Desmos on the SAT
A Desmos graphing calculator is built into the Bluebook test app for every Math question (there is no calculator in Reading and Writing); you may also bring an approved handheld calculator. Used well, Desmos saves time on many Math questions, mostly in Algebra and Advanced Math.
What Desmos can do for you
The Desmos graphing calculator is part of the test screen in the Math section. You open it with the calculator button. Type an equation and it draws the graph at once. Click a graph and gray dots appear at its important points: where it crosses the axes or another graph, and its highest or lowest point. Click a gray dot to see its coordinates. You can also hold and drag along a graph to read the coordinates of any point. You can also use a handheld calculator that the test rules allow, but Desmos does everything a normal calculator does, and it draws graphs too. Practice with it before test day, so that typing becomes automatic.
| Task | Type this | Read the answer |
|---|---|---|
| Solve an equation | left side as y = …, right side as y = … on two lines | x-coordinate of the crossing point |
| Solve a system | both equations, in any form | the crossing point (x, y) |
| Find where a graph meets the x-axis | y = the expression | click the points on the x-axis |
| Find the highest or lowest point | y = the expression | click the peak or the bottom |
| Test answer choices | type each choice as a graph or point | the one that fits all the data |
| Unknown constant k | type k in the equation | drag the slider |
Solve an equation by graphing both sides
Put the left side on one line as y = … and the right side on another. The solution is the x-coordinate of the point where the graphs cross. In the picture, the left side y = 0.6x + 1.7 and the right side y = 1.1x − 0.8 cross at (5, 4.7), so the solution of 0.6x + 1.7 = 1.1x − 0.8 is x = 5. Remember: the question asks for x, not the y-value of the crossing point.
Type carefully
Most mistakes in Desmos are typing mistakes. Use brackets for every numerator and denominator that has more than one term: type (x + 2)/4, not x + 2/4. When you type /, Desmos makes a fraction box: press the right arrow key to leave the denominator. Use ^ for powers. Inequalities need the symbols <, >, <= and >=. After typing, look at the picture: does the graph match what you expect (rising, falling, crossing the axis near where you thought)? A wrong picture usually means a typing slip.
Sliders for unknown constants
When an equation has a constant such as k, type the equation and Desmos offers to add a slider for k: click “add slider”. The slider goes from −10 to 10 at first; click its ends to change the limits. Drag the slider (or type a value) and watch the graph change. This is useful for “for what value of k does the system have no solution?” questions: move k until the two lines become parallel, then confirm with algebra or by testing an exact value.
When hand-solving is faster
Desmos is a tool, not a replacement for algebra. A simple equation such as 3x + 5 = 20 takes five seconds by hand and thirty in Desmos. Use Desmos when the numbers are ugly, when you can test answer choices, when the question is about a graph or an intersection, or when you want to check a hand solution. Use algebra when the numbers are small and nice, and when the question asks for an expression instead of a number. With about 95 seconds per Math question on average, choosing the faster tool is part of the skill.
- Type y = 0.6x + 1.7 on line 1 and y = 1.1x − 0.8 on line 2.
- Click the point where the graphs cross: Desmos shows (5, 4.7).
- The solution is the x-value: x = 5.
- Check by hand: 0.6(5) + 1.7 = 4.7 and 1.1(5) − 0.8 = 4.7 ✓
- Type both equations exactly as written, one per line.
- Click the crossing point: (6, 5).
- x + y = 11.
- Check: 0.8(6) + 0.5(5) = 4.8 + 2.5 = 7.3 ✓ and 1.2(6) − 0.4(5) = 7.2 − 2 = 5.2 ✓
- Type y = x^2 − 6x + 4.
- The graph is a parabola that opens upward, so the minimum is at the vertex.
- Click the lowest point: Desmos shows (3, −5).
- The minimum value is the y-coordinate, −5, not the x-coordinate 3.
- Check: 3² − 6(3) + 4 = 9 − 18 + 4 = −5 ✓
- Type 3x + 4y = 12 and kx + 8y = 30. Accept the slider for k.
- Drag k until the two lines do not cross: they become parallel at k = 6.
- Confirm by algebra: 2 × (3x + 4y = 12) gives 6x + 8y = 24. With k = 6 the second equation is 6x + 8y = 30, so the left sides match and the right sides differ: parallel lines, no solution ✓
- Missing brackets(x + 2)/4 is not x + 2/4. Look at the screen to see exactly what Desmos reads: a wrong typed expression gives a wrong graph.
- Giving the y-value when the question asks for xFor an equation solved by graphing, the answer is the x-coordinate of the crossing point. For a maximum or minimum value, the answer is the y-coordinate.
- Trusting a rounded decimalDesmos rounds. If a point reads 2.333, the exact value may be 7/3. In a typed answer enter 7/3 or 2.333 (use all the space you have); 2.33 is not precise enough.
- Not seeing the intersectionIf the point is off the screen, zoom out with the minus button or use the home button to reset the view. Try values of x and y that make sense for the story.
- Using Desmos for everythingIf you can solve it in two lines of algebra, do it. Use Desmos when it is clearly faster or when you want a second check.
- Forgetting that the graph shows pure numbersIn a story, the axes mean something (hours, dollars). Answer in the units asked, and think about whether a negative or fractional answer makes sense.
This whole lesson is about Desmos, so here is the minimum routine for any question: (1) decide if graphing helps; (2) type the left and right sides, or the equation, with brackets; (3) click the important point; (4) answer the question that was asked; (5) check one value by hand.
- Open Desmos with the calculator button in the Math section
- Type each expression on its own line
- Click a point to read its coordinates
- Drag a slider if there is a constant k
Practice on the Bluebook practice tests: the Desmos there is the same one you will use on test day.
Open Desmos ↗