Questions: Use the graph to write the formula for a polynomial function of least degree. f(x)=

Solution Steps

The graph crosses the x-axis at $x = -2$, $x = 3$, and appears to touch the x-axis at $x=0$. Therefore, the roots are $x=-2$, $x=0$, and $x=3$.

Since the graph crosses the x-axis at $x=-2$ and $x=3$, these roots have an odd multiplicity. The least odd multiplicity is 1. Since the graph only touches the x-axis at $x=0$, this root has an even multiplicity. The least even multiplicity is 2.

The polynomial can be written in the form $f(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2}(x - r_3)^{m_3}$, where $r_i$ are the roots and $m_i$ are their multiplicities. Thus, the polynomial is $f(x) = a(x - (-2))(x - 0)^2(x - 3) = ax^2(x+2)(x-3)$.

The graph passes through the point $(1, -2)$. Substituting $x=1$ and $f(x)=-2$ into the equation, we have: $-2 = a(1)^2(1+2)(1-3)$ $-2 = a(1)(3)(-2)$ $-2 = -6a$ $a = \frac{1}{3}$

Final Answer: