Question

which equation choice could represent the graph shown below? answer \\( f(x) = (x + 7)(x - 1)^2 \\) \\( f(x) = (x - 7)(x + 1)^2 \\) \\( f(x) = (x + 7)^2(x - 1) \\) \\( f(x) = (x - 7)^2(x + 1) \\)

Explanation:

Step1: Identify x-intercepts

The graph intersects the x - axis at \(x = - 7\) (with a double root, since the graph touches the axis and turns around) and \(x = 1\). For a polynomial function in factored form, if \(r\) is a root, then \((x - r)\) is a factor. For a double root at \(x=-7\), the factor is \((x + 7)^2\) (since \(x-(-7)=(x + 7)\)) and for the root \(x = 1\), the factor is \((x - 1)\).

Step2: Form the function

So the function in factored form should be \(f(x)=(x + 7)^2(x - 1)\)

Answer:

\(f(x)=(x + 7)^2(x - 1)\) (the option corresponding to this function)