Answer:
Explanation:
The revenue can be modeled by the polynomial function
R
(
t
)
=
−
0.037
t
4
+
1.414
t
3
−
19.777
t
2
+
118.696
t
−
205.332
where R represents the revenue in millions of dollars and t represents the year, with t = 6 corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.
Multiplicity and Turning Points
Graphs behave differently at various x-intercepts. Sometimes the graph will cross over the x-axis at an intercept. Other times the graph will touch the x-axis and bounce off.
Suppose, for example, we graph the function
f
(
x
)
=
(
x
+
3
)
(
x
−
2
)
2
(
x
+
1
)
3
.
Notice in the figure below that the behavior of the function at each of the x-intercepts is different.
Graph of h(x)=x^3+4x^2+x-6.
The behavior of a graph at an x-intercept can be determined by examining the multiplicity of the zero.
The x-intercept
x
=
−
3
is the solution to the equation
(
x
+
3
)
=
0
. The graph passes directly through the x-intercept at
x
=
−
3
. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line; it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.
The x-intercept x=2
is the repeated solution to the equation (x−2)2=0
. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept.
(x−2)2=(x−2)(x−2)