Question

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Explanation:

Step1: Identify x-intercepts

The graph crosses the x - axis at \(x=-2\), \(x = 1\), and \(x = 5\). So the roots of the polynomial are \(x=-2\), \(x = 1\), and \(x = 5\).

Step2: Form the polynomial

If \(r_1\), \(r_2\), \(r_3\) are the roots of a polynomial, the polynomial can be written in factored form as \(y=a(x - r_1)(x - r_2)(x - r_3)\). Substituting the roots, we get \(y=a(x + 2)(x - 1)(x - 5)\).

Step3: Determine the leading coefficient

We can use the y - intercept. The y - intercept is the value of \(y\) when \(x = 0\). From the graph, when \(x = 0\), \(y=- 2\). Substitute \(x = 0\) and \(y=-2\) into the equation:
\(-2=a(0 + 2)(0 - 1)(0 - 5)\)
\(-2=a(2)(-1)(-5)\)
\(-2=a(10)\)
\(a=-\frac{1}{5}\)
So the polynomial is \(y =-\frac{1}{5}(x + 2)(x - 1)(x - 5)\)

Step4: Analyze the end - behavior

The degree of the polynomial is 3 (since it's a product of three linear factors), and the leading coefficient \(a=-\frac{1}{5}<0\). For a cubic polynomial \(y = ax^3+bx^2+cx + d\), when the leading coefficient \(a<0\) and the degree \(n = 3\) (odd), as \(x
ightarrow+\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow+\infty\), which matches the end - behavior of the given graph.

Answer:

The polynomial function represented by the graph is \(y =-\frac{1}{5}(x + 2)(x - 1)(x - 5)\) (or expanded form \(y=-\frac{1}{5}x^{3}+\frac{4}{5}x^{2}+\frac{7}{5}x - 2\))