Equivalent Forms of a Function
Aligned to F-IF.C.8 — Common Core State Standards for Mathematics.
What this lesson teaches
Rewriting a function by factoring or completing the square exposes different key features. Factored form reveals the zeros, while vertex form reveals the extremum (maximum or minimum) and the axis of symmetry.
Worked example
Take f(x) = x^2 − 6x + 5. Factoring gives (x − 1)(x − 5), so the zeros are x = 1 and x = 5. Completing the square gives (x − 3)^2 − 4, so the vertex is (3, −4) and the axis of symmetry is x = 3.
Practice questions
- Factor f(x) = x^2 − 9 to find its zeros.
- Complete the square on g(x) = x^2 + 4x + 1 to find its vertex.
- Rewrite h(x) = 2x^2 − 8x + 6 in vertex form and state its minimum value.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Formas equivalentes de una función
Reescribir una función factorizando o completando el cuadrado revela distintas características clave. La forma factorizada muestra los ceros, mientras que la forma de vértice muestra el extremo (máximo o mínimo) y el eje de simetría.
Ejemplo: Toma f(x) = x^2 − 6x + 5. Al factorizar se obtiene (x − 1)(x − 5), así que los ceros son x = 1 y x = 5. Al completar el cuadrado se obtiene (x − 3)^2 − 4, entonces el vértice es (3, −4) y el eje de simetría es x = 3.
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