Question

question which equation choice could represent the graph shown below? answer \\( f(x) = (x + 4)^2(x - 2)^2 \\) \\( f(x) = (x + 4)^2(x - 2) \\) \\( f(x) = (x - 4)^2(x + 2) \\) \\( f(x) = (x - 4)^2(x + 2)^2 \\)

Explanation:

Step1: Identify x-intercepts

The graph touches or crosses the x - axis at \(x=-4\) (with a double root, since the graph touches the axis here, indicating even multiplicity) and \(x = 2\) (with a single root, since the graph crosses the axis here, indicating odd multiplicity).

Step2: Form the function from roots

If a polynomial has a root at \(x = a\), then \((x - a)\) is a factor. For a double root at \(x=-4\), the factor is \((x + 4)^2\) (since \(x-(-4)=(x + 4)\)). For a single root at \(x = 2\), the factor is \((x - 2)\). So the polynomial function is of the form \(f(x)=(x + 4)^2(x - 2)\).

Answer:

\(f(x)=(x + 4)^2(x - 2)\) (the option corresponding to this function in the given choices)