SAT Math — finding the vertex from any form: −b/2a and completing the square

Questions about the vertex of a parabola are frequent in the Advanced Math domain of the digital SAT, and the College Board's skill description asks students to choose the form of a function that displays a key feature. The difficulty usually lies in the leading coefficient: when it is not 1, completing the square and the formula x = −b/(2a) both require an extra step that is easy to forget. This material trains the vertex from every form.

From standard form you find the vertex with x = −b/(2a) when a = 2, the maximum of a downward parabola with a = −4, and the value of a that puts the vertex at x = −2. Completing the square is practiced with a leading coefficient of 3 (where the subtracted square must be multiplied by 3), with a fractional half-coefficient, to solve 2x² − 8x − 10 = 0, and to decide whether a function has a minimum or a maximum and where. From factored form you find the vertex midway between the zeros, and from two equal outputs you locate the axis of symmetry.

Two context questions ask for the minimum daily cost of a production model and the greatest height of a thrown ball, where the answer is the y-coordinate of the vertex, not the x-coordinate. One question asks which equivalent form shows the minimum value directly, and includes a form that looks right but is not equivalent.

The material offers a quiz of twelve four-option questions, each with an explanation that shows the method and names the slip behind each wrong option, and a set of flashcards that collects the rules to remember.

The content is based on the College Board's published skill descriptions for the Advanced Math domain of the digital SAT Math section. It is independent practice and is not produced or endorsed by the College Board.

• Find the vertex of y = ax² + bx + c with x = −b/(2a) for any leading coefficient
• Complete the square when the leading coefficient is not 1
• Choose the equivalent form that displays the minimum, the maximum or the zeros
• Find the axis of symmetry from zeros or from two equal outputs
• Answer maximum and minimum questions in context with the vertex

Practice material written by Zestly, based on the College Board's published description of the digital SAT Math section (Advanced Math domain: nonlinear functions — quadratic functions and the form that displays key features; equivalent expressions), as given on satsuite.collegeboard.org and in the Assessment Framework for the Digital SAT Suite.