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-intercept
y-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.
