SAT-84: Mastering Desmos II — Sliders for Unknown Constants

SAT-84: Mastering Desmos II — Sliders for Unknown Constants

Description: When a problem has an unknown constant (like k) and a condition (one solution, passes through a point, etc.), Desmos lets you create a slider for that letter and drag it until the condition is met. It turns "solve for k" into "watch the graph."

What a slider is

If you type an equation with a letter Desmos doesn't recognize as a variable (say k), it offers to add a slider. The slider lets you change k smoothly and see the graph move in real time. (Oʻzbekcha: slayder — nomaʼlum doimiyni (masalan k) silliq oʻzgartirib, grafik qanday oʻzgarishini koʻrsatadi.)

The method

1. Type the equation using the unknown letter; tap "add slider" when prompted.
2. Decide what the condition looks like on the graph (e.g. "touches the x-axis once" = one solution).
3. Drag the slider until that picture appears; read the value.

(Oʻzbekcha: shartni grafikda qanday koʻrinishini tasavvur qiling, soʻng slayderni shu holatga keltiring.)

Worked Example 1 — pass through a point

For what k does y = kx + 1 pass through (2, 7)?

• Add a slider for k and drag until the line hits (2, 7). It lands at k = 3.
• Check: 3(2) + 1 = 7 ✓. So k = 3.

Worked Example 2 — exactly one solution

For what positive k does x² + kx + 9 = 0 have exactly one solution?

• Graph y = x² + kx + 9 with a slider. One solution means the curve just touches the x-axis.
• Slide k until it touches: that happens at k = 6. (Confirms the discriminant: k² − 36 = 0.)
• So k = 6.

(Oʻzbekcha: bitta yechim — parabola x oʻqiga tegib oʻtadi; slayderni shu holatga keltiramiz.)

Worked Example 3 — match two graphs

For what c do the lines y = 2x + c and y = −x + 9 intersect on the y-axis?

• "On the y-axis" means at x = 0. The second line hits the y-axis at (0, 9).
• Slide c until y = 2x + c also passes (0, 9): that's c = 9. So c = 9.

Tip: sliders are fastest when the condition is visual — "tangent," "passes through," "same intercept." (Oʻzbekcha: shart koʻrinarli boʻlsa (urinma, nuqtadan oʻtadi), slayder eng tez yoʻl.)

Worked Example 4 — count solutions as k changes

The equation x² − 4x + k = 0 can have 0, 1, or 2 solutions depending on k. Use a slider to explore.

• Graph y = x² − 4x + k with a slider. As you raise k, the parabola lifts upward.
• It crosses the x-axis twice for small k, touches once at k = 4, and misses entirely for k > 4.
• So the number of real solutions drops from 2 to 1 to 0 as k passes 4 — you can literally watch it.

(Oʻzbekcha: slayderni surganingizda parabola koʻtariladi va yechimlar soni 2 → 1 → 0 ga oʻzgaradi.)

Set a sensible slider range

By default a slider runs from −10 to 10 in small steps, which is fine for most SAT problems. If your target value seems to sit at the very edge, widen the range by clicking the slider's endpoints and typing new limits. Narrowing the step size also lets you land on exact values like 6 instead of 5.98. Tuning the slider takes seconds and makes the read-off precise. (Oʻzbekcha: slayder oraligʻini va qadamini sozlash aniq qiymatni topishga yordam beradi.)

Practice 1

For what k does the line y = kx pass through (4, 12)?

Practice 2

For what value of b does y = x + b pass through the point (5, 2)?

Key words — Kalit soʻzlar

• Slider — slayder
• Unknown constant — nomaʼlum doimiy
• Condition — shart
• Pass through — nuqtadan oʻtish
• Tangent (touches once) — urinma (bir marta tegadi)
• Drag — surish (tortish)
• Real time — real vaqtda
• y-axis — y oʻqi
• Value — qiymat

Summary

• Type the unknown letter and add a slider to change it live.
• Translate the condition into a picture (touches once, passes through, same intercept).
• Drag until the picture appears, then read the value.