that diagram, using what you know. You know the area of the square is 36, so you can label each of the sides as 6 units long. You know that parabolas are symmetrical, so that means that the horizontal distance from the center line to points A, B, C and D is 3. That means you now have ordered pairs for the two points that are shared between the function graph and the square: point C is at (3, -6) and point D is at (-3, -6). Now that you have those, choose one of them and plug it into the given equation to find k. Problem solved!
That’s all for our function series. Hopefully this has been helpful in terms of breaking down some of the most common function problems and making them make more sense. See you next time – and until then, happy function-ing!
Alice Rothman-Hicks is a Veritas Prep SAT 2400 instructor. Since graduating from Columbia University (Magna Cum Laude, Phi Beta Kappa), Alice has been teaching and tutoring test prep, helping students achieve their own academic successes. She scored a 2350 on the SAT.