Question

a quadratic function $f(x)$ is hidden from view. you must find all intervals where $f(x)$ is negative. choose the form of the quadratic function $f(x)$ that you would like to see in order to answer the question most efficiently.

form: select a form

answer attempt 1 out of 2

submit answer
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Explanation:

To determine where a quadratic function \( f(x) \) is negative, the factored form (also called the intercept form) \( f(x) = a(x - r_1)(x - r_2) \) is most efficient. Here's why:

Key Reasoning:

• The factored form reveals the roots (\( r_1 \) and \( r_2 \)) of the quadratic, which are the \( x \)-intercepts.
• The leading coefficient \( a \) tells us if the parabola opens upward (\( a > 0 \)) or downward (\( a < 0 \)).
• With the roots and the direction of opening, we can quickly determine the intervals where \( f(x) < 0 \):
• If \( a > 0 \) (opens upward), \( f(x) < 0 \) between the two roots (\( (r_1, r_2) \) if \( r_1 < r_2 \)).
• If \( a < 0 \) (opens downward), \( f(x) < 0 \) outside the roots (\( (-\infty, r_1) \cup (r_2, \infty) \) if \( r_1 < r_2 \)).

Answer:

The most efficient form is the factored form (e.g., \( f(x) = a(x - r_1)(x - r_2) \)). When selecting from a dropdown, choose the option labeled "Factored Form" (or "Intercept Form").