Question

which equation choice could represent the graph shown below?
answer
$f(x) = (x + 6)(x^2 - 36)$
$f(x) = x(x^2 - 36)$
$f(x) = x(x^2 + 36)$
$f(x) = (x - 6)(x^2 - 36)$

Explanation:

Step1: Identify x-intercepts

From the graph, the x - intercepts are \(x=-6\), \(x = 0\), and \(x=6\). This means the roots of the function \(f(x)\) are \(x=-6\), \(x = 0\), and \(x = 6\). So the factors of the function should be \((x + 6)\), \(x\), and \((x-6)\) (since if \(x=a\) is a root, then \((x - a)\) is a factor).

Step2: Form the function from factors

We know that \((x + 6)(x - 6)=x^{2}-36\) (by the difference of squares formula \(a^{2}-b^{2}=(a + b)(a - b)\) where \(a=x\) and \(b = 6\)). So the function with factors \(x\), \((x + 6)\), and \((x - 6)\) can be written as \(f(x)=x(x^{2}-36)\) (since \((x + 6)(x - 6)=x^{2}-36\)).

Let's check the other options:

• For \(f(x)=(x + 6)(x^{2}-36)=(x + 6)(x + 6)(x - 6)=(x + 6)^{2}(x - 6)\), the root \(x=-6\) has a multiplicity of 2, so the graph should touch the x - axis at \(x=-6\) (not cross it), but in the given graph, the function crosses the x - axis at \(x=-6\), so this is incorrect.
• For \(f(x)=x(x^{2}+36)\), the roots of \(x^{2}+36 = 0\) are complex (\(x=\pm6i\)), so the only real root is \(x = 0\), which does not match the x - intercepts of the graph.
• For \(f(x)=(x - 6)(x^{2}-36)=(x - 6)(x - 6)(x + 6)=(x - 6)^{2}(x + 6)\), the root \(x = 6\) has a multiplicity of 2, so the graph should touch the x - axis at \(x = 6\) (not cross it), but in the given graph, the function crosses the x - axis at \(x = 6\), so this is incorrect.

Answer:

\(f(x)=x(x^{2}-36)\)