Question

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

Explanation:

Step1: Identify x-intercepts

The graph touches the x - axis at \(x=-4\) and \(x = 2\). When a graph touches the x - axis at a root \(r\), the multiplicity of the root is even. So the factors corresponding to the roots should be squared.

Step2: Form the equation from roots

If the roots are \(x=-4\) (i.e., \(x + 4 = 0\)) and \(x = 2\) (i.e., \(x-2=0\)) with even multiplicities (since the graph touches the x - axis at these points), the function should be of the form \(f(x)=(x + 4)^{2}(x - 2)^{2}\).

Answer:

\(f(x)=(x + 4)^{2}(x - 2)^{2}\) (the second option)