B1.4

describe key features of a polynomial function, given information about its first and/or second derivatives (e.g., the graph of a derivative, the sign of a derivative over specific intervals, the x-intercepts of a derivative), sketch two or more possible graphs of the function that are consistent with the given information, and explain why an infinite number of graphs is possible Sample problem: The following is the graph of the function $g(x)$. [The document prints the graph of y = g(x) here.] If $g(x)$ is the derivative of $f(x)$, and $f(0)=0$, sketch the graph of $f(x)$. If you are now given the function equation $g(x)=(x-1)(x-3)$, determine the equation of $f^{\prime \prime }(x)$ and describe some features of the equation of $f(x)$. How would $f(x)$ change graphically and algebraically if $f(0)=2$?