Standard, Factored, and Vertex Forms

6.4: Standard, Factored, and Vertex Forms. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 4 of 83 parts Practice Flashcards

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Read vertex form

Objective: Identify vertex coordinates and evaluate a vertex-form function.

Prerequisite skills: quadratic expansion and completing the square. In f(x) = a(x − h)² + k, the vertex is (h,k). The minus sign inside means the horizontal coordinate is h, not −h. The sign of a controls whether the vertex is a minimum or maximum. Common error: reversing the vertex horizontal coordinate. Restrictions and interpretation: Equivalent forms describe the same function and preserve all coefficients.

Formulas

f(x) = a(x − h)² + k has vertex (h,k)

Worked examples

Try itFor f(x) = 2(x − 23)² + 26, what is the x-coordinate of the vertex?Show answer

The function matches a(x − h)² + k with h=23 and k=26. The vertex x-coordinate is 23. At x=23, the squared term is zero. Therefore the requested answer is 23.

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Read vertex form

Objective: Identify vertex coordinates and evaluate a vertex-form function.

Formulas

f(x) = a(x − h)² + k has vertex (h,k)

Worked examples

Try itFor f(x) = 2(x − 23)² + 26, what is the x-coordinate of the vertex?Show answer

The function matches a(x − h)² + k with h=23 and k=26. The vertex x-coordinate is 23. At x=23, the squared term is zero. Therefore the requested answer is 23.

Convert factored form

Objective: Expand factored form to read a standard-form coefficient.

Factored form exposes zeros; standard form exposes the y-intercept directly. Expand both binomials and include any outside multiplier. Use equivalent forms according to the quantity requested. Common error: dropping the outside multiplier during expansion. Restrictions and interpretation: Equivalent forms describe the same function and preserve all coefficients.

Formulas

$a(x − r)(x − s) = ax² − a(r+s)x + ars$

Worked examples

Try itIf f(x) = 2(x − 23)(x − 26) is written as ax² + bx + c, what is b?Show answer

Expand the inner product to x² − 49x + 598. Multiply by 2: the coefficient of x is b = −2(49) = -98. Therefore the requested answer is -98.

Convert standard to vertex form

Objective: Complete a square to determine a vertex value.

For a = 1, group the x² and x terms, add and subtract the needed square constant, and rewrite. For other nonzero a, first factor a from the variable terms. The added and subtracted constants must cancel in the final expression. Common error: adding a completing constant without compensating for it. Restrictions and interpretation: Equivalent forms describe the same function and preserve all coefficients. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$x² − 2hx + c = (x − h)² + c − h²$

Worked examples

Try itFor f(x) = x² − 52x + 700, what is the y-coordinate of its vertex?Show answer

Rewrite as (x − 26)² + 700 − 676 = (x − 26)² + 24. Thus the vertex y-coordinate is 24. Therefore the requested answer is 24.

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