Question

which equation choice could represent the graph shown below?
answer
$f(x) = (x - 3)(x - 7)(x - 7)$ $f(x) = (x + 3)(x + 7)(x + 7)$
$f(x) = x(x + 3)(x + 7)$ $f(x) = x(x - 3)(x - 7)$

Explanation:

Step1: Identify x-intercepts

The graph crosses or touches the x - axis at \(x = 3\), \(x=7\) (with a double root at \(x = 7\) since it touches the axis there) and \(x = 0\)? Wait, no, looking at the graph, the x - intercepts are at \(x=3\) (crossing), \(x = 7\) (touching, so a double root) and wait, no, let's re - examine. Wait, the graph crosses the x - axis at \(x = 3\) and touches at \(x=7\) (so a repeated root) and also, wait, when \(x = 0\), what's the value? Wait, no, let's look at the roots. For a polynomial \(f(x)=(x - a)(x - b)(x - c)\), the roots are at \(x=a\), \(x = b\), \(x=c\). If the graph touches the x - axis at a root, that root has an even multiplicity. If it crosses, odd multiplicity.

Looking at the graph, it crosses the x - axis at \(x = 3\) (so multiplicity 1), touches at \(x=7\) (so multiplicity 2), and also, wait, when \(x = 0\), the graph passes through the origin? Wait, no, let's check the options.

Option 1: \(f(x)=(x - 3)(x - 7)(x - 7)\). The roots are \(x = 3\) (multiplicity 1, crosses x - axis) and \(x=7\) (multiplicity 2, touches x - axis). Let's check the end - behavior. The leading term is \(x^{3}\), so as \(x
ightarrow+\infty\), \(f(x)
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\), which matches the graph (since the left end goes down and right end goes up).

Option 2: \(f(x)=(x + 3)(x + 7)(x + 7)\). Roots at \(x=-3\), \(x=-7\) (double root). The graph would cross at \(x=-3\) and touch at \(x=-7\), but our graph has roots at positive x - values, so this is wrong.

Option 3: \(f(x)=x(x + 3)(x + 7)\). Roots at \(x = 0\), \(x=-3\), \(x=-7\). All roots are at non - positive x (except 0), but our graph has roots at positive x, so wrong.

Option 4: \(f(x)=x(x - 3)(x - 7)\). Roots at \(x = 0\), \(x = 3\), \(x=7\). The root at \(x = 0\) would mean the graph passes through the origin, but our graph does not pass through the origin (since when \(x = 0\), looking at the graph, the y - value is negative? Wait, no, let's check the end - behavior. The leading term is \(x^{3}\), same as option 1, but the roots include \(x = 0\). But in the given graph, the x - intercepts are at \(x = 3\) and \(x=7\) (with a touch at \(x=7\)), not at \(x = 0\). So option 1 is correct.

Step2: Verify end - behavior and roots

For \(f(x)=(x - 3)(x - 7)(x - 7)\), expand the leading term: \(x\times x\times x=x^{3}\). The end - behavior of a cubic with positive leading coefficient is as \(x
ightarrow+\infty\), \(f(x)
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\), which matches the graph (left end down, right end up). The roots are \(x = 3\) (multiplicity 1, crosses x - axis) and \(x=7\) (multiplicity 2, touches x - axis), which matches the graph's x - intercepts.

Answer:

\(f(x)=(x - 3)(x - 7)(x - 7)\)