Graphs of Polynomials

From the factored form of a polynomial we can obtain a very good idea of what the graph of the polynomial might look like.

Recall that the factored form (x - x1)(x - x2)...(x - xm)(a1 x^2 + b1 x + c1)... (ap x^2 + bp x + cp) of a polynomial shows us the linear and irreducible quadratic factors that make up the polynomial.

Each linear factor must have a zero, which of course must therefore be a zero of the polynomial, while irreducible quadratic factors do not have zeros and therefore contribute no zeros to the polynomial.

Since none of the factors can be zero except at their zeros, the polynomial itself cannot be zero except at zeros of its factors.

Therefore if we know the factored form of a polynomial, we can easily find its zeros:

The set of zeros of a polynomial is identical with the set of zeros of the linear factors of that polynomial.

In other words, to find the zeros of a polynomial we find its linear factors.

Furthermore, the large-| x | behavior of the polynomial, the question of whether the polynomial approaches +infinity or -infinity as the absolute value | x | of x becomes large, depends in a simple way on the number of its linear and its irreducible quadratic factors:

If a polynomial is represented in factored form as the product

p(x) = a (x - x1)(x - x2) ... (x - xm) (x^2 + b1 x + c1) (x^2 + b2 x + c2) ... (x^2 + bp x + cp)

of m linear factors and p irreducible quadratic factors, then

if m is even, the value of p(x) for both large positive x and large negative x will approach +infinity if a is positive and -infinity if a is negative

if m is odd, the value of p(x) for large positive x will approach +infinity if a is positive and -infinity if a is negative, while the value for large negative x will approach -infinity if a is positive and +infinity if a is negative.

In other words, in all cases the polynomial approaches infinity for large positive x in the direction of the sign of a (i.e., +infinity for positive a, -infinity for negative a).  For large negative x the polynomial approaches infinity in the direction of the sign of a when m is positive and in the opposite direction to the sign of a when m is negative.

These ideas will become clear through the following examples.

A polynomial of degree 3 will have factored form (x-x1)(x-x2)(x-x3), with three linear factors, or (x-x1)(ax^2+bx+c) with one linear and one irreducible quadratic factor.

Example:  y = (x-1)(x-2)(x-3)

The polynomial y = (x-1)(x-2)(x-3), with x1=1, x2=2 and x3=3, takes the value 0 whenever x = 1, 2 or 3. You easily can and certainly should check this out by substituting these values into the polynomial.

The y intercept of the polynomial occurs when x = 0, at y = (0-1)(0-2)(0-3) = -6.

When x is a large negative number, (x-1), (x-2) and (x-3) are all large negative numbers so the product y = (x-1)(x-2)(x-3) will be an extremely large negative number.

When x is a large positive number, the three factors will also be large positive numbers and their product will therefore the an extremely large positive number.

So, moving from left to right, the graph must start at large negative values, go through points (0,-6), (1,0), (2,0) and (3,0), and finally move into increasingly large positive values. Furthermore these are the only points where the graph can pass through either of the coordinate axes.

You should attempt to sketch a graph with these characteristics before reading further.  The points are depicted on the graph below.

The graph of this polynomial is shown below. It is clear that the graph rises from large negative values and passes through the four points indicated above. It is not completely clear what happens as x takes on larger and larger positive values, but it is certainly plausible that beyone x=3, where the graph passes for the last time through the x axis, the graph continues to rise faster and faster as x takes on larger and larger positive values.

It is not easy to predict how high or low the polynomial will go between zeros.   For the purposes of this course, any reasonable estimate will do.

Example:  (x-1)(x^2+2x+3)

The polynomial (x-1)(x^2+2x+3) has a single linear factor x-1, which yields the zero x = 1, and the irreducible quadratic factor x^2 + 2x + 3. You should use the quadratic formula to verify that the quadratic factor is indeed irreducible.

Note that whether x is a large positive or negative number, the quadratic factor x^2 + 2x + 3 will be a large positive number.

When x is a large positive number, x-1 is a large positive number. So when the linear factor is multiplied by the quadratic we multiply a large positive number by another large positive number and obtain a very large positive number.

When x is a large negative number, x-1 is a large negative number. So when the linear factor is multiplied by the quadratic we multiply a large negative number by a large positive number an obtain a very large negative number.

The y-intercept is y = (0-1)(0^2+2*0+3) = -3. Thus the graph starts from large negative values, passes through the points (0,-3) and (1,0), then increases in the direction of large positive values. The graph doesn't pass through either coordinate axis except at the indicated points.

You should try to sketch the graph from this information before looking at the graph below.

The graph is shown below: The way the graph levels off between large negative and large positive values makes its overall shape very close to that of the function y = x^3.

Sometimes the zeros of a polynomial are not all distinct. In the following example x1 = x2 = 1 and x3 = -2. This gives a different sort of behavior at x = x1. Rather than passing through the zero on the x axis, the graph just curves down to touch the x axis at that point before curving back up.

Example:  (x-1)(x-1)(x+2)

The large- | x | behavior is the same as for the first and second example, for much the same reasons. The y intercept is clearly at y = 2 (just substitute 0). So we are left with a function that starts at large negative values, passes through (-2,0), then (0,2) and finally (1,0) before taking off for large positive values.

The interesting behavior occurs at x = 1, where the zeros are repeated. The graph doesn't actually go through the x axis at this point. It merely comes down and touches the axis at this point, then turns and goes up, sort of like a parabola with a vertex at (1,0).

In fact the behavior of the function y = (x-1)(x-1)(x+2) near the point (1,0) is very much like that of the parabola 3(x-1)^2. This is because near x = 1, x+2 is close to 3. So (x-1)(x-1) * (x+2) is close to (x-1)(x-1) * 3, which is just 3(x-1)^2.

Example:  y = (x-1)(x-1)(x-1)

This function has just one zero, at x = 1, but this zero occurs 3 times. The function is in fact just the function y = (x-1)^3. The graph of y = (x-1)^3 is identical to the graph of y = x^3, only shifted 1 unit in the x direction.

The graph is shown below.

A polynomial of form (x-x1)(x-x2)(x-x3)(x-x4) will have four distinct zeros x1, x2, x3 and x4 as long as x1, x2, x3 and x4 are all different. At large | x |, whether x is positive or negative, the polynomial will be a product of 4 large factors of the same sign and will therefore be very large and positive.

Example:  (x-1)(x-2)(x-3)(x+1)

The zeros of this function are at x = 1, 2, 3 and -1.

This graph must start at a very large positive number, then pass through the points (-1,0), (0,-6), (1,0), (2,0), and (3,0) before returning to increasingly large positive numbers.

The only way for this to happen is for the graph to look something like that below:

A function of this form has zeros only at x=x1 and x=x2. At large x the factors x-x1 and x-x2 will be of the same sign, either both positive or both negative, while the quadratic factor a x^2 + bx + c will be dominated by the a x^2 term and hence have the same sign as a. So the function will have the same sign as a when | x | is large.

Example:  (x-1)(x-2)(x^2+2x+3)

The y intercept is at y = 6. The large-| x | behavior is positive for both positive and negative x. The only zeros are at x = 1 and x = 2.

The graph is therefore as below.

This form, in which the quadratic factors are irreducible, can never yield y = 0, since neither factor can ever be 0.

For large values of x, the first factor will be dominated by the term a1 x^2 and will be positive or negative depending on whether a1 is positive or negative. Similarly the second factor will have the same sign as a2.

The graph will therefore remain either above or below the x axis, depending on whether a1 * a2 is positive or negative.

Example:  (x^2 - x + 2)(x^2 + 2x + 3)

Both factors are irreducible, as the quadratic formula will easily verify.

Both factors therefore remain positive for all x values.

The y intercept is y = 6.

The graph therefore starts at large positive values, decreases to a minimum at some point (this point will lie between the vertices of the graphs of the two quadratic factors) either before, at or after the y-intercept (0,6), and then increase more and more rapidly.

The graph is shown below.

These are the only possible forms for degree 4 polynomials, since there are only three ways to combine linear and irreducible quadratic factors to obtain a degree of 4.

If x1, x2, ..., x5 are all distinct the polynomial will have 5 zeros.

For large negative x all five factors will be negative and will produce a very large negative value. For large positive x all factors are clearly positive and a very large positive value will result.

Example:  (x-1)(x-2)(x-3)(x+1)(x+2)

The linear factors have zeros at x = 1, 2, 3, -1 and -2. The y intercept will be y = -12, so the graph will start at large negative values and then pass in turn through (-2,0), (-1,0), (0,-12), (1,0), (2,0) and (3,0) before taking on increasingly large positive values.

This form will again be negative for large negative values of x and positive for large positive values of x.

The quadratic factor 'eats up' two of the zeros, leaving only the three zeros x = x1, x2 and x3.

Example:  (x-1)(x-2)(x-3)(x^2+2x+3)

The zeros are x = 1, 2 and 3. The y intercept is easily found to be y = -18.

The graph therefore starts with large negative values, passes through (0,-18), (1,0), (2,0) and (3,0) before moving on to large positive values.

This form has only one zero, at x = x1. Its large-| x | behavior is determined by the product a1 * a2, with large values opposite in sign to a1 * a2 when x takes on large negative values and of the same sign as a1 * a2 when x takes on large positive values.

Example:  (x-1)(x^2 - x + 2)(x^2 + 2x + 3)

The only zero is at x = 1. For large positive x all factors are positive and large so the function is extremely large and positive. For large negative x, the linear term is negative while the quadratic terms are both positive, resulting in an extremely large negative y value.

The y intercept is (0,-6). The resulting graph is shown below.

If two of the zeros are identical, as in a previous example, the function acts much like a parabola in the vicinity of those zeros, with its vertex at the zeros.

If three zeros are identical the function will act much like a y = a x^3 function with its point of symmetry shifted to the location of the zero.

In general if n zeros are identical the function in the vicinity of the zero will act much like a power function of form y = a x^n with its point of symmetry shifted to the location of the zero.

These ideas are illustrated in the example below:

Example:  y = (x-2)(x-2)(x-2)(x+1)(x+1)

The zero at x = 2 is repeated three times, with the result that near (2,0) the function acts much like (x-2)^3 (2+1)(2+1) = (x-2)^3 * 9 = 9(x-2)^3. This is just the y = x^3 function vertically stretched by factor 9 and horizontally shifted +2 units.

The zero at x = -1 is repeated twice. As a result the function near (-1,0) acts much like (-1 - 3) ^ 3 (x+1)^2 = -64 (x+1)^2. This function is just the y = x^2 parabola shifted -1 units in the x direction and vertically stretched by factor -64.

These behaviors are clearly shown on the graph below:

Note the nearly parabolic shape of the graph near x = -1, and the y = x^3 behavior near x = 2. Note also that the parabolic behavior is much sharper, due to the factor -64, which is much more influential than the factor 9 of the cubic function.

1. Sketch the graph of a degree 3 polynomial with zeros at x = -3, 1 and 2, with y taking on increasingly large positive values for large positive values of x.

Give the y = (x-x1)(x-x2)(x-x3) form of this function has well as the y = ax^3 + bx^2 + cx + d form.

2. Sketch a graph of the polynomial y = (x-1)(x+3)(x-4).

Sketch on the same set of coordinate axes the polynomial y = (1/12) * (x-1)(x+3)(x-4).

On both graphs clearly indicate and label the y intercept.

Describe how the two graphs compare.

3. Sketch a graph of the polynomial -2 (x-1)(x+2)(x-3), clearly indicating all intercepts and the large-| x | behavior of the graph.

4. What function describes the approximate behavior of the graph of y = p(x) = (x-3)(x-3)(x+4) near the point (3,0)?

Compare the values of this function and p(x) at x = 2.0, 2.8, 2.9, 3.0, 3.1, 3.2 and 4.0. How does the comparison deteriorate as x gets further and further from the zero x = 3.0?

Use your calculator (or computer) to graph 7(x-3)^2 and (x-3)(x-3)(x+4) together in a window between x = 2 and x = 4. Make the window just a little larger than necessary to show the three fundamental points of the quadratic. How do the graphs compare?

How does your graph confirm what you did previously?

In general what does the graph of a polynomial look like near a second-degree zero and why?

5. Sketch graphs of the following functions, clearly indicating and labeling all zeros and the large-| x | behavior for both positive and negative x:

y = (x-2)^2 * (x+3)^2 * (x-1)

y = -.5 * (x-3) (x+2)^3