Question

201805na - s3 - 12
the graph of the function f is shown in the xy - plane above, where y = f(x). which of the following functions could define f?
a) $f(x)=(x - 3)(x - 1)^2(x + 2)^2$
b) $f(x)=(x - 3)^2(x - 1)(x + 2)$
c) $f(x)=(x + 3)(x + 1)^2(x - 2)^2$
d) $f(x)=(x + 3)^2(x + 1)(x - 2)$

Explanation:

Step1: Identify x-intercepts

From the graph, the function touches or crosses the x - axis at \(x=-3\), \(x = - 1\) (with a double root, since the graph touches the axis here), and \(x = 2\) (with a double root, since the graph touches the axis here).

Step2: Determine the factors

For a root \(r\) of a function \(y = f(x)\), the factor is \((x - r)\). If the graph touches the x - axis at \(r\), the multiplicity of the root \(r\) is even (so the factor is squared), and if it crosses the x - axis, the multiplicity is odd.

• At \(x=-3\), the graph crosses the x - axis, so the multiplicity of the root \(x = - 3\) is 1. The factor is \((x+3)\) (since \(x-r=x - (-3)=x + 3\)).
• At \(x=-1\), the graph touches the x - axis, so the multiplicity of the root \(x=-1\) is 2. The factor is \((x + 1)^{2}\) (since \(x-r=x-(-1)=x + 1\)).
• At \(x = 2\), the graph touches the x - axis, so the multiplicity of the root \(x = 2\) is 2. The factor is \((x - 2)^{2}\) (since \(x-r=x - 2\)).

So the function should be of the form \(f(x)=(x + 3)(x + 1)^{2}(x - 2)^{2}\), which matches option C.

Answer:

C) \(f(x)=(x + 3)(x + 1)^{2}(x - 2)^{2}\)