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Polynomial Functions

Describing Graphs of Polynomial Functions

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Direct messages

As has been seen, the basic characteristics of polynomial functions, zeros and end behavior, allow a sketch of the function's graph to be made. Using other characteristics, such as increasing and decreasing intervals and turning points, it's possible to give a much more detailed description of polynomial graphs.


Increasing and Decreasing Intervals

A function is called increasing when as the x-values increase, the values of f(x) also increase. On the other hand, a function is called decreasing when, as x increases, f(x) decreases. An increasing interval is an interval of the independent variable for an increasing function. A decreasing interval is an interval of the independent variable for a decreasing function.
A point where a function has a maximum or a minimum is neither included in an increasing or a decreasing interval. The previous applet shows a function that contains two increasing intervals and one decreasing interval. Each can be described in terms of the x-values.
A point where the function switches between decreasing and increasing is known as a turning point.


Relative Minimum and Maximum

The value f(c) is a relative minimum, or local minimum, of a function if f(c) is the least output of f around x=c. Likewise, the value f(d) is a relative maximum, or local maximum, of a function if f(d) is the greatest output of f around x=d.

Relative extrema of a cubic function
If the function is continuous, the function switches from increasing to decreasing at a relative maximum or from decreasing to increasing at a relative minimum.
Sometimes the phrase relative extrema is used to refer to both relative maximums and relative minimums. Note that a function can have one or more relative extrema, or none at all.


Turning Point

Relative minimums and maximums of polynomial functions are also called turning points. This is because a function's graph turns from increasing to decreasing, or vice versa, at these points.

The graph of a polynomial function of degree n can have at most n1 turning points. The function shown is a 3rd degree polynomial and has 2 turning points.

Furthermore, a polynomial function with n real zeros must have at least n1 turning points.


Even and Odd Symmetry

If a function has a symmetry, it is either even or odd. The symmetry is even when the graph is symmetric with respect to the y-axis, and odd when it's symmetric about the origin.


Even Symmetry

If a function has even symmetry, the following rule applies:
The rule comes from the fact that even symmetry is a reflection across the y-axis. Therefore, changing the sign of the x-value does not affect the function value.
The concept applies both ways. Hence, if the rule is true for the entire domain, the function has even symmetry.


Odd Symmetry

Instead, if a function has odd symmetry, the rule it must follow is
An odd symmetry means graphically that the graph is rotated about the origin. Therefore, changing the sign of the x-value also changes the sign of the function value.
If this rule is satisfied on the entire domain, the function has odd symmetry.


Determine the symmetry of the polynomial functions

Using their function rules, determine if each of the following functions has even or odd symmetry, or neither.
Show Solution expand_more

We can determine if a function has even or odd symmetry, using its function rule and the following relationships.

To determine if any function has even or odd symmetry, f(x),f(-x), and -f(x) must be known. h(x) and g(x) are given. Thus, the following must be found.
Focusing first on h(x), we'll find h(-x).

Simplify power and product

Next, we'll find -h(x).
We can summarize the needed relationships for h(x) as follows.
h(x) h(-x) -h(x)
x5+2x -x52x -x52x
It can be seen that
Thus h has odd symmetry. The needed relationships for g can be found in the same way. This gives the following.
g(x) g(-x) -g(x)
3x3+1 -3x3+1 -3x31
Notice that
Thus, g has neither even nor odd symmetry.


Determine the key features of the function given its graph


The figure below shows the graph of a polynomial function.

Use the graph to determine

  • any zeros,
  • the y-intercept,
  • the coordinates of any turning points,
  • any relative minimums or maximums,
  • the end behavior, and
  • any symmetries.
Show Solution expand_more
Let's determine the characteristics in order from the list.



Zeros are the x-values where the value of the function is equal to 0. In other words, where the graph intersects the x-axis. In the figure, we can see that this occurs at approximately x=-6, x=1, x=3, and x=6.5.



A y-intercept is the y-value of a function where x=0. In other words, where its graph intersects the y-axis. In the figure, we can see that this occurs at approximately y=-4.


Turning points

Turning points are found where the graph turns from increasing to decreasing, or vice versa. Notice that the function has the following increasing and decreasing intervals.
Thus, the three turning points are
The rightmost minimum point is an absolute minimum point because it is where the function has its lowest value. The other minimum and maximum points are relative.


End Behavior

The end behavior shows the graph extending upward when x approaches both positive and negative infinity.
Assuming that there are no additional turning points outside the graph, we can conclude that the degree of the function is even, as the directions are the same. The leading coefficient of the polynomial must be positive, as the right-end extends upward.



Finally, we must determine if the function has any symmetries. If we look at the graph, we can see that it is not symmetric with respect to the y-axis, and also not symmetric about the origin. Thus, the function has neither even nor odd symmetry.


Sketch the polynomial function using the key features


Draw a graph of a polynomial function, f(x), having these characteristics.

  • f(x)<0 when -3<x<1 and x>3
  • f(x)>0 when x<-3 and 1<x<3
  • f(x) is decreasing when x<-1 and x>2
  • f(x) is increasing when -1<x<2
Show Solution expand_more
To draw this graph, we'll make sense of the first two bullets then the last two. If f(x)<0, the graph lies below the x-axis. Similarly, when f(x)>0, the graph lies above the x-axis. Notice that, to go from below to above the x-axis, the graph must cross it, which happens at a zero. Thus, the zeros are:
We'll plot these points in a coordinate plane.

Two of these points, (-3,0) and (3,0), lie in intervals where f(x) is decreasing. The third point, (1,0), is instead in an interval where the function is increasing. Using this, we can draw short sections of the function through the zeros.

When a function changes from increasing to decreasing, or vice versa, it does so at a turning point. From the given information, these changes occur at x=-1 and x=2. We can extend the already-drawn sections to these x-values.

Note that, because we don't have the exact coordinates of the turning point, their position is approximate. The function is decreasing both when x<-1 and when x>2. Since we know that, we can now draw the rest of the graph.

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