4 Increasing Decreasing · 2020. 9. 7. · Textbook Chapter 5.3 OBJECTIVE: Today you will learn about the end behavior of functions! A polynomial function is in STANDARD FORM if its

NOTES: INCREASING AND DECREASING DAY 4 OBJECTIVE: Today you will learn how to determine when a graph is increasing and decreasing! 1. Rollercoaster Problem: 2. Domain: ________________ Range: ________________ Relative Max: ________________ Relative Min: ________________ Absolute Max: ________________ Absolute Min: ________________ Increasing: ________________ Decreasing: ________________

3) Find the following using the following graph: Domain:________________ Range:___________________ x-intercept(s):_________________ y-intercept:__________ Local minimum:_____________ absolute min:___________ Local maximum:_____________ absolute max:___________ Increasing interval(s):_______________________________ Decreasing interval(s):________________________________

CALCULATOR GRAPHING 4. Graph x3 – 9x2 + 8x + 60 using your calculator. Sketch its graph below.

a. Determine the number of zeros for the polynomial ___________________

b. Where are the real zeros for the polynomial _________________________

c. Determine the number of turning points ____________________________

d. Where are the relative minimums or maximums? ____________________

e. Where are the absolute minimums or maximums? ____________________

f. Describe the end behavior of the graph: Right Side: _______________________________________ Left Side: ________________________________________

PRACTICE: DAY 4 KEY FEATURES OF POLYNOMIAL FUNCTIONS

Example: Increasing: __________________ __________________ Decreasing: __________________ __________________ You Try:

Increasing: ____________________________ Increasing: ____________________________ Descreasing: ___________________________ Descreasing: __________________________

1. Graph y = 3x4 + x3 - 10x2 + 2x + 7 using your calculator. Sketch its graph below.

a. Determine the number of zeros for the polynomial ________________________

b. Determine the number of real zeros for the polynomial ____________________

c. Determine the number of turning points _________________________________

d. Does the graph have relative minimums or maximums? ____________________

e. Does the graph have absolute minimums or maximums? ____________________

f. Describe the end behavior of the graph:

Right Side: _______________________________________ Left Side: ________________________________________

2. Graph y = 2x3 – 3x2 + 2 using your calculator. Sketch its graph below.

a. Determine the number of zeros for the polynomial _________________

b. Determine the number of real zeros for the polynomial ______________

c. Determine the number of turning points _________________________

d. Does the graph have relative minimums or maximums? ______________

e. Does the graph have absolute minimums or maximums? ______________

Right Side: _______________________________________ Left Side: ________________________________________

NAME: _________________________________________ DAY 4 1) State the domain and range for each of the following graphs. Then, state the intervals where the function is increasing and where the function is decreasing. a) b) D:___________ D:___________ R: ___________ R: ___________ Inc:__________ Inc:_________ Dec:_________ Dec:________ 2. Domain: ________________ Range: ________________ Increasing: ________________ Decreasing: ________________ Relative Max: ________________ Relative Min: ________________ Absolute Max: ________________ Absolute Min: ________________

3. Domain: ________________ Range: ________________ Increasing: ________________ Decreasing: ________________ Relative Max: ________________ Relative Min: ________________ Absolute Max: ________________ Absolute Min: ________________ 4.

a. Determine the number of real zeros for the polynomial ______________

b. Determine the number of turning points _________________________

c. Where does the graph have relative minimums or maximums? ______________

d. Where does the graph have absolute minimums or maximums? ______________

NOTES: END BEHAVIOR DAY 5 Textbook Chapter 5.3 OBJECTIVE: Today you will learn about the end behavior of functions! A polynomial function is in STANDARD FORM if its terms are written in descending order of exponents from left to right. Standard Form Example: f(x) = 2x3 – 5x2 – 4x + 7 Leading Coefficient_____ Degree_____ Factored Form Example: f(x) = x (x + 2) (x – 5)3 Leading Coefficient_____ Degree_____ Circle all polynomial functions. For each polynomial function, state the form and the degree.

1. f (x ) = 1

2x 4 − 3x 2 − 7

Form: __________________________ Degree: ________ LC: __________

2. f (x )= x 2 x + 3( )

Form: __________________________ Degree: ________ LC: __________

3. f (x ) = 6x 2 + 2x −1 + x Form: __________________________ Degree: ________ LC: __________

4. f (x )= 3

5x 4 + 2x + 9

Form: __________________________ Degree: ________ LC: __________

5. f (x )= −5x +12 Form: __________________________ Degree: ________ LC: __________

6. f (x ) = 22 − 19x + 2x

Form: __________________________ Degree: ________ LC: __________

7. f (x )= x(x + 3)2(x −1)3

Form: __________________________ Degree: ________ LC: __________

8. f (x )= 36x 4 − x 3 + x 2

Form: __________________________ Degree: ________ LC: __________

Describe the end behavior of the graph of the function. 1. 2.

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

Use your calculator the graph the following and determine the end behavior. 3. y = -3x5 – 6x2 + 3x – 8 4. h(x) = 6x8 – 7x5 + 4x

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

5) Describe the end behavior of the following functions: a. b.

Use your calculator to find the end behavior of each function. 6. f(x) = 3x5 + x3 + 10x2 + 4x + 1 7. f(x) = 7(x + 1)3 (x – 5) (x + 3)2

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

8. f(x) = –x6 + x4 + 10x3 + 4x2 + 1 9. f(x) = –2(x + 1)3 (x – 5)2 (x + 3)2

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

10. Choose the correct letter and answer the additional questions.

Describe the graph’s end behavior….. As x → +∞, f (x)→ ____ As x →−∞, f (x)→ ____ On what x intervals is Y (or f(x)) Increasing: ________________________________ Decreasing: _______________________________

PRACTICE: END BEHAVIOR Determine the end behavior of each of the graphs

x→∞, f (x)→ _____x→−∞, f (x)→ _____

x→∞, f (x)→ _____x→−∞, f (x)→ _____

x→∞, f (x)→ _____x→−∞, f (x)→ _____

2. Graph x3 – 9x2 + 8x + 60 using your calculator. Sketch its graph below.

a. How many zeros of the polynomial __________________________________

b. Find the real zeros of the polynomial ________________________________

c. Determine the number of turning points ______________________________

d. Where are the relative minimums or maximums? ______________________

e. Where are the absolute minimums or maximums? ______________________

f. Describe the end behavior of the graph: As x → +∞ , f (x )→ _____ as x →−∞ , f (x )→ _____

3. Graph 3x4 + x3 - 10x2 + 2x + 7 using your calculator. Sketch its graph below.

a. How many zeros of the polynomial __________________________________

b. Find the real zeros of the polynomial ______________________________

c. Determine the number of turning points ____________________________

d. Where are the relative minimums or maximums? ______________

e. Where are the absolute minimums or maximums? ______________

f. Describe the end behavior of the graph: As x → +∞ , f (x )→ _____ as x →−∞ , f (x )→ _____

NAME: ______________________________________________ DAY 5 Find the degree, leading coefficient, and end behavior. Then draw a rough sketch. 1. f(x) = 3x5 + x3 + 10x2 + 4x + 1 2. f(x) = – (x + 1)3 (x – 5)2 Degree: ___________ LC: _________ Degree: ___________ LC: _________

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

Sketch Sketch Find the degree of each function and end behavior. 3. f(x) = –3x4 + 10x3 + 4x2 + 1 4. f(x) = 2(x + 1)3 (x – 5) Degree: ___________ LC: _________ Degree: ___________ LC: _________

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

Sketch Sketch

A. End Behavior: As

As B. Identify the real zeros of the graph: __________________________________ C. Circle the turning points on the graph. Determine if they are relative maximums or minimums, absolute maximums or minimums. D. Determine the intervals where the polynomials are Increasing: ______________________________ Decreasing: _____________________________ E. Determine the domain and range .

Domain: _____________

Range: _______________

5. Use the graph to answer the questions. 6. Use the graph to answer the questions.

A. End Behavior: As

As B. Identify the real zeros of the graph: __________________________________ C. Circle the turning points on the graph. Determine if they are relative maximums or minimums, absolute maximums or minimums. D. Determine the intervals where the polynomials are Increasing: ______________________________ Decreasing: _____________________________ E. Determine the domain and range.

Domain: _____________

Range: _______________