KD Academy | Elite SAT Prep

Desmos Assistance Engine

Loading Dashboard...

Solution ID:3aecb43e

Advanced MathQuadratic Equations and FunctionsQuadratic Vertex Form & Parabola Symmetry

Core Desmos Strategy

Use the axis of symmetry x = 10: since x = 8 and x = 12 are both 2 units away from the vertex price, the number of bowls sold at $8 must equal the number sold at $12 (5 bowls).

Solved Result

Answer: 5

Extracted Problem (Vision OCR)

Complete Stem & Options

A certain bowl was sold online for prices varying between $7 and $13. The number of bowls sold can be modeled by a quadratic function based on the price, where 5 bowls were sold for $12 each and the maximum of 9 bowls were sold for $10 each. According to the model, what is the number of bowls that were sold for $8 each?

Toggle Uploaded Screenshot

Desmos Calculator Operations

Input these exact cells into your Bluebook Desmos Graphing Calculator to solve in seconds.

desmosGraphing Calculator • Step-by-Step Blueprint

Graph / FunctionDefine the quadratic model in vertex form using vertex (10, 9). Desmos creates a slider for 'a'.

$f(x) = a(x - 10)^2 + 9$

Regression (~)Solve for the leading coefficient 'a' using the known point f(12) = 5.

$5 = f(12)$

$a = -1$

EvaluationEvaluate the function at the target price x = 8.

$f(8)$

$5$

Graph Inspection Tip:Set x from 6 to 14 and y from 0 to 12 to see the vertex at (10, 9) and the two symmetric points (8, 5) and (12, 5).

Analytical & Algebraic Solution

Step-by-step mathematical derivation with KaTeX proofs

▼

Setup & Conceptual Bridge

Let $x$ represent the price of a bowl in dollars, and let $y = f(x)$ represent the number of bowls sold.

Vertex Identification: The problem states that the maximum of bowls was sold for $10 each. For a quadratic function, a maximum or minimum occurs precisely at the vertex . Therefore, the vertex of this parabola is: This tells us the axis of symmetry of the parabola is the vertical line .

Alternative Solving Methods

Alternative SAT test-taking techniques & secondary verification strategies

Parabola Symmetry Shortcut

shortcut

Because the maximum occurs at a price of $10, the line of symmetry is x = 10. The price $8 is 2 units below the vertex (10 - 2 = 8), and $12 is 2 units above the vertex (10 + 2 = 12). By quadratic symmetry, f(8) = f(12) = 5.

Vertex Form Algebraic Modeling

algebraic

Model the function in vertex form: y = a(x - 10)² + 9. Substitute the point (12, 5) to find a: 5 = a(12 - 10)² + 9 ⇒ 5 = 4a + 9 ⇒ a = -1. Then evaluate at x = 8: y = -1(8 - 10)² + 9 = -4 + 9 = 5.

Ask a Follow-Up Question

Chat with the AI Math Tutor about this specific problem or Desmos steps

Have questions about why a number was picked or how a Desmos step works? Try one of these:

Extracted Problem (Vision OCR)

Desmos Calculator Operations

Analytical & Algebraic Solution

Setup & Conceptual Bridge

Alternative Solving Methods

Parabola Symmetry Shortcut

Vertex Form Algebraic Modeling

Ask a Follow-Up Question

Other Solved Questions

Find the known root w = -5, subtract it from the total sum 37/3 to find the second root, then solve for a directly in Desmos.

Pick small consecutive integers for a, b, and c in Desmos and directly evaluate the means and their difference.

Compute cumulative frequencies to determine which interval contains the 9th restaurant (the median position).

Graph y = x^2 + (q - 2)x + 55 with a slider for q, or graph the inequality (q - 2)^2 - 220 < 0 to visually find the largest integer upper bound.

Define the factored polynomial h(x) = (x + 5)(x - 6)(x - 7) in Desmos and evaluate h(0) to find the constant term c.

Identify the contextual definitions of the coordinates: x is unit price and y is revenue, meaning an x-intercept (a, 0) corresponds to a unit price producing $0 revenue.