U3SN2: Exploring Cubic Functions & Power Functions

You should learn to: 1. Determine the general behavior of the graph of even and odd degree power functions. 2. Explore the possible graphs of cubic, quartic, and quintic functions, and extend graphical properties to higher-degree functions. 3. Generalize the key characteristics of polynomials. 4. Sketch the graph of any polynomial given key characteristics.

� �−� A polynomial function is a function that can be written in the form �(�) = ��� + ��−�� + � ⋯ ��� + ��� + �� where ��, ��−1, …�2, �1, �0 are complex numbers and the exponents are nonnegative integers. The form shown here is called the standard form of a polynomial. Really this means it is a function of the form (�) = ��� + ����� , where n is a non-negative integer.

Polynomial functions are continuous (there are no breaks in their graphs) and they only have smooth turns (there are no sharp turns). Polynomial functions also have a domain of all reals.

Example 1. Are the following Polynomials? If not, why not?

You already know that a second-degree polynomial function is called a quadratic function, and a third- degree polynomial function is called a cubic function. A quartic function is a fourth-degree polynomial function, while a quintic function is a fifth-degree polynomial function.

Example 2. Graph and compare the following.

y y y

x x x

� = � yx 3 yx 5

y y y

x x x

yx 2 yx 4 y x6

What if we have a function with more than one power? Will it mirror the even behavior or the odd behavior? y y y

x x x

� = �5 + �4 � = �3 + �8 � = �2(� + 2)

The degree of a polynomial function is the highest degree of its terms when written in standard form.

Example 3. Find the degree of each of the polynomial functions below. A. � = 4�2 + 3�5 − 12� + 2 B. �(�) = −(� − 4)2(� + 2)3 C. � = �3(� − 1)(�2 − 5)

Leading Coefficient Test: The end behavior of a graph of a function is the behavior of the graph as x approaches negative infinity and as x approaches positive infinity. DEGREE (highest power) End behavior of �(�) = ��� + ����� � is EVEN � is ODD where � is the degree of the polynomial. “�” is + In other words, � is the highest power!

Leading Leading coefficient “�” is –

Example 4. Use the degree and the leading coefficient to find the end behavior of the following functions. A. �(�) = 5�7 B. �(�) = −7�10

1 C. �(�) = ��7 D. �(�) = −27(�2 − 1)3(� + 2)7 3

Finding the �-intercepts of a polynomial function. How would we find the �-intercepts of these polynomial functions? How have we found these in the past?

Example 5. Find the x-intercepts and the zeros for each of the polynomial functions, then use the idea of end behavior to sketch the graph of the function. B. �(�) = �3 + 4� A. �(�) = �3 − �2 − 2�

y y

x x

( ) 1 ( )( )2( )3 C. ℎ(�) = �(� + 3)(� − 4)(� + 1) D. � � = 10 � + 3 � − 2 � + 1

y y

x x

So, the rule for �-intercepts is that: on the graph of a polynomial function of degree �, it can have at most “�” �-intercepts. A zero of a function, �(�), is a number x which makes �(�) = 0. The real zeros of polynomial functions are the x-values of the x-intercept points.

The polynomial function can have two types of behavior through the �-intercepts. It can “cross” or it can “bounce.” Looking at the last example, what is the determining factor of whether it crosses or bounces?

Multiplicity is how many times a particular number is a zero for a given polynomial function. If the multiplicity is even the graph will “bounce” at that zero and if the multiplicity is odd the graph will “cross” at that zero.

Example 6. Draw a rough sketch of the behavior at each of the zeros shown in the functions below. A. �(�) = (� + 4)3 B. �(�) = (� + 4)10

1 C. �(�) = �2(� + 1) D. �(�) = (� + 1)2(� − 1)2 3

Putting it all together! STEPS TO GRAPHING A POLYNOMIAL FUNCTION 1. Find end behavior determined by the degree and the “�” value.

2. Find the x-intercepts by setting �(�) = 0 and factoring or using the quadratic formula to solve. Remember, the multiplicity at each of these zeros causes our graph to “cross” if the power is odd, and “bounce” if the power is even.

3. Find the y-intercept. If it’s in standard form we can look at the constant, if it’s in another form we can find this by setting x = 0.

4. REMEMBER, the graph of a polynomial function is always smooth and continuous and beautiful, with no sharp turns. Remember the domain is all reals, so your graph should continue to the right and left FOREVER.

Example 7. Graph the following functions (rough graphs) A. �(�) = −3�3 − 12�2 y

x Degree:

End Behavior:

y-intercept:

x-intercepts:

B. �(�) = (�2 − 1)(� + 4)2 y

x Degree:

End Behavior:

y-intercept:

x-intercepts:

C. Write a polynomial function having zeros of y � = 3, and � = −3 (multiplicity 2)

x

Degree:

End Behavior:

y-intercept:

x-intercepts: