Polynomial Functions and Their Graphs Lesson

In this lesson, we will study different aspects of polynomial functions and their graphs. First, we will cover how to graph transformations of monomials. We will then discuss some of the important details involved with the graphs of a polynomial function. This will include the end behavior, the shape of a graph near a zero, and the local maxima and minima. I would note that the purpose of this section is just to get an understanding of how the graph of a polynomial function works. You may be asked to draw a few rough sketches but trying to make a perfect graph is both time-consuming and unnecessary. Using a graphing calculator is perfectly fine when you need an accurate graph of a polynomial function.

Definition of a Polynomial Function of Degree n

Let's begin with a basic definition of a polynomial function of degree n: $$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$

• n is a non-negative integer and represents the degree of the polynomial
• a_n ≠ 0
• a₀, a₁,..., a_n are known as the coefficients of the polynomial
• a₀ is known as the constant term or the constant coefficient
• a_n is known as the leading coefficient
  • The leading coefficient is the coefficient of the term where x is raised to the highest power
  • If the polynomial function is written in standard form (descending powers of x), it will be the leftmost coefficient
• a_nxⁿ is known as the leading term
  • The leading term is the term where x is raised to the highest power
  • If the polynomial function is written in standard form (descending powers of x), it will be the leftmost term

Transformations of Monomials

We know that a monomial is a polynomial with one term only. For example: $$4x^3, -2x^2, -\frac{1}{2}x^9$$ Earlier in this section, we studied the graphs of parabolas. The simplest parabola is that of f(x) = x². Desmos Link for More Detail The graph of a polynomial function f(x) = xⁿ, where n is even has the same general shape as the graph of f(x) = x². As the degree of the polynomial n gets larger, the graphs will be more flat around the origin and steeper everywhere else. We can observe this with f(x) = x⁴. Desmos Link for More Detail Additionally, we can also look at f(x) = x⁶. Desmos Link for More Detail You can clearly see when you compare the three graphs, that as n gets larger, the graph is getting more flat around the origin and steeper everywhere else. Additionally, we have previously seen the graph of f(x) = x³. Desmos Link for More Detail Similar to what we saw above, the graph of a polynomial function f(x) = xⁿ, where n is odd has the same general shape as the graph of f(x) = x³. Again, as the degree of the polynomial n gets larger, the graphs will be more flat around the origin and steeper everywhere else. We can observe this with f(x) = x⁵. Desmos Link for More Detail Additionally, we can also look at f(x) = x⁷. Desmos Link for More Detail Once again, as n gets larger, the graph is getting more flat around the origin and steeper everywhere else. We previously learned about graphing transformations. We can use that knowledge here to sketch the graph of certain polynomial functions. Let's look at an example.
Example #1: Describe the transformation from f(x) to g(x) and then graph g(x). $$f(x) = x^4$$ $$g(x) = \frac{1}{2}(x - 1)^4 - 2$$ Compared to the graph of f(x), g(x) has been vertically compressed by a factor of 2 (could also say vertically shrunk by a factor of 1/2), shifted right by 1 unit, and shifted down by 2 units. A given (x, y) on f(x) will now occur at (x + 1, y/2 - 2) on g(x). For example, the point (0, 0) on the graph of f(x) will become (1, -2) on the graph of g(x). Additionally, a point such as (2, 16) will become (3, 6), and (-2, 16) will become (-1, 6). Desmos Link for More Detail

Smooth and Continuous Definition

Polynomial functions of degree 2 or more have graphs that are smooth and continuous. When a graph is referred to as being "smooth", it means the graph has only smooth, rounded turns. In other words, the graph of a polynomial function can't have a sharp turn, which is a turn where the direction is changed suddenly. When a graph is referred to as being "continuous", it means the function can be graphed without lifting the pencil from the page. In other words, there are no breaks or holes in the graph.

Graphs of Non-Polynomial Functions

The two images above are for graphs of non-polynomial functions. In the first graph, we can see that we have a sharp turn. In the second graph, we would have to lift the pencil from the page to complete the graph.

Graphs of Polynomial Functions

The two images above are for graphs of polynomial functions. Notice the graphs are both smooth and continuous. There are no sharp turns only rounded ones. Additionally, there are no breaks or holes.

Graphs of Polynomial Functions: End Behavior

The end behavior of a polynomial function describes what happens to the function as x approaches both positive and negative infinity. $$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$ $$g(x) = a_nx^n$$ The two functions given above, f(x) and g(x), will have the same end behavior. This is due to the influence of the leading term, which is commonly referred to as the "dominating term". When x is large in terms of absolute value, the other terms are going to be insignificant in size. Let's take a look at an actual example. Below, we have f(x) and g(x) with the same leading term of x³. $$f(x) = x^3 - 2x^2 - x + 1$$ $$g(x) = x^3$$ Factor out x³ from f(x): $$f(x) = x^3\left(1 - \frac{2}{x} - \frac{1}{x^2} + \frac{1}{x^3}\right)$$ If we think about what's inside of the parentheses: -2/x, -1/x², and 1/x³ are all going to approach 0 as x becomes very large in terms of absolute value. This means what's inside of the parentheses is going to be very insignificant and the leading term x³ is going to dominate. This can also be shown with a simple table of values like we have below.

x	f(x)	g(x)
10	791	1000
50	119,951	125,000
100	979,901	1,000,000

Looking below at graphs for f(x) and g(x), we can see they have the same end behavior which is coming from the same leading term x³. Desmos Link for More Detail Since the domain of a polynomial function is the set of real numbers, we can use this end behavior to describe parts of the graph we can't sketch. For an odd-degree polynomial function, the graph will have the opposite behavior at each end, whereas, an even-degree polynomial function will have a graph with the same behavior at each end.

	n is odd	n is even
an > 0	↙ ↗	↖ ↗
an < 0	↖ ↘	↙ ↘

When end behavior is described, we will typically see the arrow notation, which is explained below. $$x → ∞$$ The above means as "x approaches positive infinity", you might also see this as "x increases without bound". $$x → -∞$$ Similarly, the above means as "x approaches negative infinity", you might also see this as "x decreases without bound". $$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$ As we discussed above, the end behavior of a polynomial function is determined by the degree n and the sign of the leading coefficient a_n. Some books refer to the process of finding the end behavior as the leading coefficient test since we are only analyzing the leading coefficient and nothing else. We will cover the four different cases below.

Odd Degree with a Positive Leading Coefficient

For our first case, the polynomial function will have an odd degree and a positive leading coefficient. $$y → ∞ \, \text{as} \, x → ∞$$ y approaches positive infinity as x approaches positive infinity. $$y → -∞ \, \text{as} \, x → -∞$$ y approaches negative infinity as x approaches negative infinity.
In other words, the graph falls to the left and rises to the right.

Odd Degree with a Negative Leading Coefficient

For our second case, the polynomial function will have an odd degree and a negative leading coefficient. $$y → ∞ \, \text{as} \, x → -∞$$ y approaches positive infinity as x approaches negative infinity. $$y → -∞ \, \text{as} \, x → ∞$$ y approaches negative infinity as x approaches positive infinity.
In other words, the graph rises to the left and falls to the right.

Summary Images for End Behavior of Polynomials when n is odd

$$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$ Example #2: Use the leading coefficient test to determine the end behavior of the graph of f(x). $$f(x) = x^3 - 3x^2 + x + 2$$ The degree of the polynomial is 3, which is odd. The leading coefficient is +1. This tells us that y will approach negative infinity as x approaches negative infinity and y will approach positive infinity as x approaches positive infinity. We could also say the graph falls to the left and rises to the right. Desmos Link for More Detail Example #3: Use the leading coefficient test to determine the end behavior of the graph of f(x). $$f(x) = -x^3 - 3x^2 + x + 2$$ The degree of the polynomial is 3, which is odd. The leading coefficient is -1. This tells us that y will approach positive infinity as x approaches negative infinity and y will approach negative infinity as x approaches positive infinity. We could also say the graph rises to the left and falls to the right. Desmos Link for More Detail

Even Degree with a Positive Leading Coefficient

For our third case, the polynomial function will have an even degree and a positive leading coefficient. $$y → ∞ \, \text{as} \, x → -∞$$ y approaches positive infinity as x approaches negative infinity. $$y → ∞ \, \text{as} \, x → ∞$$ y approaches positive infinity as x approaches positive infinity.
In other words, the graph rises to the left and rises to the right.

Even Degree with a Negative Leading Coefficient

For our fourth and final case, the polynomial function will have an even degree and a negative leading coefficient. $$y → -∞ \, \text{as} \, x → -∞$$ y approaches negative infinity as x approaches negative infinity. $$y → -∞ \, \text{as} \, x → ∞$$ y approaches negative infinity as x approaches positive infinity.
In other words, the graph falls to the left and falls to the right.

Summary Images for End Behavior of Polynomials when n is even

$$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$ Example #4: Use the leading coefficient test to determine the end behavior of the graph of f(x). $$f(x) = -2x^4 + 3x^3 - 6x^2 - 5x + 1$$ The degree of the polynomial is 4, which is even. The leading coefficient is -2. This tells us that y will approach negative infinity as x approaches both negative infinity and positive infinity. We could also say the graph falls to the left and falls to the right. Desmos Link for More Detail Example #5: Use the leading coefficient test to determine the end behavior of the graph of f(x). $$f(x) = 2x^4 + 3x^3 - 6x^2 - 5x + 1$$ The degree of the polynomial is 4, which is even. The leading coefficient is +2. This tells us that y will approach positive infinity as x approaches both negative infinity and positive infinity. We could also say the graph rises to the left and rises to the right. Desmos Link for More Detail

Shape of the Graph Near a Zero

At this point, we know how to find the zeros of a polynomial function. Recall that a zero or root of a polynomial function is a solution to the equation: $$f(x) = 0$$ The real solutions to the above equation are the x-intercepts of the graph. Let's look at an example.
Example #6: Find the x-intercepts for f(x). $$f(x) = 2\left(x - \frac{5}{2}\right)(x - 1)(x - 3)$$ Replace f(x) with 0: $$2\left(x - \frac{5}{2}\right)(x - 1)(x - 3) = 0$$ $$x = 1, \frac{5}{2}, 3$$ This tells us that the graph of our function f(x) will have x-intercepts of (1, 0), (5/2, 0), and (3, 0). Desmos Link for More Detail

Multiplicity and x-Intercepts

When discussing how to find the zeros of polynomial functions, we learned that a zero may occur more than once. For example: $$f(x) = (x - 2)^2(x - 1)$$ Here both 1 and 2 are zeros, however, 2 is a zero of multiplicity 2, which comes from the exponent of 2. To make this clear, we could rewrite the function as: $$f(x) = (x - 2)(x - 2)(x - 1)$$ When we have a zero of even multiplicity, the graph will touch the x-axis and turn around. In other words, it touches but does not cross the x-axis there. When we have a zero of odd multiplicity, the graph will cross the x-axis. Additionally, graphs will flatten out near zeros where the multiplicity is greater than one. Desmos Link for More Detail Looking at the graph above, we can see that the graph is going to cross the x-axis at (1, 0) and touch the x-axis at (2, 0) but then turn around. What causes this behavior? Recall that the graph of a polynomial function is continuous, there are no breaks or holes in the graph. This means that a polynomial function can only change its sign from positive to negative by first crossing through zero. We know that our y-value or function value is zero when x is both 1 and 2. We can use this information to set up a little table to use in conjunction with our graph to think about this further. Using both the table and graph we can see why the graph will cross the x-axis at the zero of 1 (multiplicity 1) and only touch the x-axis and then turn around at the zero of 2 (multiplicity 2). First off, we know that (x - 2)² will always be non-negative. It will be 0 at an x-value of 2 and then positive everywhere else. We also know that (x - 1) will be negative when x is less than 1, 0 at an x-value of 1, and then positive when x is greater than 1.
Let's think about what happens on the graph working left to right. For the interval (-∞, 1), we have y-values that are negative. In other words, the graph is below the x-axis. Then at an x-value of 1, we have an x-intercept, so the y-value is 0. Moving on to the interval (1, 2), the y-values are now positive. Again, this means the graph is now above the x-axis. Notice the sign changed from (-) in the interval (-∞ , 1) to (+) in the interval (1, 2). When we move from (1, 2) to (2, ∞) the y-value is 0 at an x-value of 2 but positive everywhere else. This is why the function touches the x-axis at (2, 0) but then turns around and heads back up.
In summary, when you have a zero of even multiplicity, the sign of your polynomial function is not going to change from one side of your zero to the other side of your zero. This means that the graph is going to turn around at that zero of even multiplicity. If the zero is of odd multiplicity, then the sign of the polynomial function is going to change from one side of the zero to the other side of your zero, which causes the graph to cross the x-axis at that zero.

Repeated Zeros

A factor (x - k)^m, m > 1, yields a repeated zero x = k of multiplicity m.

• When m is odd, the graph crosses the x-axis at x = k
• 
• When m is even, the graph touches but does not cross the x-axis at x = k
• 

Let's look at an example.
Example #7: State whether the graph crosses the x-axis or touches and turns around at each zero. $$f(x) = \frac{1}{8}(x - 1)^3(x + 2)^2(x - 3)$$ The zeros here are 1 (mult. 3), -2 (mult. 2), and 3 (mult. 1). The x-intercepts will be (1, 0), (-2, 0), and (3, 0). The graph will cross through at (1, 0) and (3, 0) since those zeros of 1 and 3 have odd multiplicities, 3 and 1 respectively. Then at (-2, 0), the graph will touch the x-axis and turn around. We can use the graph below for confirmation. Desmos Link for More Detail

Turning Points | Local Maxima and Minima

Polynomial functions often have turning points, where the function changes from increasing to decreasing or from decreasing to increasing. These turning points will be referred to as a local maximum point (also known as a relative maximum point) when the function takes on a greater value than any nearby points and a local minimum point (also known as a relative minimum point) when the function takes on a lesser value than any nearby points. When you hear the terms maxima and minima, these are just the plural forms of the words maximum and minimum, respectively, coming from Latin. The local maximum and minimum points on the graph of a polynomial function are called its local extrema. Note: Without the use of some basic Calculus, it is usually not possible or at least extremely impractical to find the actual turning points for a polynomial function when the degree is greater than 2. A polynomial function of degree n has at most (n - 1) turning points, with at least one turning point between each pair of consecutive real zeros. Note: Be careful here as we are only saying that a polynomial of degree n has at most (n - 1) turning points, not that it will have (n - 1) turning points. So a polynomial function of degree n may have less than (n - 1) turning points. Let's look at an example.
Example #8: Use the given graph to find the local extrema for the given polynomial function. Desmos Link for More Detail The graph above has a total of 3 local extrema. There are 2 local minimum points of (-3, -0.25) and (1, -0.25), along with 1 local maximum point of (-1, 3.75).

Creating a Rough Sketch of the Graph of a Polynomial Function

At this point, we have enough information to create a rough sketch of the graph of a polynomial function. I would again note that this process is just about understanding the concept, creating a perfect graph can be done instantly with graphing calculators such as Desmos. $$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$

• Use the Leading Coefficient Test to find the graph's end behavior
• Find the x-intercepts by solving the equation f(x) = 0
  • If k is a zero of even multiplicity, the graph touches the x-axis at x = k and then turns around
  • If k is a zero of odd multiplicity, the graph crosses the x-axis at x = k
  • If the multiplicity of k is greater than 1, the graph flattens out near (k, 0)
• Find and plot the y-intercept
  • Find the y-intercept as: (0, f(0))
• Use symmetry when available to sketch the graph
  • y-axis symmetry: f(-x) = f(x)
  • origin symmetry: f(-x) = -f(x)
• Use test points within the intervals formed by the x-intercepts to find the sign of f(x) in the interval
  • Since the graph of a polynomial function is continuous, the values of f(x) are either always positive or always negative in a given interval
  • When the values are positive, we are above the x-axis
  • When the values are negative, we are below the x-axis
• Use the maximum number of turning points of n - 1 to check the graph

Example #9: Sketch a rough graph of the polynomial function. $$f(x) = (x-1)^2(x - 4)$$ First, let's simplify: $$f(x) = x^3-6x^2+9x-4$$ 1) Use the leading coefficient test to find the graph's end behavior. As we can see the leading term is x³, which means the degree 3 is odd and the leading coefficient is positive. This means the graph falls to the left and rises to the right.
2) Find the x-intercepts. $$(x-1)^2(x - 4) = 0$$ $$x = 1, 4$$ The x-intercepts will occur at (1, 0) and (4, 0). The zero of 1 has a multiplicity of 2, therefore, the graph will touch the x-axis at (1, 0) and turn around. The zero of 4 has a multiplicity of 1, therefore, the graph will cross through the x-axis at (4, 0).
3) Find and plot the y-intercept. $$f(0) = -4$$ The y-intercept occurs at (0, -4).
4) This function is not odd or even.
5) Use test points within the intervals formed by the x-intercepts to find the sign of f(x) in the interval. Since we have x-intercepts of (1, 0) and (4, 0), this gives us three intervals to work with: (-∞, 1), (1, 4), and (4, ∞).

Interval	x	f(x)
(-∞, 1)	0.5	-0.875
(1, 4)	2	-2
(4, ∞)	4.5	6.125

Let's put what we have on the coordinate plane and see how it looks. Now we can make a rough sketch through the points. The main issue is we can see there will be a local minimum in the interval (1, 4) but what is the location of this local minimum point? Unfortunately, without Calculus it's pretty tedious to find in most cases. What you can do is plug in x-values between 2 and 4 since we already worked with 2 in the table. Look for where it stops decreasing and starts increasing. $$f(2.5) = -3.375$$ $$f(2.75) = -3.828125$$ $$f(3) = -4$$ $$f(3.25) = -3.796875$$ $$f(3.5) = -3.125$$ We can see that (3, -4) looks like a good place to turn. In this case, that is the actual turning point or local minimum. In most cases, it won't work out like that but again we are only trying to make a rough sketch. Let's put this new information into the sketch. At this point, we can make our rough sketch. It will be drawn here with a computer, so when you make one on your own keep in mind that it does not need to be perfect. Desmos Link for More Detail 6) Looking at the final graph, we can see that we have 2 turning points or local extrema. This is consistent with (n - 1) turning points since the degree is 3 and we have 2 (3 - 1 = 2) turning points. Again, we won't always have (n - 1) turning points but if you have more than (n - 1) turning points on your sketch, you know you did something wrong.

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