Explanation:
Let's analyze the graph of the function
�
(
�
)
=
�
3
−
11
�
2
+
36
�
−
36
f(x)=x
3
−11x
2
+36x−36:
Y-Intercept: To find the y-intercept, we set
�
x to 0 and calculate
�
(
0
)
f(0):
�
(
0
)
=
0
3
−
11
(
0
)
2
+
36
(
0
)
−
36
=
−
36
f(0)=0
3
−11(0)
2
+36(0)−36=−36
So, the y-intercept is at the point (0, -36).
X-Intercepts: To find the x-intercepts, we need to solve for
�
x when
�
(
�
)
=
0
f(x)=0. This means we're looking for the values of
�
x where the function crosses the x-axis.
�
3
−
11
�
2
+
36
�
−
36
=
0
x
3
−11x
2
+36x−36=0
Unfortunately, solving this cubic equation may not have simple, exact solutions. You may need to use numerical methods or a graphing calculator to find approximate values for the x-intercepts.
Shape of the Graph: To understand the overall shape of the graph, we can analyze its degree and leading coefficient. This is a cubic function (
�
3
x
3
) with a positive leading coefficient (1), which means as
�
x goes to positive or negative infinity, the function also goes to positive or negative infinity. Therefore, the graph will have an end behavior where it rises on one side and falls on the other side, similar to the shape of a cubic function.
Without specific values for the x-intercepts, we can't provide the exact locations where the graph crosses the x-axis. However, knowing that it's a cubic function with one real root (x-intercept), you can use numerical methods or a graphing calculator to approximate the x-intercepts and complete the overall picture of the graph.