In this explainer, we will learn how to investigate the graph of polynomial functions and
identify the equation of a polynomial function from its graph and vice versa.
We look at how factors correspond to -intercepts of the graph, what
happens when factors are repeated, and how the sign of the leading coefficient affects the
graph.
Consider the polynomial function
What can we tell about the graph before we can sketch
it?
Since , the constant term, we know that the
graph passes through . There is not much else! We can add
that this is the appearance of a typical degree 4 curve.
Since the degree of the
polynomial is 4, we hope that it will look like a βtypicalβ quartic curve. But which of these is typical?
At
least one detail is true of the three examples above, which we note below. The curves
all go to either or whether independent
variable goes towards or
. This behaviour is determined by the leading term, in our case
the term, which βdominatesβ when
is very large. In our case, whether or ,
the leading term contributes which is
very large and negative. The term only has 18 zeros. So the graph of
is βfacing downwardsβ as in example
(c) above.
So far, we have this picture.
The best information comes from a factored form of the polynomial
, when it is available. Here
telling
us that when one of the factors ,
, or is zero. In other words, when
Since the curve must pass smoothly through the three points and not cross the
-axis anywhere else, we can sketch the graph as shown below.
Not only do the -intercepts correspond to the distinct factors, but the
way the curve crosses the -axis is also determined by the multiplicity of
the linear factors.
The way the curve cuts the -axis
near and is βlinearβ
since we have and
.
Near , the factor makes the graph
look very much like the curve shown as the dashed
curve. Recall that the curve is just the curve
translated horizontally by 3 units.
Similarly, if is a factor of the polynomial
, then looks like a multiple of
near the intercept . This is
a vertically stretched form of or shifted
units to the left.
Example 1: Identifying the Graph of a Polynomial Given in Factored Form
Which of the following is the graph of
?
Answer
The zeros of are ,
1, and 4 from the factorization. So the choice is
between (C) or (E). As the leading coefficient of is
1, which is positive, then the correct sketch is (C).
The choice could also have been settled by noting that the -intercept
is which is above the
-axis, unlike the -intercept in (E).
Example 2: Identifying the Graph of a Quadratic Using Factoring
Solve by factoring, and hence determine which of the
following figures would be a sketch of .
Answer
In order to sketch , we begin by factoring
: so that has the single solution
. This gives as the only
-intercept of . The choice is between
(C) and (D).
The leading term here is , with leading coefficient 1, which is
positive. Therefore, as goes to , we expect
that will also go to . This tells us
that the correct choice is (C).
This is the reverse to what you should know about the characteristics of the graph of a
polynomial function determined by the characteristics of the polynomial expression itself,
which we summarize.
Relating Characteristics, Polynomial βΆ Graph
1.
Degree
Number of intercepts with horizontal line
2.
Odd/even degree
Behavior (1) as
3.
Sign of leading coefficient
Behavior (2) as
4.
Value of constant term
-Intercept
5.
Zeros/factors
-Intercepts
6.
Multiplicity of factor
Curvature at the intercept
Let us list the dictionary above.
Degree 0 and 1 polynomials give rise to straight lines: horizontal if the degree is 0, not
if it is 1. Degree 2 polynomials have graphs that are all parabolas, with mirror symmetry
along a vertical axis. In higher degree, the shapes of the graphs are more varied. What is
true is that
the number of intercepts with a horizontal line
is never more than the degree;
this number can be less than the degree.
Of course, if the polynomial has
degree 1 (linear), then every line meets the curve exactly once. If the degree is 2, a
horizontal meets the curve in either 2, 1, or 0 places.
For degree 3, we can get 1, 2, or 3, with no guarantees as for
quadratic polynomials.
Behavior (1) is whether the two βarmsβ as on
the sketched curve are both in the upper or lower half planes ( or
), which is true if and only if the polynomial has an even degree. Otherwise, whatever the degree, it must be odd.
Behavior (2) is which of quadrants 1 or 4 the graph lies in as . If the leading coefficient is positive, this is 1; otherwise, it is 4. The following figure shows the graphs of two degree 5 polynomials, and
, with leading coefficients of different signs.
A polynomial expression is a sum of multiples of powers of
and a term of βdegree 0ββthe constant term. Setting
says this constant is , which gives the
-coordinate of the point on the curve that
is also on the -axis.
The graph above cuts the -axis at the
number 4. This means that and therefore
is a factor of . Likewise,
for some polynomial
.
Knowing that a number (such as 2) is the zero of a polynomial
tells us that the graph meets the -axis at
. But they can βmeetβ in essentially three different
ways, as illustrated in the figure below.
These are as
follows:
A sharp crossing like : this is the same
as or indeed for any
that is not zero at .
A tangency
like with the entire graph to the βsame sideβ
of the -axis: this is the case for or
indeed , where is an even number,
and .
A tangency like ,
or the shown: with odd powers ,
the curve passes through but lies on both sides of the axis near
this point.
We can use these facts as in the following examples.
Example 3: Identifying the Equation of a Curve from a Sketch
Which of the following could be the equation of the given sketch?
Answer
From the intercepts , and 3, we know that ,
, and must be factors of the polynomial
used. This leaves the two choices (C) and (E).
Since the curve is in the 4th quadrant as , the leading
coefficient must be negative. So the only possibility is (C), where the coefficient of
is , while the leading coefficient in the other
option is 1.
Example 4: Identifying the Equation of a Curve from a Sketch
Which of the following could be the equation of the given sketch?
Answer
From the -intercepts at 1 and 7, we know that
and must be factors of the polynomial. The fact that the curve is
tangent at in the βparabolicβ way means that the
factor must appear an even number of times: as .
So this is either or . The behavior as (the curve in quadrant 1) tells us that it
must have a positive leading coefficient. So the answer is (C).
Example 5: Identifying the Equation of a Curve from a Sketch
Which of the following could be the equation of the given sketch?
Answer
The -intercepts at and 5 tell us that the
polynomial must have factors and
respectively.
The crossing at says the factor is to the
order 1βit is linearβwhile that at says that we have an odd power
of that is at least three.
From the given choices, we have either or
.
The behavior of as , in that , tells us that the leading coefficient has to be negative. The
solution must be which is
option (C).
In summary, this is the information we can gather from a polynomial expression in factored
form, and how we can use it in identifying the graph.
Factored Polynomial β Graph
The distinct factors give the -intercepts, with
corresponding to .
The multiplicity tells us how the curve meets
. When is even, so that the factor is one of
, ,
then the curve is entirely to one side of the -axis near
. Otherwise, for one of ,
, the curve crosses the -axis. It
is tangent only when .
The follwoing is additional information about the polynomial expression
that does not depend on the factoring but that informs
about the shape of the curve :
determines the
-intercept
.
The leading coefficient in the
expanded form is the number
. If this is positive then the curve
is in the 1st quadrant for large positive values
of . Otherwise, it is in the 4th quadrant.
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