Question

determine an equation for the pictured graph. write your answer in factored form and assume the leading coefficient is either 1 or -1, you should be able to determine which is the case by looking at the graph. y=(x + 3)^2(x - 1) question help: video message instructor

Explanation:

Step1: Identify x - intercepts

The x - intercepts are the points where the graph crosses or touches the x - axis. From the graph, the x - intercepts are \(x=-3\) and \(x = 1\). If \(x=a\) is an x - intercept, then \((x - a)\) is a factor of the polynomial. The graph touches the x - axis at \(x=-3\), so the factor \((x + 3)\) has an even multiplicity. It crosses the x - axis at \(x = 1\), so the factor \((x - 1)\) has an odd multiplicity.

Step2: Determine leading coefficient

As \(x\to+\infty\), \(y\to+\infty\). For a polynomial \(y=a(x - r_1)^{m_1}(x - r_2)^{m_2}\cdots\), when the degree of the polynomial is odd and \(a = 1\), the end - behavior is correct. Since the degree of the polynomial \(y=(x + 3)^{2}(x - 1)\) (degree \(2 + 1=3\)) and the graph goes up as \(x\to+\infty\), the leading coefficient \(a = 1\).

Answer:

\(y=(x + 3)^{2}(x - 1)\)