How do you write a polynomial function for this graph?

first off, let's notice the graph touches the x-axis at -1 and 3, namely, those are the zeros/solutions/roots of the polynomial and therefore, the factors come from those points.

now, at -1, the graph doesn't cross the x-axis, instead it simply bounces off of it, that means the zero of x = -1, has an even multiplicity, could be 4 or 2 or 6, but let's go with 2.

at x = 3, the graph does cross the x-axis, meaning it has an odd multiplicity, could be 3 or 1, or 7 or 9, but let's use 1.

y = x³ - x² - 5x - 3

the graph has roots at x = - 1 and x = 3

there is a turning point at x = - 1 indicating a double root

given x = a and x = b are the roots of a polynomial then

(x - a) and (x - b ) are factors and the polynomial is the product of the factors

here x = - 1 ( double root ) ⇔ (x + 1)² is it's factors and

x = 3 hence (x + 3) is a factor

f(x) = (x + 1)²(x - 3)

     = (x² + 2x + 1)(x - 3)

     = x³ - x² - 5x - 3 ← is a possible polynomial

step-by-step explanation:

through just a quick inspection, one can see that the graph of this cubing function starts in quadrant iii and ends in quadrant i.   when x = 0, y = 3, so the y-intercept is (0, 3).

if graphs are part of the problem statement, share them.   .