Question

which equation choice could represent the graph shown below?
answer
$f(x) = x(x + 1)(x + 3)$
$f(x) = (x + 1)(x + 3)$
$f(x) = x(x - 1)(x - 3)$
$f(x) = (x - 1)(x - 3)$
submit answer

Explanation:

Step1: Analyze the x-intercepts

The graph crosses the x - axis at \(x = 0\), \(x=- 1\), and \(x = - 3\)? Wait, no, looking at the graph, the roots (x - intercepts) can be determined by where the graph touches or crosses the x - axis. Wait, the graph shown has a root at \(x = 0\) (since it crosses the origin) and other roots? Wait, no, let's check the options. The function \(f(x)=x(x + 1)(x + 3)\) has roots at \(x=0\), \(x=-1\), \(x = - 3\). The graph is a cubic function (degree 3) because as \(x\to\pm\infty\), the function goes to \(\pm\infty\) (since the leading coefficient is positive for \(f(x)=x(x + 1)(x + 3)\), the leading term is \(x^3\)). The other options: \(f(x)=(x + 1)(x + 3)\) is a quadratic (parabola), \(f(x)=x(x - 1)(x - 3)\) has roots at \(x = 0\), \(x=1\), \(x = 3\) (which would be on the positive x - axis, but our graph has roots on the negative x - axis and at 0), \(f(x)=(x - 1)(x - 3)\) is a quadratic with roots at \(x = 1\) and \(x=3\) (positive x - axis). So the correct function should be a cubic with roots at \(x = 0\), \(x=-1\), \(x=-3\), which is \(f(x)=x(x + 1)(x + 3)\).

Step2: Verify the degree and roots

• For \(f(x)=(x + 1)(x + 3)\), it's a quadratic (degree 2), the graph of a quadratic is a parabola. But our given graph has a cubic shape (since it has two turning points, which a cubic can have). So this is not correct.
• For \(f(x)=x(x - 1)(x - 3)\), the roots are at \(x = 0\), \(x = 1\), \(x=3\). But our graph intersects the x - axis at \(x = 0\) and negative x - values, so this is incorrect.
• For \(f(x)=(x - 1)(x - 3)\), it's a quadratic with roots at positive x - values, incorrect.
• For \(f(x)=x(x + 1)(x + 3)\), it's a cubic (degree 3), roots at \(x = 0\), \(x=-1\), \(x=-3\), which matches the general position of the roots (0 and negative x - axis) and the shape of the cubic graph.

Answer:

\(f(x)=x(x + 1)(x + 3)\)