Solve any equation by graphing it
The single most useful move on the digital SAT. You almost never need to do algebra by hand, you graph each side and click where they cross.
The idea
To solve something like 3x − 7 = 2, treat each side as its own graph. Type the left side and the right side on separate lines, then click the point where the two graphs meet. The x-value of that point is your answer.
Remember the rule from earlier: if the expression contains an x, Desmos assumes y = for you, so you can just type 3x-7. A bare number needs the full y = 2.
Look at the boxes on the left of the calculator, the two functions are already typed in for you: 3x-7 and y=2. That's exactly what you'd type on the test.
Click the two lines' intersection inside the calculator, Desmos shows (3, 2). Answer: x = 3.
(x, y). Read the first number, not the second. Answer here is 3, not 2.Your turn
Solve 2x + 5 = 13 in the calculator below. Type 2x+5 in the first box and y=13 in the second, then click where the lines cross.
Show answer
(4, 13), so x = 4.Practice
Use this calculator to solve each one, type the two sides in the boxes, then click the crossing.
5x − 4 = 2x + 11.Answer
5x-4 and 2x+11. They meet at (5, 21) → x = 5.½x − 3 = 1.Answer
0.5x-3 and y=1. Cross at (8, 1) → x = 8.x satisfies x² = 3x + 10? (SAT loves “which is a solution”.)Answer
x^2 and 3x+10. Two crossings: (5, 25) and (−2, 4). Solutions x = 5 and x = −2. A parabola can meet a line twice, check both.Method 2, the one-line shortcut
Now that you're comfortable with the two-line method, here's a faster version for once you've mastered it. Instead of graphing both sides, move everything to one side and graph a single function, then click where it crosses the x-axis.
For 3x − 7 = 2, subtract 2 from both sides to get 3x − 7 − 2 = 0, and graph just the left side:
One line, one box. Click where it hits the x-axis, the point (3, 0). The x-value 3 is your answer, same as before.
Both methods always give the same answer. Use whichever feels faster on the day, the two-line method is easier to picture, the one-line method is quicker to type.
Systems of equations & special cases
Two equations at once? Graph both and click where they cross, that point solves the whole system. And the way the graphs sit tells you instantly when there's no solution or infinitely many.
Solving a system
For y = 2x + 1 and y = −x + 7, graph both and click the intersection. That single point (x, y) is the solution to both equations at once.
System solution: the lines cross at (2, 5).
Reading the three outcomes
This is the part the SAT tests constantly. Load each button above and watch what happens:
| What you see | Meaning | Example |
|---|---|---|
| One crossing point | Exactly one solution | 2x+1 & -x+7 |
| Parallel, never touch | No solution | 2x+1 & 2x+5 |
| Same line, fully overlapping | Infinitely many (identity) | 2(x+3) & 2x+6 |
Your turn
Decide how many solutions 3x − 6 = 3x + 2 has. Type 3x-6 and 3x+2 in the boxes below and look at how the lines sit.
Show answer
Practice
Use this calculator to test each system, type both equations in the boxes.
y = 3x − 4 and y = −2x + 11.Answer
(3, 5) → x = 3, y = 5.c does 4x + 6 = 4x + c have infinitely many solutions?Answer
c = 6. (Try it: only when c=6 do the graphs fully overlap.)x + y = 5 and 2x + 2y = 20 has how many solutions?Answer
x+y=10) → no solution. Watch for equations that are secretly multiples of each other.Zeros, vertex, and min / max of a parabola
Quadratics are all over the SAT. Desmos will hand you the roots and the turning point if you just click the curve, no factoring, no vertex formula.
The idea
Graph the quadratic on its own line. Then click directly on the curve. Desmos lights up its special points: where it crosses the x-axis (the zeros / roots / solutions) and the very bottom or top of the curve (the vertex, which is the minimum or maximum).
The example below is x^2 - 4x - 5, already typed in the box for you.
Click where it crosses the x-axis → (−1, 0) and (5, 0): the zeros are x = −1 and x = 5. Click the bottom of the curve → (2, −9): the vertex, so the minimum value is −9.
Your turn
Find the zeros and the vertex of x^2 - 6x + 8. Type it in the box, then click the curve's crossings and its lowest point.
Show answer
(2, 0) and (4, 0) → x = 2, 4. Vertex at (3, −1) → minimum value −1.Practice
Type each quadratic in the box and click the points you need.
x² + 2x + 5 = 0 have?Answer
x^2+2x+5. The parabola sits entirely above the x-axis, it never crosses → 0 real solutions. (A graph that doesn't touch the x-axis = no real roots.)y = 2x² − 8x + 3?Answer
(2, −5). Minimum value = −5.y = −x² + 4x + 1 has a maximum. What is it?Answer
(2, 5). Maximum value = 5.Nonlinear systems, a line meeting a curve
When one equation is a curve (a parabola or circle) and the other is a line, graph both and click every crossing. Each intersection is a solution, and there can be two, one, or none.
The idea
It works exactly like a linear system: put each equation on its own line and click where the graphs meet. The difference is that a line can cut a parabola in two places, just touch it once, or miss it entirely, so check how many crossings there are.
The example is the parabola y = x² − 1 and the line y = x + 1, already in the boxes.
Click both crossings → (−1, 0) and (2, 3). The system has two solutions: x = −1 and x = 2.
Your turn
Solve the system y = x² and y = 2x + 3. Type both in the boxes and click the crossings.
Show answer
(−1, 1) and (3, 9) → solutions x = −1 and x = 3.Practice
Type both equations and click the intersections. Remember a circle needs its full equation.
y = x² + 2 and y = x have?Answer
x^2+2 and x. The line passes below the parabola and never touches it → 0 solutions.y = 4 meet y = x²?Answer
x^2 and y=4. Crossings at (−2, 4) and (2, 4).x² + y² = 25 and y = x + 1 (a line meeting a circle).Answer
(−4, −3) and (3, 4) → two solutions.Sliders, solve for an unknown letter
When a question has an unknown constant (like c, k, or m) and a condition to satisfy, a slider lets you dial the letter until the picture is right, and read the answer off the slider.
The idea
Type the equation using the unknown letter. Desmos notices the letter isn't defined and offers to add a slider for it. Then drag the slider until the graph does what the question says, passes through a point, just touches a curve, etc. The slider's value is your answer.
Example: “The line y = 2x + c passes through the point (3, 10). Find c.” Below, the point and the line are set up, with a slider on c.
Drag the c slider until the line runs through the dot at (3, 10). It lands there at c = 4.
Your turn
The line y = mx − 1 passes through (2, 5). Use the slider on m to find it.
Show answer
(2, 5) when m = 3.Practice
Set up each one with a slider and dial in the answer.
b does y = 3x + b pass through (−2, 1)?Answer
y=3x+b (add slider) and the point (-2,1). The line hits it at b = 7.k does 4x + 6 = 4x + k have infinitely many solutions?Answer
4x+6 and 4x+k with a slider on k. The two lines overlap completely only at k = 6.y = a(x−1)² passes through (3, 8). Find a.Answer
a, plot (3,8). The curve passes through the point at a = 2.Tables, plot given data & sequences
When a question hands you a set of points or a sequence, put them in a Desmos table. You'll see the pattern on the graph, and you can read off missing values.
The idea
Start a table by typing a column name like x_1 (type x, then the underscore _, then 1). Add a second column y_1. Fill in the numbers and Desmos plots each pair as a dot, so the shape of the data appears.
The table below holds the points (0, 3), (1, 6), (2, 12), (3, 24).
The dots curve upward and each y doubles the last (3 → 6 → 12 → 24). That's exponential growth, so the next value at x = 4 would be 48.
x_1, y_1). If you just type x and y, Desmos treats it as an equation, not a table. The little _ underscore is what makes a subscript.Your turn
Put the points (1, 5), (2, 8), (3, 11), (4, 14) in a table. Is the pattern linear? What's the value at x = 6?
Show answer
Practice
Build a table for each one and look at how the dots behave.
2, 6, 18, 54, …. Table it against x = 1, 2, 3, 4, is it linear or exponential?Answer
(0, 7), (2, 3), (4, −1) lie on a line. What is the y-value at x = 6?Answer
Line of best fit (regression)
For scatter-plot questions, Desmos finds the best-fit line for you and gives the exact slope and intercept, then you use that equation to predict.
The idea
Put the data in a table (x_1, y_1), then on a new line type the model with a tilde ~ instead of an equals sign: y_1 ~ mx_1 + b. Desmos calculates the best-fit m and b and shows them. That's your line of best fit.
Below, a table of data already has the regression y_1 ~ mx_1 + b running on it.
Open the regression line's row and Desmos shows m ≈ 1.95, b ≈ 1.15. To predict at x = 10, type 1.95(10)+1.15 ≈ 20.6.
~, not =. With an equals sign Desmos thinks you're defining an equation and won't fit anything. Also make sure the variables in the model match your column names exactly (x_1, y_1).Your turn
Fit a line to (1, 4), (2, 5), (3, 9), (4, 10), (5, 13) and predict the value at x = 8.
Show answer
m ≈ 2.3, b ≈ 1.5. At x = 8: 2.3(8)+1.5 ≈ 19.9. (Exact numbers appear in the regression row.)Practice
Enter each dataset as a table, add y_1 ~ mx_1 + b, and read m and b.
(0, 2), (1, 4), (2, 6), (3, 8), what are the slope and intercept of the best-fit line?Answer
m = 2, b = 2. (When data is exactly linear, the fit is exact.)y = 1.8x + 3. What does it predict at x = 20?Answer
1.8(20)+3 → 39.Exponential growth & decay
Exponentials appear all over the SAT, population, money, half-life. Graphing y = a·bˣ shows instantly whether it grows or decays, and lets you read off any value.
The idea
An exponential is y = a·bˣ. The a is the starting value (where it sits at x = 0) and b is the multiplier each step. If b > 1 the curve grows; if b is between 0 and 1 it decays. Type it in and the shape tells you which.
The example is y = 3·2ˣ (starts at 3, doubles each step), already in the box.
Click the curve at x = 0 → (0, 3), the starting value. Each step right doubles it: 3 → 6 → 12. Base 2 > 1, so it's growth.
b = 1.05; “decays 10% per year” → b = 0.90. Don't type 0.05 or 0.10 as the base, you add or subtract from 1 first.Your turn
A 100 mg sample halves each hour: f(x) = 100·(0.5)ˣ. How much is left after 3 hours? Define the function and evaluate f(3).
Show answer
f(3) = 12.5 mg. Base 0.5 < 1, so it decays, halving from 100 → 50 → 25 → 12.5.Practice
Type each function in the box; click the curve or evaluate with f(number).
y = 5·(0.8)ˣ grow or decay?Answer
f(x) = 2·(3)ˣ, what is f(4)?Answer
f(x)=2(3)^{x}, type f(4) → 162 (2 × 81).Answer
500(1.04)^{10} → about 740. The base is 1.04, not 0.04.Method 2, exponential best fit
Once linear regression (Lesson 7) is comfortable, the same tilde trick fits an exponential curve to data. Put the points in a table and type y_1 ~ ab^(x_1), Desmos finds the best a and b.
For this doubling data the fit gives a = 5, b = 2, i.e. y = 5·2ˣ. Use the tilde ~, exactly like the line of best fit.
Circles & conics
Circle questions become easy once you graph them: the center and radius are right there. This is the topic we started with, now it's a full lesson.
The idea
A circle is (x − h)² + (y − k)² = r², with center (h, k) and radius r. Type the full equation (a circle needs both sides, you can't drop the = r²). Graph the center as its own point to see it, then count squares out to the edge for the radius.
The example is (x-2)^2 + (y-3)^2 = 25 with its center (2, 3) plotted.
Right side is 25, so radius = √25 = 5. Count 5 squares from the center dot (2, 3) to the edge to confirm.
= 25 means radius 5, not 25. And Desmos never marks the center for you, you have to plot the point yourself to see it.Your turn
Graph (x+1)^2 + (y-4)^2 = 9. What are its center and radius?
Show answer
(−1, 4) (signs flip inside the brackets), radius √9 = 3.Practice
Graph each circle. Plot the center point too so you can see it.
x² + y² = 49?Answer
(0, 0), radius √49 = 7.(x−5)² + (y+2)² = 16.Answer
(5, −2), radius √16 = 4.Method 2, the “messy” general form
Once the standard form is comfortable, here's how to handle the general form like x² + y² − 6x − 8y = 0, where the center isn't obvious. Just graph it, Desmos still draws the circle, then plot points around it, or use the fact that the center is the middle of the widest span.
Desmos draws it; the center turns out to be (3, 4) and the radius 5. Plot (3, 4) and count to the edge to confirm.
Inequalities & shaded regions
Type an inequality and Desmos shades every point that satisfies it. For a system, the overlap of the shading is your solution region, perfect for “which point works” questions.
The idea
Write inequalities just like equations but with <, >, <=, or >=. Desmos shades the region. Put two inequalities in and the darker overlapping area is where both are true.
Below: y ≤ −x + 4 and y ≥ x − 2, shaded.
Any point in the doubly-shaded wedge satisfies both. Test a point like (0, 0), it's in the region, so it's a solution.
≤ or ≥); a dashed line means they don't (< or >). If a question's answer sits right on the line, this decides it.Your turn
Shade y > 2x − 3 and y < −x + 6. Is the point (1, 1) a solution to both?
Show answer
(1, 1): it lands inside the overlap (1 > −1 and 1 < 5), so yes, it satisfies both.Practice
Type the inequalities and read the shaded region. Plot a test point when a question asks about one.
(2, 5) a solution to y ≥ x + 1?Answer
y ≥ x+1 and plot (2, 5). Since 5 ≥ 3, the dot is in the shaded area → yes.y < 3x and y > −2, name one point in the solution region.Answer
y = −2 and below y = 3x. For example (2, 0) works (0 < 6 and 0 > −2). Any point Desmos shows in the doubly-shaded area is valid.Time-saver tricks
A few small moves that quietly save minutes: evaluating a function at many values, and getting statistics instantly.
Trick A, define a function, then plug in
Define the function once with f(x) = …, then on a new line just type f(number) and Desmos prints the value. No re-typing the whole expression for each input.
Below, f(x) = 2x² − 3x + 1 is defined, and f(4) is evaluated.
Desmos shows f(4) = 21. Change the input to f(−1), f(0), anything, the answer updates instantly.
g(x) and asks for a specific output, you don't graph and hunt, define the function and type the value directly. Faster and no misreading the graph.Trick B, instant statistics
Desmos has built-in stats functions. Type mean(…), median(…), stdev(…), or total(…) with a list of numbers in brackets, and it computes it.
Your turn
Find the mean and median of 4, 8, 15, 16, 23, 42. Type mean([4,8,15,16,23,42]) and median([4,8,15,16,23,42]).
Show answer
Practice
g(x) = x² − 5x, what is g(6)?Answer
g(x)=x^2-5x, type g(6) → 6.12, 15, 21, 24?Answer
mean([12,15,21,24]) → 18.SAT Desmos cheat sheet
Everything from the demo lessons on one page. Use the print button to save it as a PDF.
| Goal | What to type in Desmos | How to read the answer |
|---|---|---|
| Solve an equation for x | leftSiderightSide (each own line) | Click the intersection; take the x-value. |
Skip typing y = | Allowed if it contains x (e.g. 3x-7) | A bare number needs y = 2. |
| One-line solve | y = left − right | Click where it crosses the x-axis (the zero). |
| Solve a system | Both equations, each own line | Intersection (x, y) solves both. |
| No solution | , | Lines are parallel, never cross. |
| Infinitely many (identity) | , | Lines fully overlap (looks like one line). |
| Zeros / roots of a quadratic | x^2-4x-5 | Click where it crosses the x-axis. |
| Line meets a curve (nonlinear system) | Both equations, each own line | Click each crossing; count them for “how many”. |
| Min / max (vertex) | x^2-4x-5 | Click the turning point; value = y-coord. |
| Solve for an unknown letter | y=2x+c + slider on c | Drag until condition met; read the slider. |
| Plot data / spot a pattern | Table with x_1, y_1 columns | Look at the dots: line vs curve. |
| Line of best fit | y_1 ~ mx_1 + b (tilde!) | Read m and b Desmos computes. |
| Exponential growth / decay | 3*2^x, base >1 grows, <1 decays | Click curve; a = value at x=0. |
| Exponential best fit | y_1 ~ ab^(x_1) | Read a and b. |
| Circle center & radius | (x-h)^2+(y-k)^2=r^2 | Center (h,k); radius = √(right side). |
| Inequality region | y <= -x+4 | Shaded area; overlap = both true. |
| Evaluate a function | f(x)=… then f(4) | Desmos prints the value. |
| Statistics | mean([…]), median([…]) | Value shown instantly. |
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