SAT Regression on Desmos: Practice Problems, Solved

To run a regression on Desmos, put the data in a table, then type the model on a new line with a tilde in place of the equals sign, such as y_1 ~ mx_1 + b. You can type every line on this page into the College Board version of Desmos on our Desmos on the SAT page. Desmos finds the parameters that fit your data and stores them, so you can use them on the next line.

Bluebook has a Desmos calculator built in for the whole Math section, so this skill works on test day too. This page solves five SAT-style regression problems: a linear model, a quadratic with a known maximum, and three practice problems with answers.

How to set up a regression on Desmos

Desmos's help article on regressions gives the method in three moves. Add your data to a table. On a new line, write the model with ~ where the equals sign would go. Inside the model, use the table's column names, x_1 and y_1. Typing x_1 gives you x with a small 1 below it.

The hard part on the SAT is picking the model. The question tells you which one to use.

Desmos keeps the fitted values of letters like m, a and b. Start each new problem on a blank graph so you don't mix one problem's a with the last one's. In the College Board version, Desmos uses Log mode for applicable exponential, logarithmic and power regressions. With exact data like the tables on this page, the fitted numbers match what you would get by hand.

Worked example: a linear model (I do)

A linear model estimates the population of a city from 1995 to 2019. The model estimates the population was 55 thousand in 1995, 217 thousand in 2015, and x thousand in 2019. To the nearest whole number, what is the value of x?

From TTA's question bank, record 102805. Choices: 225, 235, 240, 249.

Pick the model. The question says "linear model," so fit a line. Two points are enough to fix a line.

Pick the x-values. Years like 1995 make clumsy numbers. Let t be years after 1995 and P be the population in thousands. Then 1995 is t = 0, 2015 is t = 20, and 2019 is t = 24.

Line 1 is a table:

2  y_1 ~ mx_1 + b
3  f(x) = mx + b
4  f(24)

Line 2 fits m = 8.1 and b = 55. Line 3 reuses those stored values, and line 4 gives 249.4. Round to the nearest whole number: x = 249.

Check by hand. The slope is (217 - 55) / 20 = 162 / 20 = 8.1 thousand people per year. Then 8.1(24) + 55 = 194.4 + 55 = 249.4, so the answer is 249, the stored key. The question asks for x in "x thousand," so 249 is the answer, not 249,000.

Guided example: a quadratic with a known maximum (we do)

A quadratic function gives the estimated length of daylight d(t), in hours, in a certain city t months after March 1, where 0 ≤ t ≤ 7. According to the function, the estimated length of daylight is 12.66 hours 6 months after March 1 and the maximum estimated length of daylight is 13.91 hours 3.5 months after March 1. Based on this function, what is the estimated length of daylight, in hours, on March 1?

From TTA's question bank, record 131101. Choices: 10.41, 11.46, 12.66, 13.91.

Step 1 (shown). The question gives two points: (6, 12.66) and (3.5, 13.91). Two points alone do not fix a quadratic. For example, y = -0.5x² + 4.25x + 5.16 passes through both, but its peak is at x = 4.25. The word "maximum" adds the missing fact: the vertex is (3.5, 13.91). So write the model in vertex form, with a as the only unknown.

Step 2 (shown). Put both points in the table and fit only a:

2  y_1 ~ a(x_1 - 3.5)^2 + 13.91

Desmos fits a = -0.2. A negative a means the parabola opens down, which a maximum needs.

Step 3 (you finish). March 1 is t = 0. On line 3, define f(x) with the stored a. On line 4, find f(0). Then pick the choice.

Check your answer: f(x) = a(x - 3.5)² + 13.91, so f(0) = -0.2(3.5)² + 13.91 = -2.45 + 13.91 = 11.46 hours, the stored key. The function only covers 0 ≤ t ≤ 7, so don't use it for the rest of the year. For more problems built on a vertex, see SAT vertex form problems on Desmos.

Your turn: three SAT-style practice problems (you do)

We wrote these three for this page. Solve each one on Desmos first. The answers are at the end of this section.

Practice 1 (exponential). The table shows four values of y. Which equation models the data?

A) y = 250(0.8)^x
B) y = 250(0.2)^x
C) y = 200(0.8)^x
D) y = 250(1.2)^x

Practice 2 (quadratic). A quadratic function f has f(0) = 5, f(2) = 21 and f(5) = 15. What is the maximum value of f(x)?

Practice 3 (line of best fit). A student records hours of practice x and quiz scores y: (2, 61), (4, 66), (6, 74) and (8, 79). Based on the line of best fit for these data, what score does the line predict for 7 hours of practice, to the nearest whole number?

Answers

1. A. y_1 ~ ab^{x_1} fits a = 250 and b = 0.8. Each step keeps 80% of the last value, which is a 20% decrease. Check: 250(0.8)³ = 128. B drops too fast (250, 50, 10, 2), C starts at 200, and D grows.
2. 23. Three points fix a quadratic, so y_1 ~ ax_1^2 + bx_1 + c fits a = -2, b = 12 and c = 5. The vertex is at x = -b/(2a) = -12/(-4) = 3, and f(3) = -2(9) + 12(3) + 5 = 23. In Desmos, type f(x) = ax^2 + bx + c, then f(-b/(2a)).
3. 76. y_1 ~ mx_1 + b fits m = 3.1 and b = 54.5. Then 3.1(7) + 54.5 = 76.2, which rounds to 76. These four points are not on one line, so this line is a best fit, not an exact one. Seven hours sits inside the data (2 to 8 hours), so the prediction stays within the range you measured.

For a harder exponential table, try exponential functions with Desmos regression.

When algebra is faster

College Board's calculator policy says some Math questions are better solved without a calculator. Record 102760 in TTA's bank is one of them: y is 90 when x = 0, and y grows by 20% each time x goes up by 1. A 20% increase means multiplying by 1.20, so the answer is y = 90(1.20)^x, the stored key. A fit on (0, 90), (1, 108) and (2, 129.6) returns a = 90 and b = 1.2, but you had the answer before you built the table.

For which handheld calculators College Board allows and bans, including the CAS rule, see our SAT calculator policy guide.

Mistakes to avoid

• Typing = instead of ~. Desmos's help article says the tilde takes the place of the equals sign in a regression. Look at line 2 before you read any numbers.
• Counting from the wrong year. In the population problem, 2019 is t = 24, not t = 4. Using 4 gives 87.4, which isn't close to any choice. Write down what t = 0 means before you fill the table.
• Fitting a general quadratic to two points. Two points leave one coefficient free, so a three-letter fit can't give you the question's function. Look for the third fact in the stem, such as a maximum, a minimum or a y-intercept.

Practice next

Learn the model choice in the free lesson notes Growth Factors, Doubling Time and Linear-vs-Exponential Models. Then try Scatterplots: Notes and Practice 1 for lines of best fit. For more SAT-style math sets, open the TTA SAT question bank.

When two equations must both hold, you need an intersection, not a fit: see how to solve systems of equations on Desmos. For the calculator itself, read our College Board Desmos guide and SAT quadratics on Desmos.

Checked against Desmos's Regressions help article, Desmos's College Board calculator sheet and College Board's calculator policy on September 25, 2026. The Test Advantage is independent of College Board and Desmos.