Regression from a supplied table

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Lesson progressPractice problems 0/4

Difficulty: Intermediate
Estimated time: 30 minutes
Domains: Problem Solving and Data Analysis Advanced Math
Techniques: TablesRegressionLinear-regressionQuadratic-regressionExponential-regressionPrediction

What you’ll learn

Put the input variable in x₁ and the output variable in y₁.
Choose an automatic linear, quadratic, or exponential regression.
Read and use the fitted parameters reported by Desmos.
Find the requested equation, coefficient, or prediction.
Preserve the units and precision requested by the question.
Recognize when a short pattern calculation is faster than regression.

Prerequisites

Why this matters on the SAT

Turn a table into an SAT answer

The SAT can give you a table of paired values and ask for an equation, a coefficient, or a prediction from a model. A regression fits an equation to paired data, and Desmos can do it in seconds. Your job is to put the variables in the right columns, choose the model the question calls for, and use the output to answer the exact question.

See the advantage on a typical SAT question.

SAT example

The amount of a medicine in a sample is modeled by an exponential function. The table shows the amount, $A(t)$ , at several times $t$ .

$t$ (hours)	$0$	$2$	$4$
$A(t)$ (milligrams)	$500$	$320$	$204.8$

According to the model, what is the predicted amount of medicine, in milligrams, at $t=5$ , rounded to the nearest whole number?

Fast Desmos solution

Put $t$ in x_1 and $A(t)$ in y_1, then select Add Regression and Exponential. Desmos finds $A(t)=500(0.8)^t$ . Evaluating the model at $t=5$ gives $163.84$ , which rounds to $\boxed{164}$ milligrams.

This lesson focuses on Desmos's automatic linear, quadratic, and exponential regressions. Later lessons cover custom regression, regression with several conditions, and harder decisions about exact fit versus best fit.

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The regression finds the exponential model and predicts the next value.

When is regression the right method?

You should already know how to create a table, keep paired values in the same row, and use a calculated column.

Regression is a strong choice when a question:

supplies several paired values;
names a linear, quadratic, or exponential model;
asks for a line or curve that best fits measured data;
asks for a prediction according to a model;
gives answer choices whose coefficients are difficult to recover mentally.

Use a shorter method when the table shows an immediate pattern. If every output increases by $7$ when the input increases by $1$ , the slope is already visible. If every output is multiplied by $0.5$ , the exponential multiplier is already visible. Desmos can verify either result, but entering the full table may take longer than the observation.

Check your understanding:

A table has only two rows, and the output increases from $18$ to $23$ while the input increases from $4$ to $5$ . The question asks for the rate of change. Should you begin with regression?

1. Put the variables in the right columns

Each row of the Desmos table must represent one pair. Put the input, or explanatory variable, in x_1. Put the output, or response variable, in y_1.

If a table gives time and population, time usually belongs in x_1 because population is being modeled as a function of time. The wording decides the direction:

“model $y$ as a function of $x$ ” means $x$ is the input and $y$ is the output;
“predict sales from years of experience” means experience is the input and sales is the output;
“model temperature as a function of time” means time is the input and temperature is the output.

Check your understanding:

A question asks you to model fuel remaining as a function of distance traveled. Which variable belongs in x_1, and which belongs in y_1?

Common mistake:

Copying the visual left-to-right order without reading which quantity is being predicted. Reversing the columns fits a different model.

2. Choose the regression the question calls for

After entering the table, select Add Regression beside the table and choose the requested model:

Model	Automatic form	What Desmos reports
Linear	$y_1\sim mx_1+b$	slope $m$ and intercept $b$
Quadratic	$y_1\sim ax_1^2+bx_1+c$	coefficients $a$ , $b$ , and $c$
Exponential	$y_1\sim ab^{x_1}$	initial value $a$ and multiplier $b$

The tilde, $\sim$ , tells Desmos to fit a model to the table. It is not the equals sign. The automatic menu creates the regression for you, so you do not need to memorize these entries yet.

Use the model family stated in the prompt. If the question does not name one, use the context, answer choices, and table pattern. At equal input intervals:

roughly constant differences support a linear model;
roughly constant second differences support a quadratic model;
roughly constant ratios support an exponential model.

Do not choose a more complicated curve merely because it can pass through a small table. Model choice must come from the question and the pattern, not from the number of parameters.

Check your understanding:

At equal 2-hour intervals, a quantity changes from $80$ to $60$ to $45$ to $33.75$ . Which automatic regression should you choose, and what pattern supports it?

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Use the equal intervals and output pattern to decide which regression model to select.

3. Turn the regression output into an SAT answer

Desmos stores the fitted parameter values, so you can use them on another expression line. After the opening exponential regression reports $a$ and $b$ , for example, enter a*b^5 to evaluate the model at $t=5$ . Use the parameter letters directly instead of retyping displayed decimals, so Desmos calculates with the fitted values rather than rounded copies.

Before submitting, reread what the question asks for:

If it asks for the slope, submit $m$ , not the full equation.
If it asks for the exponential multiplier, submit $b$ , not the percent increase.
If it asks for a prediction, evaluate the model at the requested input.
If the output is measured in thousands of dollars, keep that unit in mind.
If the data are measurements and the prompt says “best estimate,” report an appropriate estimate rather than claiming an exact value.

A regression display is numerical. When a table represents an exact function, verify the candidate equation against the supplied rows before treating rounded output as exact.

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Using the fitted parameter letters preserves their precision in the prediction.

4. Example: Fit noisy data and make a prediction

Worked example

A company recorded each salesperson's years of experience, $x$ ,

and average monthly sales, $y$ , in thousands of dollars.

$x$ (years)	$1$	$2$	$3$	$4$	$5$
$y$ (thousands of dollars)	$4$	$7$	$9$	$13$	$15$

A linear model is used to predict monthly sales from years of experience.

According to this model, what is the best estimate of the monthly sales, in thousands of dollars, for a salesperson with $6$ years of experience?

A
$16$
B
$18$
C
$20$
D
$24$

Step 1

Orient the table

Years of experience is the input, so put those values in x_1. Monthly sales is the predicted output, so put those values in y_1.

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Each row keeps one experience value paired with its corresponding sales value.

Step 2

Run the requested regression

Select Add Regression, then keep Linear. Desmos fits

y_1\sim mx_1+b

and reports

m=2.8,\qquad b=1.2.

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The linear regression reports the slope and intercept of the fitted model.

Step 3

Evaluate the requested input

The question asks for the model's prediction at $x=6$ . Enter m*6+b:

2.8(6)+1.2=18.

The model predicts monthly sales of $18$ thousand, so the answer is B.

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Using the fitted parameters predicts monthly sales at six years of experience.

Common mistake:

Reporting $2.8$ , the slope, or interpreting $18$ as $18$ instead of $18{,}000$ . The table's unit is thousands of dollars.

Finish the solution

The table is entered and a quadratic regression is selected. Use the fitted coefficients to choose the equation.

Match the fitted quadratic

Finish the solution

For the quadratic function $f$ , the table shows three values of $x$ and the corresponding values of $f(x)$ .

$x$	$-1$	$0$	$1$
$f(x)$	$10$	$14$	$20$

Which equation defines $f$ ?

First steps

Put the $x$ -values in x_1 and the $f(x)$ -values in y_1.
Select Add Regression, then choose Quadratic.

Finish it

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Match the fitted quadratic coefficients to the equation choices.

Practice problems

SAT practice problems

The calculator remains available for every problem. Start from a clean table, run the requested regression, and submit the exact equation or estimate the question asks for.

Fit a roughly linear trend

Practice problem

A technician records the screen size $x$ , in inches, and mass $y$ , in grams, of six smartphone models. The data follow a roughly linear trend.

$x$ (inches)	$4.7$	$5.0$	$5.5$	$6.1$	$6.5$	$6.8$
$y$ (grams)	$135$	$145$	$155$	$170$	$182$	$190$

Which equation best models the relationship between $y$ and $x$ ?

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Enter the paired data and fit the linear model named in the problem.

Predict with exponential decay

Practice problem

A tank contains $y$ liters of water $x$ hours after draining begins. The measured amounts are rounded to the nearest whole liter.

$x$	$1$	$4$	$7$	$10$
$y$ (liters)	$510$	$313$	$193$	$118$

The amount of water is modeled by an exponential function. According to this model, what is the predicted amount of water, in liters, at $x=13$ , rounded to the nearest whole number?

Your answer

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Enter the paired data, fit an exponential model, and evaluate the prediction.

Read a fitted coefficient

Practice problem

A wildlife center records the number of sea turtles, $P(t)$ , in a rehabilitation pool $t$ days after observations begin. The number of turtles is modeled by a quadratic function.

$t$ (days)	$0$	$4$	$8$
$P(t)$ (turtles)	$120$	$152$	$120$

The model can be written as

P(t)=at^2+bt+c,

where $a$ , $b$ , and $c$ are constants. What is the value of $b$ ?

Your answer

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Enter the paired data and read the requested coefficient from the fitted quadratic.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Put the input in x_1, the output in y_1, and keep each pair in one row.
Use the automatic linear, quadratic, or exponential regression named or supported by the question.
Read each fitted value by its parameter letter.
Use the fitted model to answer the requested equation, coefficient, prediction, unit, and precision.
Treat measured-data models as estimates and verify exact-function candidates against the table.
Skip regression when a constant difference, constant ratio, or other short pattern gives the answer faster.

Next lesson

Mean, median, and frequency tables

Use Desmos for one-variable data and weighted frequency tables.

Start next lesson

Free Practice

What you’ll learn

Prerequisites

Turn a table into an SAT answer

Fast Desmos solution

When is regression the right method?

1. Put the variables in the right columns

2. Choose the regression the question calls for

3. Turn the regression output into an SAT answer

4. Example: Fit noisy data and make a prediction

Orient the table

Run the requested regression

Evaluate the requested input

Finish the solution

Match the fitted quadratic

First steps

Finish it

Practice problems

SAT practice problems

Fit a roughly linear trend

Predict with exponential decay

Read a fitted coefficient

4 practice examples left

Quick recap

Mean, median, and frequency tables

Free question bank practice problems