Desmos SAT Launchpad

What visual condition in Desmos means "no solution"?

Value of \(c\):

L4 Level 4: The Equivalence Check

Overlapping Graphs

Question 5 (Rational Expression Equivalence) Medium

Which of the following is equivalent to \(\frac{3x^2 - 12}{x - 2}\) for all \(x \neq 2\)?

A) \(3x - 6\)

B) \(3x + 6\)

C) \(3x^2 - 6\)

D) \(x + 6\)

How did you confirm the correct answer in Desmos?

Correct Choice:

Strategic Debrief: Pencil vs. Desmos

1. Which question above was vastly FASTER on Desmos than by hand? Why?

2. Name one situation where using Desmos might actually WASTE your time:

Digital PSAT/SAT Prep Lab • Mathematics Page 2 of 2 • Desmos SAT Sprint

4. Answering What is Asked:

College Board purposely asks for \(x+y\), \(x_2 - x_1\), or \(2x\). Remind Navigators that their primary job is verifying the final prompt question before writing.

Desmos SAT Launchpad • Teacher Facilitation Guide Page 1 of 2

ANSWER KEY & SOLUTION PROTOCOLS | Exact Keystrokes & Explanations

All 5 Sprint Items

Calibration Run: \(3x + 2y = 18\) and \(y = 2x - 5\) Answer: 7

Desmos Action: Type both lines directly. Click intersection at \((4, 3)\).

Trap Alert: Students write 4 or 3. The prompt asks for \(x + y = 4 + 3 = \mathbf{7}\).

Q1: \(y = x^2 - 4x + 3\) & \(y = 2x - 2\) Correct: C (Value = 4)

Desmos Keystrokes: Line 1: y = x^2 - 4x + 3 | Line 2: y = 2x - 2

Intersections Found: \((1, 0)\) and \((5, 8)\). Thus \(x_1 = 1\) and \(x_2 = 5\).

Calculation: \(x_2 - x_1 = 5 - 1 = \mathbf{4}\). Choice C.

Algebraic Check: \(x^2 - 4x + 3 = 2x - 2 \implies x^2 - 6x + 5 = 0 \implies (x-1)(x-5) = 0\).

Q2: \(4.5x - 1.5y = 12\) & \(3.0x + 2.5y = 31\) Answer: 6.4 or 32/5

Desmos Keystrokes: Type both lines without converting decimals to fractions.

Intersection: Gray dot appears at \((4.8, 6.4)\).

Target Value: Question asks for \(y = \mathbf{6.4}\) or \(\mathbf{32/5}\).

Time Saved: ~90 seconds compared to hand elimination with decimal multiplication.

Q3: \(h(t) = -16t^2 + 96t + 20\) Answer: 164 feet

Desmos Keystrokes: y = -16x^2 + 96x + 20 (Desmos prefers \(x\) and \(y\)).

Window Adjustment: Zoom out on the y-axis until the parabola apex is visible.

Vertex Dot: Gray dot at \((3, 164)\). Maximum height = \(\mathbf{164}\).

Trap: \(t = 3\) is the time to reach maximum, NOT the maximum height itself.

Q4: \(cx - 6y = 15\) & \(4x - 8y = 21\) (No Solution) Answer: c = 3

Desmos Slider Method: Type Line 1 with letter \(c\). Click add slider: c. Type Line 2.

Visual Rule: "No solution" means lines are parallel (same slope, different y-intercepts). Drag slider until lines never intersect: occurs exactly when \(c = \mathbf{3}\).

Algebraic Proof: Slope 2 = \(4/8 = 1/2\). Slope 1 = \(c/6\). Setting \(c/6 = 1/2 \implies c = 3\). (Intercepts: \(-21/8 \neq -15/6\)).

Q5: Equivalent to \((3x^2 - 12)/(x - 2)\) Correct: B (3x + 6)

Desmos Overlap Method: Enter prompt into Line 1. Enter choices A, B, C, D into subsequent lines.

Result: Choice B (\(y = 3x + 6\)) overlays perfectly onto Line 1 everywhere (except a point hole at \(x = 2\)).

Algebraic Proof: \(\frac{3(x^2 - 4)}{x - 2} = \frac{3(x-2)(x+2)}{x - 2} = 3(x+2) = 3x + 6\).

Support / Scaffolding:

For students struggling with window scaling on Q3, write standard SAT window settings on the board: \(X: [-10, 10]\), \(Y: [-20, 200]\). Pair struggling students with a confident Desmos user.

Fast Finisher Extension:

Challenge fast pairs to find the equation of a line passing through \((3, 14)\) and \((7, 30)\) using a Desmos table and linear regression: y1 ~ mx1 + b.

Desmos SAT Launchpad • Teacher Facilitation Guide Page 2 of 2