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 “typical” 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 “dominates” 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 “facing downwards” 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 “linear” 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 ⟶ 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.

- Behavior (1) is whether the two “arms” 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 “degree 0”—the 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 “meet” 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 “same side” 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 “parabolic” 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 1—it is linear—while 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 → 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.