Final answer:
The standard form of a quadratic function is expressed as ax² + bx + c, where a ≠ 0. The correct option is A, stating that the standard form of a quadratic function is f(x) = a(x - h)² + k.
Step-by-step explanation:
When dealing with quadratic functions, which are second-order polynomials, it's important to understand the various forms they can be written in and how these forms are associated with different aspects of their algebraic properties.
The standard form of a quadratic equation is f(x) = ax² + bx + c, where a ≠ 0. However, another common form of the quadratic function is f(x) = a(x - h)² + k, which is known as the vertex form because it directly shows the vertex of the parabola as the point (h, k). The vertex form highlights the maximum or minimum value of the quadratic function, which is the y-coordinate of the vertex, k.
The option that states the vertex form is used for direct calculation of roots is incorrect; in fact, the standard form is normally converted into factored form or the quadratic formula is used to calculate the roots. The quadratic formula, √(b² - 4ac)/2a, is indeed derived from the standard form and is vital for finding the x-intercepts of the parabola when it is not easily factorable.
The standard form and factored form (f(x) = a(x - r1)(x - r2)) serve different purposes; factored form clearly shows the roots or x-intercepts of the parabola while standard form provides the coefficients that are used when utilizing the quadratic formula. Therefore, the statement that claims standard form is the same as factored form is incorrect.
In conclusion, based on the options provided, the correct option in the final answer is A) The standard form of a quadratic function is f(x) = a(x - h)² + k.