Describe the graph of the function f(x) = x3 - 18x2 + 101x - 180.

Question

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

Hi Angie,

 

Let's begin by finding the y-intercept (value for y when x = 0):

    y = x³ - 18x² + 101x - 180

    y = 0³ - 18(0)² + 101(0) - 180

    y = -180

y-intercept = (0,-180)

 

Next, let's determine the x intercepts (values for x when y = 0).  We expect three such values since the order of our equation is cubic (highest exponent for x is 3).  Below are the steps to determine these three values of x:

    1.   The Rational Root theorem states that the possible real zeroes for any higher order function greater than 2 will be all the positive/negative factors of the constant term divided by all the positive/negative factors of the coefficient of the leading (highest) exponent term:

              x³ -18x² + 101x - 180

                  constant term:  -180

                  coefficient of leading exponent term:  1

                  possible reals zeroes for x:

                      +/- 180,90,60,45,30,20

                      +/- 15,12,9,6,5,4,3,2,1

     2.  Trying out the possibilities beginning with the lowest factor (since we have a high negative constant term) and working our way up - we see that y will be 0 when x = 4:

              y = x³ - 18x² + 101x - 180

              y = (4)³ - 18(4)² + 101x - 180

              y = 64 - 18(16) + 101(4) - 180

              y = 64 - 288 + 404 - 180

              y = -224 + 404 - 180

              y = 180 - 180

     3.  Since x = 4 is a zero of the function then

(x - 4) must be a factor, which means if we divide

(x - 4) into (x³ - 18x² + 101x - 180) we should get a quadratic factor and be able to solve it for the remaining two zeroes.

     4.   The quadratic factor and its solution:

               x² - 14x + 45 = 0

               (x - 9)(x - 5) = 0

               x = 9; x = 5

x-intercepts:  (4,0); (5,0); (9,0)

 

Below is the link to the graph of our cubic function 

(x³ - 18x² + 101x - 180):

 

https://dl.dropbox.com/s/jx7rfscwauap4qo/Graph_of_Cubic_Function.png?raw=1

 

All intercepts are shown confirming our algebraic solution above:

    y-intercept:    (0,-180)

    x-intercepts:  (4,0); (5,0); (9,0)

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.

 

Answer 2

f(x) factors into (x-5)(x-4)(x-9)

 

If you set each of these factors equal to zero, you see that x=4, x=5, x=9  are the zeroes of the polynomial, which means they are the x-intercepts of the graph.

 

Remember that the y-intercept occurs when x=0. If we put 0 in place of x, we find that f(0) = -180.

 

x-intercepts:  (4,0), (5,0), and (9,0)

 

y-intercept     (0,-180)

 

So, the graph looks essentially like the drain pipe under the sink.  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). It 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.