Sliders & Parameters
Turn algebra problems into visual ones. When a question asks "for what value of $k$...", a slider answers it in seconds.
How sliders work
When you type an equation using any letter that isn't $x$ or $y$, Desmos recognizes it as a parameter. A prompt appears: "Add slider: a", click it. The slider bar drops in, and dragging it live-updates the graph.
- Default slider range: $-10$ to $10$.
- Change the range: click the slider endpoints to type custom min/max (e.g., $0$ to $100$).
- Step size: click the step button to set increments (e.g., $0.1$ for fine tuning, $1$ for integer-only).
- Multiple parameters: use letters like
a, b, k simultaneously, each gets its own slider.
Worked example: "one solution" systems
Problem: For what value of $k$ does the system below have exactly one solution?
$\quad y = x^2$
$\quad y = 4x + k$
- Type
y=x^2.
- Type
y=4x+k. Click "Add slider: k".
- Drag the $k$ slider. Watch the line rise and fall across the parabola.
- One solution = the line is tangent to the parabola (touches at exactly one point). This happens around $k = -4$.
- Refine the step to $0.1$ and fine-tune. Exact answer: $k = -4$.
Algebra path: set $x^2 = 4x + k$, discriminant = 0, solve. Works but takes ~90 seconds. Slider: 15 seconds.
Worked example: vertex translation
Problem: The graph of $y = (x - h)^2 + 3$ passes through $(5, 7)$. What is $h$?
- Type
y=(x-h)^2+3. Add slider for $h$.
- Type the point:
(5, 7). A dot appears on the graph.
- Drag $h$ until the parabola passes exactly through the point. Around $h = 3$, the fit is exact.
- Check: $(5-3)^2 + 3 = 4 + 3 = 7$. ✓
Answer: $h = 3$.
Worked example: exponential growth
Problem: A population is modeled by $P = 200 \cdot b^{t}$. If $P = 450$ when $t = 3$, what is $b$? (Round to 2 decimals.)
- Type
y=200*b^x. Add slider for $b$, range $1$ to $2$, step $0.01$.
- Type the point:
(3, 450).
- Drag $b$ until the curve hits the point. $b \approx 1.31$.
Answer: $b \approx 1.31$.
When sliders shine
- "For what value of $k$...", single parameter.
- "For what values of $a$ and $b$...", two parameters, adjust each.
- "How does changing $c$ affect the graph of...", conceptual questions. Drag the slider and describe what changes.
- Discriminant questions (one / no / two solutions), the tangency case is visual.
Common traps
Watch for these gotchas
- Answer needs to be exact, not a slider-visual estimate. Fine-tune step size down to $0.01$ or read the intersection coordinates instead of eyeballing.
- Letter collisions. Don't use $x$ or $y$ as parameters, they're reserved for axes. Use $a$, $b$, $c$, $k$, $m$, $h$.
- Slider range missed the answer. If the slider hits its max without solving, click the endpoint to expand the range.
Try it yourself
Practice problem
For what value of $k$ does the system $y = x^2 + 4x$ and $y = -2x + k$ have exactly one solution? Use a slider for $k$ and drag until the line is tangent to the parabola.
Show answer
Steps: Type y=x^2+4x, then y=-2x+k and add a slider for $k$. Drag until the line just kisses the parabola at a single point. Fine-tune step to $0.1$, then $0.01$. Answer: $k = -9$.
Algebraic check: set $x^2 + 4x = -2x + k \Rightarrow x^2 + 6x - k = 0$. Discriminant $= 36 + 4k = 0 \Rightarrow k = -9$. ✓
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