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If you need to refresh your skills regarding positive/negative, increasing/decreasing, maximum/minimum, and/or transformations (as they relate to graphing), see the Refresher portion of this section.
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A polynomial function is a function which is defined by a polynomial expression.
Examples: f(x) = x2 + x - 6; P(x) = x3 - x2 - 12x; y1 = x2 + 4x + 4 |
In Algebra 2, additional emphasis will be placed on the topics of zeros,
multiplicity, end behavior, and transformations as they relate to graphing.
Roots (or Zeros):
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If you plug in r (some real number) for x in a polynomial function, P(x), and get an answer of 0, the number, r, is called a root, or zero, of the polynomial. |
In Algebra 1, you learned that the fastest way to find roots (or zeros), is to factor the polynomial, and then set the factors equal to zero. This process utilizes the zero factor principle which states that "if a • b = 0, then either a = 0 and/or b = 0."
 The real numbers that create the roots (or zeros) of a polynomial correspond to the x-intercepts of the graph of the polynomial function.
This is valuable information when it comes to creating the graph of a polynomial (without a graphing calculator). It is also valuable if you are given the graph and are attempting to create a possible equation.
Note: A polynomial of degree 2 will have two roots (zeros), a polynomial of degree 3 will have three roots (zeros), and so on.
The Fundamental Theorem of Algebra states that a polynomial of degree n will have exactly n zeros, allowing for repeated roots and complex roots.
Keep in mind that not all polynomial functions have roots that are real numbers. Remember those quadratic graphs that float completely above the x-axis (never crossing or touching the x-axis), because their roots are complex numbers.
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Example:
Knowing how to find the zeros of a polynomial is essential for graphing polynomial functions.
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What happens when a factor repeats?
Multiplicity of Roots (or Zeros): We just saw that the real roots (zeros) of a polynomial correspond with the x-intercepts of the polynomial graph. In some situations, the graph will "cross" the x-axis at these points. In other situations, the graph may simply "touch" (be tangent to) the x-axis at these points. Let's see if we can determine, before we draw the graph, whether it will "cross" the x-axis at each root, or simply "touch" (be tangent to) the x-axis at each root.
Consider the example at the right. The polynomial is of degree two, so there will be two roots (zeros). The factor of (x + 3) is repeated twice, and can also be written as (x + 3)2.
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The number of times a factor appears in a polynomial is referred to as its multiplicity. |
In the example at the right, the factor (x + 3) has a multiplicity of 2, since it appears twice. It creates a "repeated root".
Multiplicity EVEN: When the multiplicity (the number of times a factor repeats) is an even number, the graph will just "touch" (be tangent to) the x-axis at that point. |
Why? "EVEN" multiplicities are factors that occur an even number of times, and form squares. Since squares are always positive, the graph near the root (zero) will not change signs from positive (above the x-axis) to negative (below the x-axis), or vice versa. The graph will "touch", or "bounce off", the x-axis at the root (zero) but remain on the same side of the x-axis. |
Example degree 2:
"just touching" the x-axis
may also be referred to as
"bouncing off" the x-axis.
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What if the multiplicity is ODD?
Consider the example at the right. The polynomial is of degree three, so there will be three roots (zeros). The factor of (x - 1) appears three times, and can be written as (x - 1)3.
Multiplicity ODD: When the multiplicity (the number of times a factor repeats) is an odd number, the graph will "cross" the x-axis at that point. |
Remember:
If you see a factor such as (x - 1)3, the multiplicity is 3.
If you see a factor such as (x + 2)2, the multiplicity is 2.
If you see a factor such as (x + 3), the multiplicity is 1.
Note: When you factor a polynomial, the sum of the multiplicities equals the degree of the polynomial.
x3 + x2 - 5x + 3 = (x - 1)2(x + 3) = (x - 1)2(x + 3)1
degree = 3
sum of multiplicities = 2 + 1 = 3 |
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Example degree 3:
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End Behavior: Looking at a few aspects of a polynomial will tell us what is happening at either end of the graph of the polynomial function. We will be looking at the DEGREE of the polynomial and the SIGN of the leading coefficient to determine what is happening to the graph.
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End behavior refers to the appearance of a graph as it is followed indefinitely in either horizontal direction. |
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Even Degree Polynomial
("ends" behave similar to a quadratic) |
Leading coefficient POSITIVE: both "ends" are UP.
Leading coefficient NEGATIVE: both "ends" are DOWN.
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Odd Degree Polynomial
("ends" behave similar to a cubic) |
Leading coefficient POSITIVE: left end is DOWN and right end is UP.
Leading coefficient NEGATIVE: left end is UP and right end is DOWN.
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SUMMARY: The End Behavior of a Polynomial Function with Leading Term axn :
End Behavior |
n is Even (not zero) |
n is Odd |
a is positive |
Both Ends UP |
Left Down, Right Up |
a is negative |
Both Ends Down |
Left Up, Right Down |
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Let's see how to pull together the degree, roots, multiplicities, and end behavior
of a polynomial function to create its graph or sketch.
Depending upon the information given, it may be possible to only produce a "rough sketch" of the graph instead of an actual graph such as you would see on your graphing calculator. In some cases, there will be insufficient information to determine the y-values of coordinates. In such situations, the "sketch" will be produced using only the x-axis as a reference line.
Note: the process we will be using to create sketches, assumes that all of the zeros are real numbers.
If there are complex zeros, this process may miss this information and may not yield a reliable sketch.
Further analysis will be needed to deal with complex roots.
 (No Graphing Calculator!) |
Given a positive leading coefficient, with root x = -5 with multiplicity of 2, root x = 1 with multiplicity of 1 and root x = 4 with multiplicity of 1, produce a sketch of this polynomial function. Assume all roots are given.
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Note: This will be a rough sketch since without knowing "a", the leading coefficient, we will be unable to obtain a "definite" equation for this situation.
Analysis:
Degree: Since this polynomial function has 4 roots (one repeated), we will be dealing with degree 4.
End
Behavior: Even degree with positive leading coefficient tells us that BOTH ends point up.
Multiplicity: Root x = - 5 with multiplicity 2 will just "touch" (or "bounce off") the x-axis. Roots x = 1 and x = 4, each with multiplicity of 1, will "cross" the x-axis.
Preparing Graph:
Plot the roots first. Using your end-behavior information, start the graph from the left and intersect with each root until the last root is completed, and the right side end-behavior is drawn.
Sketch:
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Comments:
We know that this graph will have a relative minimum at (-5,0), We do not know exactly where the relative maximum between x = -5 and x = 1 will occur, or how high it will reach on the y-axis. Similarly, we do not know the exact location of the relative minimum between 1 and 4.
We do, however, get a feel for what the graph is doing.
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What if:
If we assume that the leading coefficient is ONE, we could produce the graph shown at the right by graphing
f(x) = (x + 5)(x + 5)(x - 1)(x - 4). Notice the different scales on the x and y axes. The x-axis is using a scale of 1, but the relative maximum at x = -1.5 is showing a y value of 168.5 and the relative minimum at x = 2.8 is showing a y value of -131.4. Remember, this is only ONE possible graph. A common factor other than ONE will vertically stretch or compress the locations of the relative maximum at x = -1.5 and the relative minimum at x = 2.8.
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 (No Graphing Calculator!) |
Given the polynomial function sketch shown below, describe what you know about the equation of this polynomial based upon the degree, the roots, and the end behaviors. Assume all roots are reals. |
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One Possible Analysis:
Roots: Real roots appear at x = -1,
x = 1, x = 3 and x = 4.
Multiplicity: Since the graph "crosses" the x-axis at -1, 1 and 4, these roots have a multiplicity that is odd (let's say 1). The graph only "touches" the x-axis at 3, so the multiplicity is even (let's say 2).
Degree: Since the degree is the sum of the multiplicities, for our choices the degree is 5.
End
Behavior: The degree is odd and the pattern is LEFT DOWN and RIGHT UP. The leading coefficient of this equation will be positive.
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Equation Information:
We know this could be a 5th degree equation of the form P(x) = (x + 1)(x - 1)(x - 3)(x - 3)(x - 4). We know that the leading coefficient is positive (as indicated by the end behavior), but we don't know its actual value.
This theoretically could also be the rough sketch of a graph where the multiplicities are larger odd and even values, making the degree larger, such as
P(x) = (x + 1)3(x - 1)5(x - 3)4(x - 4)3. |
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Sketch the graph of P(x) = x3 + 3x2 - 4. (No Graphing Calculator!) |
This problem differs from the previous examples in that it has supplied the actual equation of the polynomial function. Since we know the equation, we will be able to make a more reliable sketch.
Analysis:
Roots: Factoring this equation will supply the roots.
P(x) = x3 + 3x2 - 4 = (x + 2)(x + 2)(x - 1).
Since the constant term is -4, the possible factors are ±1, ±2 or ±4. Use the Polynomial Remainder Theorem to determine the factors.
The roots are -2, -2, and 1.
Multiplicity: Since the factor (x + 2) appears twice, its multiplicity is 2 telling us the graph will only "touch" the x-axis at -2. Since the multiplicity of (x - 1) is 1, the graph "crosses" the x-axis at 1.
End
Behavior: The degree is odd and the leading coefficient is positive, so the end behavior pattern is LEFT DOWN and RIGHT UP.
Additional Information: Since we know the actual equation, we can obtain a few actual points on the graph to make our sketch more reliable. For example, picking points between the roots and picking the y-intercept are usually helpful.
The y-intercept (where x = 0) will be at (0,-4).
Test x = 0.5: (0.5,-3.125)
Test x = -1: (-1,-2) |
Root points in blue.
Test points in red.
If you also plot points to the right of
x = 1 and to the left of x = -2, you
will be able to see how fast the graph
is increasing (to show how steep
to make the end behaviors).
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FYI:
Graph behavior near Roots:
We know that the graph of a polynomial function will intersect the x-axis at its root values, and we know that if the multiplicity of that root is even, the graph will "bounce off" the x-axis and remain on the same side of the x-axis. If the multiplicity of the root is odd, the graph will "cross" the x-axis at that root. But, are there any changes in appearance of the graph surrounding these intersections as the multiplicities get larger?
Surrounding EVEN multiplicities:
As the degree of the equation increases, there may be subtle differences in the shape of the graph surrounding the root values.
As even multiplicities increase, the graph will become increasingly "flatter" near the root value.
Consider this example raising factor (x - 2) to increasingly larger even powers. |
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While the subtleties may not be observable in a standard view of the graphs (on the left), the zoomed view (on the right) shows that as the multiplicity (the power) increases, the graph gets increasingly "flatter" surrounding the root (the x-intercept). |
Surrounding ODD multiplicities:
A similar subtle change occurs with odd number multiplicities.
As odd multiplicities increase, the graph will again become increasingly "flatter" near the root value.
Consider this example raising factor (x - 2) to increasingly larger odd powers. |
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Again, while the standard view of the graphs (on the left) may not emphasize what is happening surrounding the root value, the zoomed view shows that as the multiplicity (the power) increases, the graphs again get increasingly "flatter" surrounding the root (the x-intercept).
Why is this occurring?
As the multiplicities of these factors are increasing, the behaviors of the graphs at the x-intercepts are resembling the "parent function's" graphs. For example, the behavior of the graph surrounding the root corresponding to the factor (x - 2)2 is resembling the graph of y = x2.
The changes you see occurring to the graphs shown above resemble the shapes of the graphs of the functions y = x2, y = x4, y = x8 and y = x, y = x3 , y = x7. |
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