To find a slope or rate of change on Desmos, put the points in a table, then type y_1 ~ mx_1 + b on a new line. Desmos fits m, the slope, and b, the y-intercept, and stores both for later lines. You can type every line on this page into the College Board version of Desmos on our Desmos on the SAT page.
For a linear function, the rate of change is the slope. It is the same number between any two points on the line. Bluebook has Desmos built in, and calculators are allowed on the whole Math section, so this method works on test day too. This page works four SAT-style problems, then gives you four practice problems with answers.
How to set up a linear regression on Desmos
Desmos's help article on regressions gives the method. Add your data to a table. On a new line, write the model with ~ where the equals sign would go, and use the table's column names, such as x_1 and y_1.
- Add a table. Put the x-values in the first column and the y-values in the second.
- On a new line, type
y_1 ~ mx_1 + b. - Read the values Desmos fits for m and b. Desmos stores them, so a later line can use them.
Two points fix a line, so a two-point table gives the exact slope and intercept. With three or more points that don't sit on one line, the same fit gives the line of best fit instead. Start each new problem on a blank graph so one problem's m doesn't get mixed up with the next one's.
Rate of change and slope
The average rate of change of f from x = a to x = b is
(f(b) - f(a)) / (b - a)
For a linear function, this equals the slope for any a and b. For a curve, it is the slope of the line through (a, f(a)) and (b, f(b)). When an SAT question asks for the positive difference between two rates of change, it wants the larger rate minus the smaller one.
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Start 1-day free trial →Worked example 1: compare two rates of change
SAT-style question. A car starts from rest. From 0 to 8.00 seconds, its speed increases from 0 to 11.0 meters per second. From 8.00 to 14.0 seconds, its speed increases from 11.0 to 23.0 meters per second. What is the positive difference, in meters per second per second, between the average rate of change of the car's speed from 0 to 8.00 seconds and the average rate of change from 8.00 to 14.0 seconds?
Interval 1. Time is the input and speed is the output. Add a table with the points (0, 0) and (8, 11):
| x1 (seconds) | y1 (meters per second) |
|---|---|
| 0 | 0 |
| 8 | 11 |
On a new line, type y_1 ~ m_1x_1 + b_1. The fit gives m1 = 1.375. By hand, the speed rose 11 meters per second in 8 seconds, and 11/8 = 1.375.
Interval 2. Add a second table with (8, 11) and (14, 23). Desmos gives this table its own column names, so read them from the table's header. If the header shows x2 and y2, type y_2 ~ m_2x_2 + b_2. New parameter names, m2 and b2, give the second fit its own slope instead of reusing m1.
The fit gives m2 = 2. By hand, (23 − 11)/(14 − 8) = 12/6 = 2.
Subtract. On a new line, type m_2 - m_1. The result is 0.625.
Answer: 0.625 meters per second per second, which is 5/8.
Worked example 2: find p + w in 165x + py = w
SAT-style question. Line g is defined by 165x + py = w, where p and w are constants. Line g passes through the points (−9, 12) and (0, 1). What is the value of p + w?
Step 1. Fit the line. Add a table with (−9, 12) and (0, 1), then type y_1 ~ mx_1 + b. The fit gives m = −11/9, about −1.222, and b = 1. So line g is y = −(11/9)x + 1.
Step 2. Match the question's form. Move the x-term to the left side: (11/9)x + y = 1. The question's x-coefficient is 165, so multiply every term by the number that turns 11/9 into 165:
165 ÷ (11/9) = 165 × 9/11 = 135
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That gives 135 × (11/9)x + 135y = 135, which is 165x + 135y = 135. So p = 135 and w = 135.
Step 3. Let Desmos do the scaling. Desmos keeps the fitted m and b, so you can type p = -165/m and w = -165b/m on two new lines. Both come out to 135.
Check by hand. 165(−9) + 135(12) = −1,485 + 1,620 = 135, and 165(0) + 135(1) = 135. Both points satisfy 165x + 135y = 135.
Answer: p + w = 135 + 135 = 270.
Worked example 3: the y-intercept from two points (your turn to finish)
SAT-style question. A line passes through (2, 5) and (6, 13). What is the y-intercept of the line?
Add a table with (2, 5) and (6, 13) and type y_1 ~ mx_1 + b. The question asks for the y-intercept, so the number you want is b, not m. Read b before you look below.
Check your answer: the fit gives m = 2 and b = 1, so the line is y = 2x + 1 and the y-intercept is 1. By hand, the slope is (13 − 5)/(6 − 2) = 8/4 = 2, and 5 = 2(2) + b gives b = 1. At x = 6, 2(6) + 1 = 13, which matches the second point.
Worked example 4: how many solutions does the system have?
SAT-style question. Line j is y = 3x − 4. Line k is 6x − 2y = 8. How many solutions does the system of these two equations have?
Type each equation on its own line:
1 y = 3x - 4
2 6x - 2y = 8You see one line, because the second graph lies on top of the first. By hand, 6x − 2y = 8 gives 2y = 6x − 8, so y = 3x − 4, the same equation as line j.
Answer: infinitely many solutions.
Read a two-line graph this way. Lines on top of each other mean infinitely many solutions. Parallel lines that never meet mean no solution. Lines that cross once mean exactly one solution. When the system has a missing constant, our guide to when a system has infinitely many solutions shows how to find it.
When algebra is faster
College Board's calculator policy says some Math questions are better done without a calculator. For a slope from two clean points, (y2 − y1) / (x2 − x1) takes one line of arithmetic. Reach for the table when the question gives three or more points, awkward decimals, or asks you to reuse the slope in a later step.
Linear function question types and what to type
| The question asks for | What to do on Desmos |
|---|---|
| A slope from two or more points | Table, then y_1 ~ mx_1 + b; read m |
| A y-intercept | The same fit; read b |
| Same line, parallel lines, or one crossing | Graph both equations and look |
| The difference between two rates of change | Two tables, fit m1 and m2, then type m_2 - m_1 |
| Constants such as p and w | Fit y = mx + b, then rescale to the question's form |
Word problems: which number is x?
When a linear question comes with a context, such as speed, cost, distance or temperature, decide first which quantity is the input and which is the output. The input goes in the x1 column and the output in the y1 column. In worked example 1, time is the input and speed is the output. Once the numbers are in the right columns, a speed problem and a cost problem run through the same steps. For more setups, see our guide to SAT word problems on Desmos.
Mistakes to avoid
- Typing = instead of ~. Desmos's help article says the tilde takes the place of the equals sign in a regression. Check the line before you read m and b.
- Reusing m and b across fits. In worked example 1, the second interval needs its own names, m2 and b2. Otherwise you can't subtract one slope from the other.
- Dropping the sign of a rate. In practice problem 3 below, depth goes down while temperature goes up, so the rate of change is negative.
Practice problems
- A line passes through (0, −3) and (4, 9). Write its equation in slope-intercept form.
- Line r has equation 4x + 2y = 10. Line n is perpendicular to line r and passes through (2, 1). What is an equation of line n?
- Water temperature rises from 15°C to 35°C as depth decreases from 200 meters to 50 meters. What is the average rate of change of temperature with respect to depth?
- How many solutions does the system y = 2x + 5 and 4x − 2y = −10 have?
Answers. 1) y = 3x − 3. The slope is (9 − (−3))/(4 − 0) = 12/4 = 3. 2) y = (1/2)x. Line r is y = −2x + 5, so line n has slope 1/2, and 1 = (1/2)(2) + b gives b = 0. 3) −2/15 degrees Celsius per meter, about −0.133. The change is (35 − 15)/(50 − 200) = 20/(−150). 4) Infinitely many. The second equation rewrites to y = 2x + 5, the same line as the first.
Practice next
Work more slope and linear function questions in the free SkillDrill Linear Functions: Notes and Practice 1, then the leveled set in Lines and Linear Relationships. For mixed SAT-style math, open the TTA SAT question bank.
Related guides: find the y-intercept on Desmos, find x- and y-intercepts on Desmos, SAT regression practice problems on Desmos, and how to solve systems of equations on Desmos.
Checked against Desmos's Regressions help article, College Board's calculator policy and College Board's digital SAT launch announcement on September 26, 2026. The Test Advantage is independent of College Board and Desmos.
Frequently Asked Questions
How do I find slope on Desmos?
Put the points in a table. On a new line, type y_1 ~ mx_1 + b, using the table's column names. Desmos fits m, the slope, and b, the y-intercept, and stores both so later lines can use them.
What is the line through (2, 5) and (6, 13)?
y = 2x + 1. The slope is (13 - 5)/(6 - 2) = 2, and 5 = 2(2) + b gives b = 1.
Is rate of change the same as slope?
For a linear function, yes. The rate of change is the slope, and it is the same between any two points. For a curve, the average rate of change from x = a to x = b is (f(b) - f(a))/(b - a), the slope of the line through those two points.
How do I find the positive difference between two rates of change?
Find each rate as a slope, then subtract the smaller from the larger. If a car's speed goes from 0 to 11 m/s in 8 seconds and from 11 to 23 m/s in the next 6 seconds, the rates are 1.375 and 2, so the positive difference is 0.625.
When do two linear equations have infinitely many solutions?
When they describe the same line. On Desmos the two graphs lie on top of each other. By hand, one equation rewrites into the other: 6x - 2y = 8 becomes y = 3x - 4.