How do I fit a quadratic model to a table in the calculator?

Type the x values into one column and the y values into the next, named x_1 and y_1. On a new row type y_1 ~ a x_1^2 + b x_1 + c. On the test-day calculator, press the right-arrow key after each subscript 1, or the rest of the row is typed into the subscript. The calculator finds a, b and c and lists them under the row.

Does the question have to name the model before I can fit it?

Yes. A fit needs a model to find constants for. If the question does not name one, choose it first: constant differences mean linear, constant second differences mean quadratic, and a constant ratio means exponential.

What if the table already shows the constant I need?

Read it. The y value at x equals 0 is the starting amount, and for evenly spaced x values a common difference between the y values is the slope. Reading is quicker than fitting.

Why might my fitted constant differ slightly from the answer choices?

When the data is a scatter of measurements rather than exact values, a fitted constant can differ slightly between calculators. Read the choices for the nearest one instead of expecting an exact match. Data that lies exactly on a model gives clean numbers.

How do I read an exponential fit?

Fit y_1 ~ a b^x_1, pressing the right-arrow key after the first subscript 1 on the test-day calculator. It fits this model in log mode by default, which gives the same constants when the values lie exactly on the model. The constant a is the value at x equals 0. The base b is a multiplier for one step in x, not a rate: a base of 0.9 means 90 percent is kept and the value falls by 10 percent each step.

Fit a linear, quadratic or exponential model to a table of values

Type the table into the calculator and fit the model the question names, with a tilde: $y_1 \sim mx_1 + b$ for a line. The fit row lists the constants. Read the one the question asks for.

1.Type the table into the calculator: the $x$ values in one column named $x_1$ , the $y$ values in the next named $y_1$ . Copy the values exactly.
2.On a new row fit the model the question names: $y_1 \sim mx_1 + b$ , $y_1 \sim ax_1^2 + bx_1 + c$ or $y_1 \sim ab^{x_1}$ . On the test-day calculator, press the right-arrow key after each subscript 1.
3.Read the constant the question asks for from the fit row. Every constant arrives together, so check the letter before you copy the number.
4.Put one $x$ from the table into the model to check it, then give the constant in the form the question wants: a rate, a starting amount or a percent.

Try it on a question

Work it below before you open the working. The steps say what to do, and you can do each one yourself.

A food truck owner tries five prices for a lunch box and records the weekly profit at each price. A quadratic model $P = ap^2 + bp + c$ fits all five values exactly, where $p$ is the price in dollars and $P$ is the weekly profit in dollars.

\begin{array}{c|ccccc} p & 2 & 3 & 4 & 5 & 6 \\ P & 220 & 320 & 380 & 400 & 380 \end{array}

What is the value of $c$ ?

Choice A: $-100$
Choice B: $-20$
Choice C: $220$
Choice D: $400$

1
Type the prices into a column named $x_1$ and the profits into a column named $y_1$ . Use the values exactly as the table gives them.
Type: a table of x_1 and y_1
2
On the next row type $y_1 \sim a x_1^2 + b x_1 + c$ . The tilde asks the calculator for the $a$ , $b$ and $c$ whose curve passes through the points, and it draws that curve.
Type: y_1 ~ a x_1^2 + b x_1 + c
On the test-day calculator, press → (the right-arrow key) where it is marked: y_1 → ~ a x_1 → ^2 + b x_1 → + c
3
Find the answer under the fit. The constants it found are listed under that row.

Open in the full calculator

Loading the calculator…

The full working, and why each wrong choice is there

Type the table into the calculator and fit the model the question names: $y_1 \sim a x_1^2 + b x_1 + c$ . The fit reads $a = -20$ , $b = 200$ and $c = -100$ , so $P = -20p^2 + 200p - 100$ . The question asks for $c$ only.

Check one row before you answer. At $p = 4$ : $-20(16) + 200(4) - 100 = -320 + 800 - 100 = 380$ , which is the table's value.

By hand this is three equations in three unknowns. The fit is one row.

Choice A: $-100$ the answer
The fit gives $P = -20p^2 + 200p - 100$ . The constant $c$ is the profit the model predicts at a price of zero: a loss of 100 dollars.
Choice B: $-20$
This is $a$ , the number on $p^2$ . The fit row lists all three constants, so check the letter beside the value.
Choice C: $220$
This is the first profit in the table, at a price of 2. The constant $c$ is the model's value at a price of 0, which the table does not show.
Choice D: $400$
This is the largest profit in the table, at a price of 5. It is the greatest value the data shows, not the constant $c$ .

Use it when

A table of $x$ and $y$ values, and the question names the model (linear, quadratic or exponential) and asks for one of its constants.
A quadratic. It has three constants, and finding them by hand takes three equations. The fit finds all three at once.
The $x$ values are uneven, or the table does not start at $x = 0$ , so the constant is not sitting in the table to be read.
The constants are decimals or the values are large, where a slip in hand arithmetic is easy.

Do not use it when

The table already shows the constant. The $y$ value at $x = 0$ is the starting amount, and a common difference between evenly spaced rows is the slope. Read it, do not fit.
The data is a scatter of measurements rather than exact values. A fitted constant can differ slightly between calculators, so read the choices for the nearest one instead of expecting your number to match exactly.
The question has not named the model and asks which one fits. Naming the family is a different job: test the differences and the ratios first, and fit only once the model is known.

Five questions on this move

Untimed, and nothing is saved. A wrong first answer gets one more try before the working is shown.

Question 1 of 5

The depth of water in a fountain basin is measured at four times after a pump is switched on. A linear model $d = mt + b$ fits all four values exactly, where $t$ is the number of minutes since the pump was switched on and $d$ is the depth in centimetres.

\begin{array}{c|cccc} t & 2 & 5 & 6 & 10 \\ d & 8.5 & 10.75 & 11.5 & 14.5 \end{array}

What is the value of $m$ ?

Five is a start

In your account there is a focused set of ten on Two-variable data: models and scatterplots from the question bank, with the same one more try. And after each practice test, the drills your plan sets for the skills you missed name the moves for them.

Practise Two-variable data: models and scatterplots in your account

How this page was checked

Every maths answer, by our own calculator
The answer to each of the 6 questions on this page is worked out again by our calculator engine in an automated test, which fails when a key disagrees with the engine.
In the calculator the test uses
Claude (an AI assistant) typed this page's calculator steps into the College Board version of the Desmos calculator on 14 October 2026 and read the same answers. All six tables on this page were entered and fitted, and each gave the keyed constant. Each fit row needs the right-arrow key after a subscript there; it is marked. The exponential fit runs in log mode there and gives the same constant.
The whole page, by a person
Not yet. Every question and every step of its walk-through was worked through in a browser by Claude, an AI assistant, on 14 October 2026. The page was published on 14 October 2026 before a person had read it. If something here is wrong, tell us and it is put right.

The questions were drafted with AI and are original: none is a College Board question, and none comes from the question bank behind our practice tests.

Where this comes from

“Let the table find the constants” is one card from the Preptics lesson Choosing the model: linear, quadratic or exponential, which sits under the Digital SAT skill Two-variable data: models and scatterplots. The lesson has the other cards around it, and its questions come back for review on a schedule. The lessons are part of the Digital SAT track.

Common questions

How do I fit a quadratic model to a table in the calculator?: Type the x values into one column and the y values into the next, named x_1 and y_1. On a new row type y_1 ~ a x_1^2 + b x_1 + c. On the test-day calculator, press the right-arrow key after each subscript 1, or the rest of the row is typed into the subscript. The calculator finds a, b and c and lists them under the row.
Does the question have to name the model before I can fit it?: Yes. A fit needs a model to find constants for. If the question does not name one, choose it first: constant differences mean linear, constant second differences mean quadratic, and a constant ratio means exponential.
What if the table already shows the constant I need?: Read it. The y value at x equals 0 is the starting amount, and for evenly spaced x values a common difference between the y values is the slope. Reading is quicker than fitting.
Why might my fitted constant differ slightly from the answer choices?: When the data is a scatter of measurements rather than exact values, a fitted constant can differ slightly between calculators. Read the choices for the nearest one instead of expecting an exact match. Data that lies exactly on a model gives clean numbers.
How do I read an exponential fit?: Fit y_1 ~ a b^x_1, pressing the right-arrow key after the first subscript 1 on the test-day calculator. It fits this model in log mode by default, which gives the same constants when the values lie exactly on the model. The constant a is the value at x equals 0. The base b is a multiplier for one step in x, not a rate: a base of 0.9 means 90 percent is kept and the value falls by 10 percent each step.

Keep going

Every move: the full set. The calculator on its own: the graphing calculator.

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