Desmos for the SAT · Technique 12 of 23

Which Equation Represents This Situation?

Do not build the equation. Type the four choices and keep the one that matches every number the question gave you.

SolvesWhich equation represents this situation?Which equation defines the function?Find the line through two points

Watch the lesson

The whole move worked on one real SAT question, narrated.

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A second example: four candidate functions, all named g in the question, tested against two facts with one list each.

A large group of SAT questions never asks you to solve anything. They describe a situation, or hand you two points, or name a slope and a starting value, and then ask which of four equations represents it. The usual route is to build the equation yourself and see which choice you landed on, which means doing the algebra with no way to check it.

There is no need. The four choices are already on the page, and each fact in the question is something the right equation has to reproduce. So stop building and start checking: give each choice its own name in Desmos, then ask all four for their value at the number the question gave you. One line, four answers, and the ones that miss are gone.

One thing to plan for, because it is the whole difficulty of this move. Testing one fact is usually not enough. The SAT builds at least one wrong choice that agrees with the first thing you check and falls apart on the second, so expect to test twice and keep going until exactly one choice is standing.

Nova

You do not have to build the equation. All four candidates are printed right there, and Desmos can check every one of them against the question's own numbers in about two lines.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps finding the slope, then solving for the intercept.

The question

$h$ is linear, with $h(28)=15$ and $h(26)=22$

Which of the four choices defines h?

The algebra way

Slope first: (22 - 15) / (26 - 28), which is 7 / -2.
Put one point back in: 15 = (-7/2)(28) + b, so 15 = -98 + b.
Solve: b = 113.
Three steps, and the order you subtract in decides the sign of everything after it.

The Desmos way

Type the four choices as a, b, c and d.
Type [a(28), b(28), c(28), d(28)] and read the four values.
Two of them return 15, so type [a(26), d(26)] and keep the one that returns 22.

The payoff: The algebra asks you to produce the answer and gives you nothing to check it against. Testing asks the four answers you were already handed to prove themselves, and a choice that disagrees with a number in the question is wrong no matter how the algebra felt.

Quick reference

What the question gave you, and the line that tests it

The question gives you	What you type	The right choice
A function value, like $h(28)=15$	`[a(28), b(28), c(28), d(28)]`	Returns 15
A point on the graph, like $(2,18)$	`[a(2), b(2), c(2), d(2)]`	Returns 18
A slope, or a per-step increase	`[a(1)-a(0), b(1)-b(0), c(1)-c(0), d(1)-d(0)]`	Returns the number the question named
A starting amount, like $q(0)=14$	`[a(0), b(0), c(0), d(0)]`	Returns 14
A percent change per step	`[a(0), b(0), c(0), d(0)]`, then the same list at 1	Moves by that percent, in that direction

Step by step

The whole move, all 5 steps

Type each answer choice on its own line, giving each one its own name: a(x) = ..., b(x) = ..., c(x) = ..., d(x) = .... Never reuse the letter the question used.
List the facts the question gave you. Two function values, a slope and a point, a starting amount and a rate: each one is a number the right equation has to produce.
Test all four at once on a single line: [a(k), b(k), c(k), d(k)], where k is the x-value the question handed you. Desmos prints the four results side by side, in the order you typed them.
Rule out every choice whose value missed. If exactly one is left, that is the answer.
If two are still standing, test the second fact, listing only the survivors: [a(k), d(k)] at the new value. Keep that list short, because four long decimals collapse into "4 element list" instead of showing you the numbers.

The question

In the linear function $h$ , $h(28)=15$ and $h(26)=22$ .

Which equation defines h?

A $h(x)=-\frac{2}{7}x+23$
B $h(x)=-\frac{2}{7}x+113$
C $h(x)=-\frac{7}{2}x+23$
D $h(x)=-\frac{7}{2}x+113$

Step 1 of 5

Type the first choice, but call it a, not h: a(x) = -2/7 x + 23. Press the right arrow after the 7 to come back out of the fraction before you type the x.

In the calculator: a(x)=-2/7→x+23

Renaming is not optional here. All four choices define h, and Desmos cannot hold four different definitions of the same letter.

Now try it on a real SAT question

Head or scratch paper. The sentence is already the equation: a daily rate times the number of days, plus a fee that happens once. Writing that down is faster than typing four choices, and knowing which questions to read instead of type saves as much time as the typing does.

Sean rents a tent at a cost of 11 dollars per day plus a onetime insurance fee of 10 dollars.

Which equation represents the total cost c, in dollars, to rent the tent with insurance for d days?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

Straight from Desmos

Desmos Help Center: Functions and Function NotationNaming a function and calling it, which is the whole mechanic here.

The Desmos trap: when to close the calculator

Give each choice its own name. All four are usually written as the same letter, and typing them that way breaks it: Desmos says "You have defined h in more than one place" and stops returning values for any of them.
One test is usually not enough. Every question on this page has at least one wrong choice that matches the first fact, so keep testing facts until exactly one choice is left standing.
Pick your test value with a little care. In the tent question both $11d+10$ and $10d+11$ return 21 at one day, and only split at two days. A value where the numbers in the choices can swap places without changing the answer tells you nothing.
When the sentence translates directly, translate it. A onetime fee plus a daily rate is two seconds of reading, and typing four choices is slower.
The mirror image of this question, one equation and four numbers to test, is Build a Table to Test Several x-values at Once: the same move, with the choices down a column instead of across a list.
If a question gives you the graph rather than the numbers, read two clean points off it first, then test the choices against those.

If the calculator does something you did not expect

Keep the second list short. Four long decimals collapse in the panel into "4 element list", so test only the choices that are still alive.

Nova

Read the sentence first though. If it says 11 dollars per day plus a onetime fee of 10, you can write $c=11d+10$ faster than you can type four choices. Save this for the ones where the choices look alike and the algebra has a sign in it you could drop.

All Desmos techniques

Put this move into real practice

"Which Equation Represents This Situation?" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

Practice free

SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.