Sliders & animation
Sliders turn a static plot into a tiny interactive simulation. They are the single best way to understand a function — instead of guessing what each constant does, you watch.
Creating a slider
Type a variable definition on its own line:
a = 1
DesmosGraph automatically recognises that a is a free parameter and shows a slider with a default range of −10 to 10. Now you can use a anywhere in your other expressions:
y = a*x^2
Drag the slider and the parabola changes width and direction in real time.
Customising the range
Each slider has small min and max input boxes on either side. Change them to focus on the range you care about. For phase variables a typical choice is 0 to 2π; for amplitudes maybe −5 to 5.
Multiple sliders together
Combine sliders for more powerful exploration:
y = a*sin(b*x + c) + d a = 1 b = 1 c = 0 d = 0
Now you have full control over amplitude (a), frequency (b), phase shift (c) and vertical shift (d) — the four classic transformations of a sinusoid.
Animation
Click the small triangle next to a slider's label. The parameter sweeps back and forth between min and max automatically. This is invaluable for:
- Showing how a graph morphs as a parameter changes.
- Demonstrating what "phase shift" looks like in real life.
- Creating mesmerising visuals for class.
Click the icon again to pause.
Lesson ideas for teachers
Quadratic transformations
y = a*(x - h)^2 + k a = 1 h = 0 k = 0
Ask students to predict the effect of each slider before they touch it. Then verify.
Linear regression intuition
y = m*x + b m = 1 b = 0
Plot a few sample points (you can add point coordinates as (2, 5)-style entries via implicit pairs in future updates) and let students adjust the line to fit visually.
The unit circle and trig
x^2 + y^2 = 1 y = sin(t) t = 0
Animate t and discuss why sin(t) is the y-coordinate of the unit-circle point at angle t.
Smart use of sliders
- Use a slider for integer parameters by setting the step (we use a sensible default; very small steps slow animation slightly).
- Combine an animated slider with implicit equations — the heart curve scales with a parameter, for instance.
- For demonstrations, use a wider min/max so the change is visually dramatic.
Ready to dive deeper? Browse examples or read about our latest articles.