(04.04) Describe the graph of the function f(x) = x3 − 18x2 + 101x − 180. Include the y-intercept, x-intercepts, and the shape of the graph.

Answer and Explanation :Given : Function [tex]f(x) = x^3 -18x^2 + 101x -180[/tex]To find : Describe the graph of the function. Include the y-intercept, x-intercepts, and the shape of the graph?Solution :   Function [tex]f(x) = x^3 -18x^2 + 101x -180[/tex]First we plot the graph with the help of graphing tool. Refer the attached figure below.1) x-intercept x-intercept is the point where graph intersects the x-axis or y=0.In the graph, three point touches the x-axis i.e. (4,0), (5,0) and (9,0).So, x-intercept are x=4,5,9.2) y-intercepty-intercept is the point where graph intersect the y-axis or x=0.In the graph, one point touches the x-axis i.e. (0,-180).So, y-intercept is y=-180.3) The shape of the graph.In the given function the leading coefficient is 1 and degree is 3.End behavior is [tex]x\rightarrow -\infty \ \ \ \ \ \ f(x)\rightarrow -\infty\\x\rightarrow \infty \ \ \ \ \ \ \ f(x)\rightarrow -\infty[/tex]On the left, it comes up from below and crosses the y-axis at  (0,-180).It continues up and crosses the x-axis at (4,0). Then turns around again, and heads down, crossing the x-axis again at (5,0).Then turns around one last time, crossing the x-axis for the last time at (9,0), and continuing upward to the right forever.