How to use Desmos sliders for "value of k" questions on the SAT
Updated
"For what value of ..." questions are where many students lose time on SAT math: the algebra means comparing slopes, matching coefficients, or using the discriminant. With Desmos, you don't need to remember which. Give the constant a slider, drag it, and watch for the moment the graph does what the question describes.
This guide shows how sliders work, four SAT-style examples, and how to get an exact answer when dragging lands near one.
The method in three steps
- 1Type the equations with the constant in them, like . Desmos shows an add slider button for . Click it.
- 2Drag the slider and watch the graph move. Press the play button to animate it if you like.
- 3Stop when the graph matches the question: passing through a point, lines parallel, lines on top of each other, or a curve just touching a line or the -axis.
Example 1: through a point
Hit the point
Type and add the slider. Then type on its own line so Desmos plots the point. Drag until the line runs through the point: at .
Check: .
Answer:
Example 2: infinitely many solutions
Make the lines overlap
Type both equations and add a slider for . Drag until the second line lands exactly on top of the first. That happens at .
Infinitely many solutions means the two equations describe the same line. Algebra confirms it: multiplying by 3 gives .
Answer:
Example 3: exactly one solution
Slide until they just touch
Graph each side: and . Add a slider for . Raising lifts the parabola. At first it crosses the line twice; slide up until it touches the line at a single point. That's .
Check with the discriminant: has one solution when , so .
Answer:
Example 4: an answer that isn't a whole number
Drag close, then type it
Type and , and add a slider for . Dragging gets the curve close to the point somewhere between 2 and 3. Click the slider's value, type , and the curve passes right through it.
Check: .
Answer:
Mistakes that cost points
- An answer outside the slider's range. If nothing works between and , widen the range by clicking its end numbers.
- Trusting a dragged value. Dragging gets you close. Type the exact candidate into the slider to confirm it.
- No solution vs. infinitely many. Parallel lines that never meet: no solution. The same line twice: infinitely many.
- Using a slider when algebra is one step. If plugging in a point gives the constant right away, as in Example 1, that can be faster.
Practice this move
Practice on real SAT-style problems
Intersect Prep puts the same Desmos calculator beside 42 SAT-style problems that use this move, with a step-by-step strategy for each one. Create a free account to try a full lesson with the calculator; a pass unlocks the rest.
Questions
- How do you make a slider in Desmos?
- Type an expression with a letter Desmos doesn't recognize, like k in y = kx − 3. Desmos shows an "add slider" button; click it. You can also type k = 1 on its own line.
- Can you use sliders on the digital SAT?
- Yes. The Desmos calculator in the Bluebook app supports sliders, just like the free Desmos website.
- How do I change a slider's range or step?
- Click the numbers at either end of the slider. You can set the minimum, maximum, and step size, or click the current value and type an exact number.
- Which SAT questions are sliders good for?
- Any question with an unknown constant: "for what value of k" questions about solutions of a system, a graph passing through a point, or a curve touching a line or the x-axis.