Question

a quadratic function $f(x)$ is hidden from view. you must find the $x$-intercept(s) of $f(x)$ and write the answer(s) in the form $(x, y)$. 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
select a form
factored form
standard form
vertex form
answer
there are no $x$-intercepts
submit answer

Explanation:

Step1: Understand x - intercepts and quadratic forms

To find the \(x\)-intercepts of a quadratic function \(y = f(x)\), we set \(y=0\) and solve for \(x\). The factored form of a quadratic function is \(f(x)=a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots (or \(x\)-intercepts when \(y = 0\)). If we have the factored form, we can directly set each factor equal to zero (\(x-r_1 = 0\) or \(x - r_2=0\)) to find the \(x\)-intercepts \((r_1,0)\) and \((r_2,0)\).

The standard form \(f(x)=ax^{2}+bx + c\) requires us to solve the quadratic equation \(ax^{2}+bx + c = 0\) (using quadratic formula or factoring, but factoring may not always be easy), and the vertex form \(f(x)=a(x - h)^{2}+k\) requires us to set \(a(x - h)^{2}+k=0\) and solve for \(x\), which involves more steps (isolating the squared term, taking square roots, etc.). So the factored form is the most efficient to find \(x\)-intercepts.

Step2: Choose the form

Among the given forms (Factored Form, Standard Form, Vertex Form), the Factored Form is the most efficient to determine the \(x\)-intercepts because it directly shows the roots (the \(x\)-values where the function crosses the \(x\)-axis) when we set \(f(x) = 0\).

Answer:

Factored Form