Question

(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 1

Answer and Explanation :

Given : Function
`f(x) = x^3 -18x^2 + 101x -180`

To find : Describe the graph of the function. Include the y-intercept, x-intercepts, and the shape of the graph?

Solution :

Function
`f(x) = x^3 -18x^2 + 101x -180`

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

`x\rightarrow -\infty \ \ \ \ \ \ f(x)\rightarrow -\infty\\x\rightarrow \infty \ \ \ \ \ \ \ f(x)\rightarrow -\infty`

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.