Polynomials I
43
Unit 4 – Polynomials I
Study Notes 1 – Graphing & Finding the Zeros of Cubic Functions
Objective: Connect the characteristics and behaviors of cubic functions to its factors. Build cubic functions from linear and quadratic functions. Graph the basic cubic function using reference points and symmetry. Dilate, reflect and translate cubic functions. Identify the relative maximum and relative minimum on a graph and find the maximum or minimum using a calculator. Identify the zeros of a cubic function by factoring. By the End of Class I Should… A cubic function is a function that can be written in the standard form: where 0. A relative maximum is the highest point in a particular section of a graph. A relative minimum is the lowest point in particular section of a graph. 2
4
8 on your calculator and copy it, to the best of your Example 1: Graph ability, on the following graph. A. Find the coordinates of the y‐intercept. B. Find the coordinates of the relative maximum. C. Find the coordinates of the relative minimum. D. Identify the type of zeros of this function. E. What are those zeros? (Look at the x‐intercepts) F. Write the zeros as linear factors. G. Multiply out the linear factors. Domain: Range: H. What do you end up with? Cubic functions can either be written in standard form or in factored form ) as we observed in part (f) of Example 1. Example 2: Multiply out each function in factored form and write it in standard form, then use your calculator to verify your product is correct. (Hint: The graphs should be identical) A. 1
1
3 B. 1 2
3
5 44
Previously, you graphed linear & quadratic functions using the following notation. You can use this notation to identify or perform transformations on any function, including cubic functions. Graphing Adjustments Graphing Cubic Equations:
Example 3: Graph the following cubic functions. A. Parent Function: x y 1
C. E. 2
1
3 Domain: Range: B. 2 D. 2 F. 4 Finding the Zeros of Cubic Functions:
Example 4: Find the zeros of the following cubic functions. A. 4
25
100 B. 4
20
25 45
Study Notes 2 – Power Functions & Polynomials
Objective: Determine whether a function is even, odd, or neither based on graphs and algebraic patterns. Determine the general behavior of the graph of even and odd degree power functions using the Leading Coefficient Test. Define polynomials. By the End of Class I Should… If a graph is symmetric about a line, then the line divides the graph into two identical parts. A function is even if its graph is symmetric to the y‐axis & odd if its symmetric to the origin. If is symmetric to the x‐axis it is not a function. y
y
y x, y
( x, y )
 
 x, y 
  x, y 
x
x
x
  x,  y 
( x,  y )
Example 1: Use your calculator to determine whether each function is even, odd, or neither. Name specific points. A. 4
2 B. | | 1 Absolute Value on the Calculator Think About It: Describe what you observe about the , coordinate. This can be summarized as the following. Even Function The sign of the ____ coordinate changes. Odd Function The sign of the ____ & the ___ coordinates change. You can check if a function is odd, even, or neither algebraically by finding a few ordered pairs of solutions that have opposite x‐values and look for symmetry. You can also graph the function to visually determine if the function is even or odd. Example 2: Determine whether each function is even, odd, or neither without graphing. A. 1 B. 1 C. 46
A power function is a function of the form , where n is a non‐negative integer. Examples: 3 ,
3
9
1,
3
2
4
9 Example 3: Graph and compare the following. y = x y  x 3 y  x5 y  x 2 y  x 4 y  x 6 Think About It: Describe some of the patterns you observe. Look back to Example 3. Instead of drawing the entire graph just draw little arrows to represent what the ends of the graph are doing for each function. (Hint: Each arrow either goes up or down). If each is multiplied by a negative, what are the ends of each graph doing now? The arrows are describing the end behavior of the graph of a function. The end behavior of a graph describes the eventual rise or fall of the graph. In other words what does the graph do as x approaches infinity and as x approaches negative infinity? Based on your observations above, fill in the following table that summarizes the patterns of end behavior. End behavior of
Exponent is Even Exponent is Odd Leading Coefficient is + Leading Coefficient is – y
y
y
y
y
y
x
x
x
x
x
x
47
To find the end behavior of functions examine its leading term. This is often referred to as the Leading Coefficient Test. Example 4: Find the end behavior of the following functions. A. 2
1 B. 2
2
5
1 C. 5
3
2
1 D. 5
4 E. 2
1
2 F. 3
3
5 The leading coefficient test only works if the function is in standard form. How can we determine the leading term without having to multiply the factors together? Example 5: Find the end behavior of the following functions without foiling. A. 3
1
2
3 B. 2
1
6 In Example 4 we started working with functions that have an exponent higher than those in previous units. Name the following functions we have learned up to this point: If we keep going up further is quartic is quintic Continually naming functions with higher and higher exponents could get a bit ridiculous. For this reason we have a generalized term: polynomial functions. A polynomial function is represented by: ⋯
where the coefficients are real numbers and the exponents are nonnegative integers. Side Note: Define real number. Define nonnegative integer. Example 6: Identify the following as an example of a polynomial or not. A. 5
5
3 B. 3
5
9 C. 3 D. 1 E. F. 20
9
Other attributes of polynomials include:  Exactly one y‐intercept.  The graph of a polynomial of degree n has at most “n” x‐intercepts.  The polynomial includes exactly n zeros.  The graph of the polynomial of degree n has at most 1 relative extrema. o Relative extrema are the relative min & relative max; that is, the turning points. 48
Study Notes 3 – Graphing Higher Order Polynomials
Objective: Extend graphical properties to higher‐degree functions. Generalize the key characteristics of polynomials. Sketch the graph of any polynomial given key characteristics. By the End of Class I Should… Last time we discussed some key characteristics (attributes) of polynomial functions. Today we will be learning to how to graph higher order polynomials. The graph of a polynomial is continuous and only has smooth turns. There are no breaks in the graph and no sharp turns. Example 1: Are the following polynomials? If not, why not? When graphing a polynomial there are 4 things you will need to find: 1. End Behavior 2. Y‐intercept 3. Zeros 4. Multiplicity Before we put all the steps together, let’s practice each one separately. Example 2: Identify the end behavior of the following functions. A. 3
7
1 B. 1
2 C. 9
3
5 Example 3: Find the y‐intercept of the following functions. (Hint: 0) 6
8 B. 1
3 C. 2
2
6 A. 3
Example 4: Find the zeros of the following functions. (Hint: 0) A. 2
1 B. 6
13
6 C. 2
3
6
9 D. 3
2
1 49
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 (x‐intercept) and if the multiplicity is odd the graph will “Cross” at that zero (x‐intercept). Example 5: Determine the multiplicity for each zero for the following functions. 3
1
A. 1
2 B. 1
3 C. 5 Let’s try putting all the pieces together and begin graphing polynomials. Example 6: Sketch a rough graph of the following polynomials. A. 1
2
3 End Behavior: y‐intercept: Zeros: Multiplicity: B.
2
5
End Behavior: y‐intercept: Zeros: Multiplicity: C.
4
4
8
1 32 End Behavior: y‐intercept: Zeros: Multiplicity: 1.
2.
3.
4.
5.
6.
7.
Steps to Graphing a Polynomial Function
Find the end behavior determined by the Leading Coefficient Test. ↑↑, ↓↓, ↓↑,
↑↓ Find the y‐intercept by setting 0. Find the x‐intercept(s)/zero(s). Find the multiplicity by examining the exponent of each factor. Plot the x‐intercepts & y‐intercept on the coordinate plane. Label on your graph the multiplicity for the zero. (Even – Bounce, Odd – Cross) Connect the dots with a smooth curve. 50
Study Notes 4 – Graphing & Writing Equations of Higher Order Polynomials
Objective: Review graphing higher order polynomials. Write equations for higher order polynomials given their zeros. By the End of Class I Should… Review: List the necessary information for graphing a polynomial function. 1. 2. 3. 4. Example 1: Sketch a rough graph of the following polynomials. A.
3
1
6 B.
2
1 4
3
5 9
9 C.
D.
3
9
27 51
Example 2: Find a polynomial function with the given zeros, multiplicities, and degree. (There are many correct answers). A. Zero: 2, Mult: 2 B. Zero: 3, Mult: 1 C. Zero: 5, Mult: 3 Zero: 1, Mult: 1 Zero: 2, Mult: 3 Zero: 0, Mult: 2 Degree: 3 Degree: 4 Degree: 7 Falls to the Right Falls to the Right Falls to the Left Rises to the Left Example 3: Sketch the graph of a polynomial function that satisfies the given conditions. If not possible, explain your reasoning. (There are many correct answers). B. Fourth‐degree polynomial with three A. Third‐degree polynomial with two distinct real zeros and a negative leading distinct real zeros and a positive leading coefficient. coefficient. D. Fourth‐degree polynomial with two C. Fifth‐degree polynomial with two distinct real zeros and a positive leading distinct real zeros and a negative leading coefficient. coefficient. 52
Example 4: Choose the possible graph(s) for each given polynomial function f(x). A.
2 6 B.
C. 1 53
Partner Activity:
1st: Write the equation of any polynomial function in factored form. 2nd: You will switch equations with three of your classmates. You will then sketch, by hand, the graph of the equation that your partner wrote. When you are finished, use a graphing utility to check each other’s work. Partner 1’s Name: ______________________ Equation: End Behavior: y‐intercept: zeros: Multiplicity: Checked Graph on the Calculator: Did it Match? If No, fix what was incorrect. Then describe in words what went wrong. Partner 2’s Name: ______________________ Equation: End Behavior: y‐intercept: zeros: Multiplicity: Checked Graph on the Calculator: Did it Match? If No, fix what was incorrect. Then describe in words what went wrong. Partner 3’s Name: ______________________ Equation: End Behavior: y‐intercept: zeros: Multiplicity: Checked Graph on the Calculator: Did it Match? If No, fix what was incorrect. Then describe in words what went wrong. 54
Study Notes 5 – Higher Order Polynomial Inequalities & Piecewise Functions
Objective: Solve higher order polynomial inequalities. Review graphing piecewise functions. By the End of Class I Should… To solve Higher Order Polynomials Inequalities: 1. If needed, subtract from both sides so that one side is zero. 2. Factor (if not already). 3. Find the zeros (x‐intercepts). 4. Put the x‐intercept(s) on the x‐axis. 5. Draw a sketch of the graph (with the help of end behavior and multiplicity). 6. Determine the piece(s) of the graph that makes the inequality true. Example 1: Solve the following polynomial inequalities. A.2
5
12 B.
1
5
0 C.
9 9 3
32
48 D. 2
E.
1
3
6
0 55
A piecewise function is a function that is defined by two or more equations over a specified domain. 1,
4
Example 2: Graph 4 ,
4
Find: 5 0 6 2 2
,
1
,
1
2 Example 3: Graph 3 ,
2
Find: 2 1 0 1 2 Example 4. Write the equation for the piecewise function. 56