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)^2 \\)
\\( f(x) = (x + 7)^2(x + 1)^2 \\)

Explanation:

Step1: Identify x-intercepts

The graph touches or crosses the x - axis at \(x=-7\) and \(x = - 1\). At \(x=-7\) and \(x=-1\), the graph touches the x - axis (has a "bounce" or a repeated root), which means the multiplicity of the roots at these points is even.

Step2: Analyze the form of the function

If a function has a root at \(x = a\) with multiplicity \(n\), then the factor is \((x - a)^n\). For the root \(x=-7\), the factor is \((x + 7)\) (since \(x-(-7)=x + 7\)) and since the graph touches the x - axis here, the multiplicity is 2, so the factor is \((x + 7)^2\). For the root \(x=-1\), the factor is \((x + 1)\) (since \(x-(-1)=x + 1\)) and since the graph touches the x - axis here, the multiplicity is 2, so the factor is \((x + 1)^2\). So the function is of the form \(f(x)=(x + 7)^2(x + 1)^2\).

Answer:

\(f(x)=(x + 7)^2(x + 1)^2\) (the last option)