Desmos for the SAT · Technique 12 of 15
Use a Slider to Test an Unknown Constant
Drag a value around until the graph does what the question describes.
Some questions give you an equation with an unknown constant, like a in ax + 4y = 20, and ask for the value that makes some condition true, like passing through a given point.
Type the equation in with a letter standing in for the unknown, and Desmos offers to turn that letter into a slider. Drag it until the graph matches the condition, and read the value off the slider.
When the condition is about the number of solutions, read the intersections as you drag: one grey dot where the graphs touch is exactly one solution (tangent), graphs that never touch is none, and two graphs that merge into one is infinitely many.
Nova
This one feels like cheating in the best way. When a question hands you an equation with a mystery letter in it, you don't have to solve for it, you just make it a slider and drag until the graph does exactly what the question asks. Watching the line swing into place as you slide is honestly kind of fun.
The algebra way vs the Desmos way
This technique sidesteps solving for the unknown constant algebraically.
The algebra way
- Substitute the point into the equation:
a(2) + 4(3) = 20. - Simplify:
2a + 12 = 20, so2a = 8. - Divide by 2:
a = 4. One slip moving the12across and the point is gone.
The Desmos way
- Type
ax + 4y = 20and add a slider for the lettera. - Plot the point
(2, 3)and dragauntil the line passes through it, landing ona = 4.
The payoff: Dragging the slider until the line hits the point is quicker than the substitution, but the slider only estimates, so confirm the exact value with the algebra.
See the move, live
That same example, already loaded in the real calculator, the one built into Bluebook on test day. Try the move yourself.
Before you start: set up Desmos
Convert decimals to fractions
Desmos shows decimals like 1.333. SAT choices are usually fractions. Click the round fraction-converter button next to a value to turn 1.333 into 4/3.
Widen the axis bounds when the graph looks empty
Desmos opens on a small window from -10 to 10 on both axes. If a crossing, vertex, or data point sits outside that box, the graph can look empty even when an answer exists. Open the wrench icon and widen the x and y bounds until the point you need comes into view.
Step by step
Nova
Follow along one step at a time, hit Next as you go. I'll flag the spots where people slip.
Type the equation using a letter other than x or y for the unknown constant, e.g. a x + 4 y = 20.
Syntax: Use a letter other than x or y, e.g. a x + 4 y = 20, then click "add slider: a" and drag a to test the condition.
Now try it on a real SAT question
The equation ax + 4y = 20 is graphed in the xy-plane, where a is a constant. The graph passes through the point (4, 2).
What is the value of a?
Pick an answer to get instant coaching from Nova.
Whatever you do, do not name your slider x or y, Desmos reserves those for the axes and will not let you. Pick literally any other letter and drag it until the line lines up with the point you are testing.
Substituting the point: a(4) + 4(2) = 20, so 4a + 8 = 20, and 4a = 12, giving a = 3. Dragging the slider a until the line ax + 4y = 20 passes through (4, 2) lands on the same value.
In the app, drills like this adapt to your level, and your AI coach can walk you through any step you get stuck on.
Nova
Whatever you do, do not name your slider x or y, Desmos reserves those for the axes and will not let you. Pick literally any other letter and drag it until the line lines up with the point you are testing.
Set it up yourself in the blank calculator, then read the answer straight off the graph. Muscle memory beats watching.
Watch it
The Desmos trap: when to close the calculator
- If the constant only needs to satisfy one clean substitution, like this one, solving it algebraically by plugging in the point takes about as long as dragging a slider. Skip the slider when the algebra is that direct.
- Sliders are best for conditions you can only test visually, like two graphs staying tangent or never intersecting, not for values you could solve for with one substitution.
- A slider can reveal two values that both fit, like k = 4 and k = -4. Reread the question for a constraint like k > 0 to choose the right one.
- Sliders default to a range of -10 to 10, so a large constant like 25 or 50 sits past the end. Click the small numbers at the ends of the slider to widen the range.
- If the condition is a solution count, confirm the exact value with the discriminant: b2 - 4ac = 0 means exactly one solution (tangency).
Nova
One honest catch: sliders eyeball it, they don't nail the exact value. So on the constant questions where the answer is 'no solution' or 'infinitely many solutions,' don't trust the drag. Set the slopes or matching coefficients equal by hand instead, because being off by a hair changes the whole answer.
Put this move into real practice
"Use a Slider to Test an Unknown Constant" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.
Practice freeSAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.