Desmos for the SAT · Technique 23 of 23

Use Regression to Find an Unknown Constant

Swap the equals sign for a tilde and Desmos solves for the letters. One setup line is required, and leaving it out fails silently.

SolvesUnknown constantsEquivalent expressionsCompleting the squareVertex form

Watch the lesson

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

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One tilde line solves for the constant, with the setup line that fails silently if it is left out.

This is a regression move pointed at algebra instead of data, and it is not really a graphing move at all. Desmos has a regression engine, and a regression is "find the values of these letters that make two sides agree". That is also exactly what an equivalent-expressions question asks.

So when a question says $2x^{2} - 8x + 6$ can be rewritten as $2(x - h)^{2} + k$ and asks for k, you do not have to complete the square. Type both sides with a tilde (~) where the equals sign goes, keeping the unknown constants as letters, and Desmos reports what each letter has to be.

One line makes or breaks it. The tilde needs data to fit against, so you must give it a list of x-values first: x_1 = [1, 2, 3, 4, 5], and every x in the equation has to be written as $x_{1}$ . Leave that line out and Desmos does not complain. It returns numbers that look perfectly reasonable and are wrong. That failure is silent, which is why it leads this lesson instead of sitting in a footnote.

Nova

Desmos will solve for an unknown constant if you ask it the right way: put a tilde where the equals sign goes, leave the constants as letters, and it tells you what they have to be. Completing the square, matching coefficients, finding a and k, all of it becomes one line. There is exactly one setup line you cannot skip, and this lesson leads with it because skipping it fails silently.

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 completing the square.

The question

$f(x) = 2x^{2} - 8x + 6$ , rewritten as $f(x) = 2(x - h)^{2} + k$

What is the value of k?

The algebra way

Factor the 2 out of the first two terms: 2(x^2 - 4x) + 6.
Halve the -4 and square it to get 4, then add and subtract it inside: 2(x^2 - 4x + 4 - 4) + 6.
Pull the -4 out through the 2, which turns it into -8: 2(x - 2)^2 - 8 + 6.
Combine: k = -2. The step where the -4 becomes -8 is where this goes wrong.

The Desmos way

Type x_1 = [1, 2, 3, 4, 5].
Type 2(x_1 - h)^2 + k ~ 2x_1^2 - 8x_1 + 6.
Read k = -2 off the screen.

The payoff: Completing the square disappears entirely, along with the factor-out-and-multiply-back step that causes most of the errors. The tilde asks the same question the algebra asks, and answers it numerically.

Quick reference

What the tilde solves, and what it hands back

The question gives you	What you type	What you read
$3x^{2} + bx$ is equivalent to $x(3x - 5)$	`3x_1^2 + b x_1 ~ x_1(3x_1 - 5)`	$b = -5$
$x^{2} + 6x + 2$ as $(x + h)^{2} + k$	`(x_1 + h)^2 + k ~ x_1^2 + 6x_1 + 2`	$h = 3$ , $k = -7$
$3x^{2} + 12x + 7$ as $a(x - h)^{2} + k$	`a(x_1 - h)^2 + k ~ 3x_1^2 + 12x_1 + 7`	$a = 3$ , $h = -2$ , $k = -5$
$x^{2} - 7x + 10$ as $(x - a)(x - b)$	`(x_1 - a)(x_1 - b) ~ x_1^2 - 7x_1 + 10`	$a = 5$ , $b = 2$ , in either order

Step by step

The whole move, all 5 steps

Type the list of x-values first, on its own line: x_1 = [1, 2, 3, 4, 5]. Five values is plenty.
Type the identity with a tilde in place of the equals sign, writing every x as x_1 and leaving the unknown constants as plain letters. Type x then 1 with no underscore and Desmos closes the subscript itself. If you type the underscore, the cursor stays inside the subscript and the next character lands there; press the right arrow to leave it.
Read the fitted values Desmos prints under the expression, in the block headed REGRESSION PARAMETERS, and pick out the letter the question asked for.
Sanity-check by eye: put the original and your completed version on the graph and confirm the curves sit on top of each other.
If two constants play symmetric roles, like the two roots in (x - a)(x - b), Desmos may assign them either way round, so only trust the pair, not which letter got which.

The question

The function f is defined by $f(x) = 2x^{2} - 8x + 6$ , and can be rewritten as $f(x) = 2(x - h)^{2} + k$ .

What is the value of k?

Step 1 of 4

Type the list of x-values first: x₁ = [1, 2, 3, 4, 5]. Remember the subscript happens on its own, so you type x then 1 with no underscore.

In the calculator: x_1=[1,2,3,4,5]

Type this line first, every time. It is the data the tilde fits against, and without it Desmos still prints numbers, just wrong ones.

The actual values do not matter much. Any handful of distinct numbers gives the fit enough to work with.

Syntax: x₁ = [1, 2, 3, 4, 5] on one line, then the identity with a tilde: 2(x₁ - h)² + k ~ 2x₁² - 8x₁ + 6. Type x then 1 for the subscript, with no underscore. The subscript holds on to a caret, an opening parenthesis and a tilde, so press the right arrow after x₁ before any of those three. A plus or minus steps out on its own.

Now try it on a real SAT question

Head or scratch paper. Expand the right-hand side and a is the middle coefficient. The tilde is more than this question needs, which makes it a good first test of the method: you already know what it should return.

The expression $2x^{2} + ax$ is equivalent to $x(2x + 7)$ for some constant a.

What is the value of a?

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: RegressionsThe tilde is documented here as a data-fitting tool; using it on an algebraic identity is the same machinery pointed at a different problem.

The Desmos trap: when to close the calculator

Every x in the identity has to be written as x₁, matching the list. A stray plain x makes the fit meaningless.
When two constants play interchangeable roles, like the roots in (x - a)(x - b), Desmos may assign them either way round. Only trust the pair, which is fine when the question asks for a sum or a product.
For a single obvious constant, expanding by hand is faster. Use this move for completing the square, vertex form, and problems with several unknown constants at once.
The tilde fits, it does not prove. If a question asks you to show two expressions are equivalent for all x, graphing both and seeing one curve is the confirmation worth doing.

If the calculator does something you did not expect

The x₁ list is not optional, and leaving it out does not produce an error. Desmos returns numbers that look plausible and are wrong. The tell: x₁ itself appears under REGRESSION PARAMETERS with a value next to it, because Desmos has treated it as one more unknown to fit. If you see x₁ listed there, the list is missing.
Read the results as the numbers they are approaching. A fit is numerical, so 2 can appear as 1.9999999999999998.

Nova

If the question is 2x^2 + ax equals x(2x + 7), distributing takes two seconds and the tilde takes two lines. Save this for completing the square, for vertex form, and for anything with two or three unknown constants at once, where the by-hand route is long.

All Desmos techniques

Put this move into real practice

"Use Regression to Find an Unknown Constant" 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.

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