Question

1. the graph of a quintic function g is shown on the xy - coordinate plane. write an equation for g.

Explanation:

Step1: Identify x - intercepts

The x - intercepts of the graph are \(x = 0\), \(x=a\), \(x = b\), \(x=c\), \(x = d\) (assume non - zero roots are \(a,b,c,d\)). A quintic function has the general form \(y = k(x - r_1)(x - r_2)(x - r_3)(x - r_4)(x - r_5)\), where \(r_i\) are the roots. Since the graph passes through the origin \((0,0)\), one root is \(x = 0\).

Step2: Determine the leading - coefficient sign

As \(x\to+\infty\), \(y\to+\infty\), so the leading coefficient \(k>0\). Let's assume the other non - zero roots are \(x = 4\), \(x = 8\), \(x = 12\), \(x = 16\) (by observing the x - values where the graph crosses the x - axis). A possible equation is \(y=k x(x - 4)(x - 8)(x - 12)(x - 16)\). To simplify, we can take \(k = 1\).

Answer:

\(y=x(x - 4)(x - 8)(x - 12)(x - 16)\)