Polynomial Functions Chapter 7 Algebra 2B

A polynomial function is a function of the form f (x) = a n x n + a n – 1 x n – 1 +· · ·+ a 1 x + a 0 Where a n  0 and the exponents are all whole numbers. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right. For this polynomial function, only one variable (x) a n is the leading coefficient, a 0 is the constant term, and n is the degree. a n  0 anan anan leading coefficient a 0a 0 a0a0 n n degree descending order of exponents from left to right. n n – 1

DegreeTypeStandard Form You are already familiar with some types of polynomial functions. Here is a summary of common types of polynomial functions. 4Quartic f (x) = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 0Constantf (x) = a 0 3Cubic f (x) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 2Quadratic f (x) = a 2 x 2 + a 1 x + a 0 1Linearf (x) = a 1 x + a 0

One way to evaluate polynomial functions is to use direct substitution. Another way to evaluate a polynomial is to use synthetic substitution. Value of a function: f (k)

Real Zeros of a polynomial function: Maximum number of real zeros is equal to the degree of the polynomial. Real zeros: where the graph crosses the x-axis. How many (total) zeros do the following functions have? f(x) = x + 2 g(x) = x 2 – 4 h(x) = x 3 – 2x 2 – 10x + 20 p(x) = x 4 + 8x one: –2 two: 2, –2 three: -3.16, 2, 3.16 four: -1, 1 & 2 complex The zeros may be real or complex... The Fundamental Theorem of Algebra: Counting complex and repeated solutions, an nth degree polynomial equation has exactly n solutions.

END BEHAVIOR OF P OLYNOMIAL F UNCTIONS The end behavior of a polynomial function’s graph is the behavior of the graph, which is f(x), as x approaches infinity (+  ) or negative infinity (–  ). The expression x +  is read as “x approaches positive infinity.”

A B C D

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. f (x) = x 2 – 3x 4 – S OLUTION The function is a polynomial function. It has degree 4, so it is a quartic function. The leading coefficient is – 3. Its standard form is f (x) = – 3x 4 + x 2 –

Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. Identifying Polynomial Functions The function is not a polynomial function because the term 3 x does not have a variable base and an exponent that is a whole number. S OLUTION f (x) = x x

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. S OLUTION f (x) = 6x x – 1 + x The function is not a polynomial function because the term 2x – 1 has an exponent that is not a whole number.

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. S OLUTION The function is a polynomial function. It has degree 2, so it is a quadratic function. The leading coefficient is . Its standard form is f (x) =  x 2 – 0.5x – 2. f (x) = – 0.5 x +  x 2 – 2

f (x) = x 2 – 3 x 4 – Identifying Polynomial Functions f (x) = x x f (x) = 6x x – 1 + x Polynomial function? f (x) = – 0.5x +  x 2 – 2

Your Turn 1: What are the degree and leading coefficient? a) 3x 2 - 2x 4 – 7 + x 3 b) x x 7 c) 4x 2 – 3xy + 16y 2 d) 4x 6 + 6x 4 + 8x 8 – 10x

Value of a function Using Direct Substitution Use direct substitution to evaluate f (x) = 2 x 4  8 x x  7 when x = 3. Solution: f (3) = 2 (3) 4  8 (3) (3)  7 = 98

Value of a function Using Synthetic Substitution Use synthetic substitution to evaluate f (x) = 2 x 4 +  8 x x  7 when x = 3.

Polynomial in standard form Using Synthetic Substitution 2 x x 3 + (–8 x 2 ) + 5 x + (–7) The value of f (3) is the last number you write, In the bottom right-hand corner. The value of f (3) is the last number you write, In the bottom right-hand corner. 20–85 –720–85 –7 Coefficients 3 x -value 3 S OLUTION Polynomial in standard form

Your Turn 2: Use direct substitution. a)f(x)= 2x 2 - 3x + 1 f(-4)

Your Turn 3: Use direct substitution. b)f(x)= x 2 – 4x – 5 f(a 2 -1)

Your Turn 4: Use Synthetic Substitution Find f(2) a.3x 2 - 2x 4 – 7 + x 3 Your Turn 5: Use Synthetic Substitution Find f(-5) b.100 – 5x x 4

G RAPHING P OLYNOMIAL F UNCTIONS END BEHAVIOR f(x)=

Your Turn 6: Degree: ___ End behavior: # real zeros: _____ For each graph below,  describe the end behavior,  determine whether it represents an odd-degree or an even-degree function, and  state the number of real zeros.

Your Turn 7: Degree: ___ End Behavior # real zeros: _____ For each graph below,  describe the end behavior,  determine whether it represents an odd-degree or an even-degree function, and  state the number of real zeros.

Your Turn 8: Degree: ___ End behavior: # real zeros: _____ For each graph below,  describe the end behavior,  determine whether it represents an odd-degree or an even-degree function, and  state the number of real zeros.

Your Turn 9: Your own: Degree: ___ End behavior: # real zeros: _____ For each graph below,  describe the end behavior,  determine whether it represents an odd-degree or an even-degree function, and  state the number of real zeros.

Closure 7.1

Lesson 7.1 Check for understanding: Closure 7.1 Homework: Practice 7.1

Warm-up 7.1

7.2 Graphs of Polynomial Functions

We have learned how to graph functions with the following degrees: 0 Example: f(x) = 2 horizontal line 1Example: f(x) = 2x – 3 line 2Example: f(x) = x 2 + 2x – 3 parabola How do you graph polynomial functions with degrees higher than 2?

Graphs of Polynomial Functions: are continuous (there are no breaks) have smooth turns with degree n, have at most n – 1 turns Follows end behavior according to n (even or odd) and to a n (positive or negative). We’ll make a table of values, then graph...

END BEHAVIOR FOR POLYNOMIAL FUNCTIONS > 0even f (x)+  f (x) +  > 0odd f (x)–  f (x) +  < 0even f (x)–  f (x) –  < 0odd f (x)+  f (x) –  a n n x –  x +  A B C D End behavior of a polynomial function:

x f (x) –3 –7 –2 3 – – Graphing Polynomial Functions Graph: f(x) =x 3 + x 2 - 4x - 1 The degree is odd and the leading coefficient is positive, so f (x) – as x – and f (x) + as x +. n: __ a n : ___ # turns: at most ____ # total zeros: ____ # real zeros: at most _____ # Real Zeros: ___ between ___ and ___ x = _________ Relative Maxima value: x = ____ Relative Minima value: x = ____ leftright

Example 2 : Graph by making a table of values and find the zeros. f (x) = –x 4 – 2x 3 + 2x 2 + 4x. n: __ a n : ___ # turns: at most ____ # total zeros: ____ # real zeros: at most _____ End Behavior: ____________ as ________ ____________ as _______ # Real zeros: ______ Zeros: ______ (exact) & _______(exact) between ____ and ____ ( x = _____) between ____ and _____ ( x = _____) Relative Maxima: ________ ________ Relative Minima: ________ x f(x)

Your Turn 1 : Graph by making a table of values and find the zeros. f (x) = 3x 3 – 9x + 1 n: __ a n : ___ # turns: at most ____ # total zeros: ____ # real zeros: at most _____ End Behavior: ____________ as ________ ____________ as _______ # Real zeros: ______ Zeros: ______ (exact) & _______(exact) between ____ and ____ ( x = _____) between ____ and _____ ( x = _____) Relative Maxima: ________ ________ Relative Minima: ________

Your Turn 2 : Graph by making a table of values and find the zeros. f (x) = -x 3 + 4x. n: __ a n : ___ # turns: at most ____ # total zeros: ____ # real zeros: at most _____ End Behavior: ____________ as ________ ____________ as _______ # Real zeros: ______ Zeros: ______ (exact) & _______(exact) between ____ and ____ ( x = _____) between ____ and _____ ( x = _____) Relative Maxima: ________ ________ Relative Minima: ________

Lesson 7.2 Check for understanding: Closure 7.2 Homework: Practice 7.2

Closure 7.2

Warm-up/Review Lesson 7.2 a) Real Zeros: check table ___________________________________ b) x-coord. ______ (max/min?) Value: ______ x-coord. ______ (max/min?) Value: ______ a) Real Zeros: use 2nd calc ___________________________________ b) x-coord. ______ (max/min?) Value: ______ x-coord. ______ (max/min?) Value: ______

Warm-up/Review Lesson 7.2 a) Real Zeros: ___________________________________ b) x-coord. ______ (max/min?) Value: ______ x-coord. ______ (max/min?) Value: ______ a) Real Zeros: ___________________________________ b) x-coord. ______ (max/min?) Value: ______ x-coord. ______ (max/min?) Value: ______

Zeros: -.879; 1.347; Zeros: ; 0.160; x = 0 value: -3 x = 2 value: x = -1.0 value: 4.5 x = 2 value: -9

Zeros: ; x = -2 value: -4 x = 0 value: 4 Zeros: ;.414 x = -1 value: -4 x = 1 value: 2.75

7.3: Solving Polynomial Equations by using quadratic techniques

Vocabulary Quadratic form:Quadratic formula: u

Example 1: Write the given expression in quadratic form, if possible Answer:

Example 2: Write the given expression in quadratic form, if possible The answer will have to look like: Answer:

Example 3: Write the given expression in quadratic form, if possible The answer will have to look like: Answer:

Example 4: Write the given expression in quadratic form, if possible The answer will have to look like: Answer:

____________________________________________________________________________ In your own words: What is necessary for an expression to be written in quadratic form?

You look at the two terms that are not constants and compare the exponents on the variable. If one of the exponents is twice the other, the trinomial can be written in quadratic form. In your own words: What is necessary for an expression to be written in quadratic form?

Your Turn 1: Write the given expressions in quadratic form, if possible a) 2x 4 + x 2 + 3b) x c) x 6 + x 4 + 1d) x - 2x 1/2 + 3

Your Turn 1: Write the given expressions in quadratic form, if possible. Show u a)2x 4 + x u = x 2 Answer: 2u 2 + 1u + 3 b) x u = x 6 Answer: 1u c) x 6 + x Answer: Not possible d) x – 2x 1/2 + 3 u = x 1/2 Answer: u 2 – 2u + 3

Example 5: Solve

Example 6: Solve

Your Turn 2: Solve each equation

Example 7: Solve Answers

Example 8: Solve Answers

Your Turn 3: Solve Answer: -27 & -8

Closure 7.3

Warm-up/Review Lesson 7.3

Warm-up 7.4 Pg. 15

7- 4: Polynomial Division, Factors, and Remainders

Objectives:  Determine whether a binomial is a factor of a polynomial by using synthetic substitution

Review of Synthetic Division Example 1: (2x 2 + 3x – 4) ÷ (x – 2) Example 2: (p 3 – 6) ÷ (p – 1)

Review of Synthetic Division & Long Division Example 3: (2x 3 – 7x 2 – 8x + 16) ÷ (x – 4) Example 4: (5x 3 + x 2 – 7) ÷ (x + 1)

Example 5: Method 1: Direct substitution Method 2: Synthetic substitution

Given that (x+2) is a factor of f(x), find the remaining factors of the polynomial Example 6: Remaining factors: ______________________________

Your Turn 1

Lesson 7.4 Check for understanding: Closure 7.4 Homework: Practice 7.4

Closure 7.4

Warm-up/Review Lesson 7.4

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7-5: Roots & Zeros of Polynomial Functions  Find all the exact zeros of a polynomial function by: 1) graphing calculator 2) Synthetic substitution 3) Quadratic formula

Vocabulary Complex zeros always in pairs! A polynomial function may have __ or __ or __ …or any _____ number of complex zeros. Examples: _________ & _________(its conjugate) _________ & _________(its conjugate)

Example 1:Find all exact zeros Step 1: Use your graphing calculator to find at least one real zero & use synthetic substitution to get the depressed polynomial. Step 2: Once you get a polynomial with degree 2 you can solve the quadratic equation (by the method of your choice!) Step 3: Give the Answer: the Zeros are ________________________________

Example 2: Find all the exact zeros of Step 1: Graphing calculator & Synthetic substitution Try another zero until you get a depressed polynomial with degree 2. Step 2: Solve the quadratic equation! Step 3: Zeros are _________________________________

Your Turn 1: Find all the zeros of Step 1: Step 2: Step 3: Answer _________________

Example 3: Write a polynomial function of least degree with integer coefficients whose zeros include 4 & 7i  _________(its conjugate) Remember: Imaginary roots always come in pairs If p & q are roots of an equation, then (x-p) and (x-q) are factors!!! So, because there are ___ zeros, the least degree will be: ____. And we get the polynomial function with the least degree by multiplying:

Warm-up/Review Lessons 7.3 & 7.5 pg 24 Solve the following equations. Give exact answers. You may use any method. 1. total # of zeros: _____ # Real zeros: _____ 2. total # of zeros: _____ # Real zeros: _____

Warm-up/Review Lessons 7.3 & Give exact answers! total # of zeros: _____ # Real zeros: _____

Closure Lesson 7.5

Warm-up/Review Lesson 7.5 (pg. 24)

7.7 Function Operations Learning Targets: I can find the sum, difference, product, and quotient of functions. I can find composition of functions.

Vocabulary

Composition of Functions There is a 40% off sale at Old Navy and as an employee you receive a 10% discount, how much will you pay on a $299 jacket? You do not get 50% off…...this is an example of a composite function. You will pay 90% of the cost (10% discount) after you pay 60% (40% discount).

Composition of Functions There is a 40% off sale at Old Navy and as an employee you receive a 10% discount, how much will you pay on a $299 jacket. You will pay 90% of the cost (10% discount) after you pay 60% (40% discount). The two functions look like this… f(x) = 0.6x g(x) = 0.9x We can put these together in a composite function that looks like this… f(g(x)) “f of g of x”

Work from the inside out (find g of 299 first)... f(g(299)) = f( ) = f(269.1) Now, find f of = ( ) = The jacket will cost $ f(x) = 0.6x and g(x) = 0.9x What is f(g(x)) when x = 299?

Function Operations You can perform operations, such as addition, subtraction, multiplication, and division, with functions… restriction: g(x) = 0 because:_____________________

Your Turn 1: Don’t forget the restriction since the denominator can’t ever be equal to ___!

Example 2:

Your Turn 2: Find and

Example 3: If f(x) = x 2 – 5 and g(x) = 3x find f[g(2)] and g[f(2)]

Your Turn 3: Find and

Closure Lesson 7.7

Warm-up Lesson 7.8

-2x 2 + 3x + 3 6x 3 + 4x 2 - 3x - 2 6x x x + 7

7.8 Inverse Functions & Relations

Learning Targets I can find the inverse of a function or relation. I can determine whether two functions or relations are inverse functions by using composition of functions.

Inverse relation – just think: switch the x & y-values. the inverse of an equation: switch the x & y and solve for y.

the inverse of a graph: the reflection of the original graph in the line y = x. y = x Is this relation a function? _____________________

Example 1: Find the inverse Step 4 Replace y with f --1 (x)

Your Turn 1: How are the two lines related? ________________________________________________________________

Use composition of functions.

Your Turn 2: Use composition of functions! You may graph to double check!

Closure Lesson 7.8

Warm-up Lesson 7.9

(15 x 18 / 9)17.99 = 539.7

7.9 Square Root Functions

Learning Targets: I can graph and analyze square root functions (state domain & range). I can graph square root inequalities.

First, let’s look at the parent graph. Domain: Range:

Domain: Range:

xy Domain: Range:

xy Domain: Range:

xy Domain: Range: - 2 CHANGE!! Add this!

Graphing inequalities:

Your Turn:

Closure 7.9

REVIEW chapter 7 Lessons

7a. 7b. 7c.7d. restriction

8a. 8b.

9a.9b.

Find the inverse if it exists. 10a. 10b.

Determine whether the following are inverse functions. Use composition of functions.

Domain: __________________ Range: ___________________ 12a. Graph the square root function. State Domain and Range. y =

12b. Graph the square root function.