Question

example: write the equation in factored form
you try it: write the equation in factored form

Explanation:

Step1: Identify x-intercepts and multiplicities

The graph crosses or touches the x - axis at \(x=-2\), \(x = - 1\), and \(x = 1\). At \(x=-2\), the graph crosses the axis, so the multiplicity is 1 (corresponding factor \((x + 2)\)). At \(x=-1\), the graph crosses the axis, so multiplicity 1 (factor \((x + 1)\)). At \(x = 1\), the graph touches the axis (vertex - like), so multiplicity 2 (factor \((x - 1)^2\)).

Step2: Determine the leading coefficient

We know a point on the graph, for example, when \(x = 0\), \(y=2\). Let the equation be \(y=a(x + 2)(x + 1)(x - 1)^2\). Substitute \(x = 0\), \(y = 2\):
\[

$$\begin{align*} 2&=a(0 + 2)(0+1)(0 - 1)^2\\ 2&=a\times2\times1\times1\\ 2&=2a\\ a&=1 \end{align*}$$

\]

Step3: Write the factored form

Using the leading coefficient \(a = 1\) and the factors from the x - intercepts, the equation is \(y=(x + 2)(x + 1)(x - 1)^2\).

Answer:

\(y=(x + 2)(x + 1)(x - 1)^2\)