SAT-83: Mastering Desmos I — Graphing and Intersections

SAT-83: Mastering Desmos I — Graphing and Intersections

Description: The digital SAT gives you the Desmos graphing calculator. Many algebra questions become one-click problems if you graph them. This lesson covers the core move: type equations, then read the answer off the graph.

The big idea: solutions are points on a graph

• The solution(s) of one equation = where the graph crosses the x-axis (the x-intercepts, also called roots or zeros).
• The solution of a system = where the two graphs intersect.

So instead of solving by hand, you graph and click the special points; Desmos shows their coordinates. (Oʻzbekcha: tenglama yechimi — grafik x oʻqini kesgan nuqta; sistema yechimi — ikki grafik kesishgan nuqta.)

How to do it

1. Type each equation on its own line exactly as written (y = …, or the full equation).
2. Click the curve where it crosses the x-axis, or where two curves meet; Desmos labels the point.
3. Read the coordinate the question asks for (often just the x-value).

(Oʻzbekcha: har bir tenglamani alohida qatorga yozing, soʻng kerakli nuqtani bosib koordinatani oʻqing.)

Worked Example 1 — solve a quadratic

Solve x² − 5x + 6 = 0 with Desmos.

• Type y = x² − 5x + 6. The curve crosses the x-axis at x = 2 and x = 3.
• Solutions: x = 2 and x = 3 — no factoring needed.

Worked Example 2 — a linear system

Solve the system y = 2x + 1 and y = −x + 7.

• Type both lines. They intersect at (2, 5).
• Solution: x = 2, y = 5.

(Oʻzbekcha: ikki chiziq (2, 5) nuqtada kesishadi — bu sistema yechimi.)

Worked Example 3 — messy numbers

How many real solutions does x² + 3x + 5 = 0 have?

• Graph y = x² + 3x + 5. The curve never touches the x-axis.
• So there are 0 real solutions (matches the negative discriminant from SAT-34).

Tip: enter equations as "y = …". If a question is given as "f(x) = …", graph "y = f(x)" or type the expression directly. (Oʻzbekcha: tenglamalarni "y = ..." koʻrinishida kiriting.)

Worked Example 4 — solve an equation by graphing both sides

Solve 2x + 1 = x² − 2 using graphs.

• Graph y = 2x + 1 and y = x² − 2 separately; their intersections give the solutions.
• They meet at x = −1 and x = 3, so the equation's solutions are x = −1 and x = 3.

This "graph both sides" trick turns any equation into an intersection problem — very handy when one side is messy. (Oʻzbekcha: tenglamaning ikki tomonini alohida grafik qilib, kesishmalarni topamiz.)

Common setup mistakes in Desmos

A few errors waste time: forgetting to type the exponent with the caret (x^2, which Desmos renders as x²), writing "2x" as "2 x" with a stray space, or leaving off "y =" so nothing graphs. Also remember to zoom out if a curve seems to vanish — the interesting points may be off the default screen. A quick habit of checking that something actually plotted prevents most of these. (Oʻzbekcha: darajani ^ bilan yozing, "y =" ni unutmang, va grafik koʻrinmasa masshtabni kichraytiring.)

Practice 1

Use Desmos-style reasoning: the graph of y = x² − 9 crosses the x-axis where?

Practice 2

Lines y = 3x − 2 and y = x + 4 intersect at what point?

Key words — Kalit soʻzlar

• Graphing calculator — grafik kalkulyator
• x-intercept — x oʻqini kesish nuqtasi
• Root / Zero — ildiz / nol
• Intersection — kesishish nuqtasi
• System — sistema
• Curve — egri chiziq
• Coordinate — koordinata
• Plot / Graph — chizish
• Solution — yechim

Summary

• Solutions of one equation = x-intercepts; solution of a system = intersection point.
• Type each equation as y = …, then click the key points to read coordinates.
• Great for quadratics, systems, and "how many solutions" questions.