G10 Math Q2 Week 1- Graph of Polynomial Functions.pptx

polynomial functions

1. GRAPHS OF POLYNOMIAL FUNCTIONS

2. MOST ESSENTIAL LEARNING COMPETENCY Understands, describes, and interprets graph of polynomial functions (MELC 13).

3. LEARNING OBJECTIVES At the end of the lesson, learners are expected to: 1. Understand the properties of the graph of a polynomial function. 2. Describes and interprets graph of a polynomial function. 3. Appreciate graphs of polynomial functions as applied in some real-life situations.

4. Do you remember when an equation is a polynomial function? What makes an equation not a polynomial? An equation is a polynomial function if: There is no negative exponent in one of the variables There is no fractional exponent There is no radical exponent. It is considered a polynomial function if the exponents of the variable are all positive integers

5. TRUE OR FALSE Directions : Write the word TRUE if the statement is correct and FALSE if it is not. 1. The number of turning points in the graph of a polynomial function is the same as the degree of a polynomial function. 2. The leading term of the polynomial function in standard form: f(x) = 5x 4 - x 3 +2x -2 is 5x 4 3. The Leading Coefficient Test is used to determine the end behaviors of the graph of a polynomial function. 4. X-intercepts are points where the graph intersects the x-axis, that is x= 0. 5. If the degree of the polynomial is odd and the leading coefficient is positive, the graph falls to the left and rises to the right.

6. TRUE OR FALSE Directions : Write the word TRUE if the statement is correct and FALSE if it is not. 1. The number of turning points in the graph of a polynomial function is the same as the degree of a polynomial function. 2. The leading term of the polynomial function in standard form: f(x) = 5x 4 - x 3 +2x -2 is 5x 4 3. The Leading Coefficient Test is used to determine the end behaviors of the graph of a polynomial function. 4. X-intercepts are points where the graph intersects the x-axis, that is x= 0. 5. If the degree of the polynomial is odd and the leading coefficient is positive, the graph falls to the left and rises to the right. FALSE TRUE TRUE TRUE FALSE

7. LET’S COMPARE Observe and determine the similarities the figure of an electromagnetic wave and the graph of a polynomial function.

8. TRIVIA: Electromagnetic waves or EM waves are waves that are created as a result of vibrations between an electric field and a magnetic field. In other words, EM waves are composed of oscillating magnetic and electric fields. They are also perpendicular to the direction of the EM wave. EM waves appearance is somehow similar to a graph of polynomial function. They are both curved form in appearance and both figures have ups and downs. In the electromagnetic wave this are called the crest and trough and in polynomial functions these are called turning points.

9. LET’S INVESTIGATE Complete the table. This activity will help you understand how to use the Leading Coefficient Test to describe the behavior of the graph

10. POLYNOMIAL FUNCTION LEADING COEFFICIENT (+ OR -) DEGREE (EVEN 0R ODD)   GRAPH  BEHAVIOR OF THE GRAPH (RISING OR FALLING) Relation of the LC and degree to the Behavior of the Graph Left Hand Right Hand f(x) = x 2

11. POLYNOMIAL FUNCTION LEADING COEFFICIENT (+ OR -) DEGREE (EVEN 0R ODD)   GRAPH  BEHAVIOR OF THE GRAPH (RISING OR FALLING) Relation of the LC and degree to the Behavior of the Graph Left Hand Right Hand f(x)= -x 2

12. POLYNOMIAL FUNCTION LEADING COEFFICIENT (+ OR -) DEGREE (EVEN 0R ODD)   GRAPH  BEHAVIOR OF THE GRAPH (RISING OR FALLING) Relation of the LC and degree to the Behavior of the Graph Left Hand Right Hand f(x) = x 3

13. POLYNOMIAL FUNCTION LEADING COEFFICIENT (+ OR -) DEGREE (EVEN 0R ODD)   GRAPH  BEHAVIOR OF THE GRAPH (RISING OR FALLING) Relation of the LC and degree to the Behavior of the Graph Left Hand Right Hand f(x) = -x 3

14. POLYNOMIAL FUNCTION LEADING COEFFICIENT (+ OR -) DEGREE (EVEN 0R ODD)   GRAPH  BEHAVIOR OF THE GRAPH (RISING OR FALLING) Relation of the LC and degree to the Behavior of the Graph Left Hand Right Hand f(x) = x 2 + even rising rising If the LC is + and the degree is even, then, the graph of the polynomial function rises to the left and rises to the right

15. POLYNOMIAL FUNCTION LEADING COEFFICIENT (+ OR -) DEGREE (EVEN 0R ODD)   GRAPH  BEHAVIOR OF THE GRAPH (RISING OR FALLING) Relation of the LC and degree to the Behavior of the Graph Left Hand Right Hand f(x) = -x 2 - even falling falling If the LC is - and the degree is even, then, the graph of the polynomial function falls to the left and falls to the right

16. POLYNOMIAL FUNCTION LEADING COEFFICIENT (+ OR -) DEGREE (EVEN 0R ODD)   GRAPH  BEHAVIOR OF THE GRAPH (RISING OR FALLING) Relation of the LC and degree to the Behavior of the Graph Left Hand Right Hand f(x) = x 3 + odd falling rising If the LC is + and the degree is odd, then, the graph of the polynomial function falls to the left and rises to the right

17. POLYNOMIAL FUNCTION LEADING COEFFICIENT (+ OR -) DEGREE (EVEN 0R ODD)   GRAPH  BEHAVIOR OF THE GRAPH (RISING OR FALLING) Relation of the LC and degree to the Behavior of the Graph Left Hand Right Hand f(x) = -x 3 - odd rising falling If the LC is - and the degree is odd, then, the graph of the polynomial function rises to the left and falls to the right

18. Read and study Things to Remember of the Supplementary Learning Material

19. Things to Remember: To sketch the graph of a polynomial function we need to consider the following: Leading term- This is the term in the polynomial function with the highest exponent. It also tells the degree of the function. Leading coefficient- This is the numerical coefficient of the leading term.

20. End Behavior of the Graph Leading Coefficient Test is used to determine the right-hand and the left-hand behaviors of the graph of a polynomial function. The graph of a polynomial function is: Rising to the extreme left and rising to the extreme right if the degree n is even and the leading coefficient a n > 0 (positive) Falling to the extreme left and rising to the extreme right if the degree n is odd and the leading coefficient a n > 0 (positive) Falling to the extreme left and falling to the extreme right if the degree n is even and the leading coefficient a n < 0 (negative) Rising to the extreme left and falling to the extreme right if the degree n is odd and the leading coefficient a n < 0 (negative)

21. X- intercepts -These are the points where the graph intersects the x-axis. These are the values of x when y =0 Multiplicity of roots -It is the number of times the same root has been obtained. If there is an even multiplicity of roots, the graph is tangent to a point on the x-axis. If there is an odd multiplicity of roots, then the graph crosses the x-axis. Y-intercept - This is the point where the graph intersects the y-axis. Number of turning points - these are points where the graph shifts from decreasing to increasing function value, or vice versa. The number of turning points is strictly less than the degree of the polynomial. It is at most n-1 number of turning points.

22. To show and describe the graph let us follow these steps: f(x) = 2x 3 +6x 2 -2x -6 or f(x) = (2x+6) (x -1) (x+1). Step 1: Identify the leading term and leading coefficient of the polynomial function Leading Term: 2x 3 (the exponent is 3 so n is odd) Leading Coefficient: 2 (leading coefficient is positive so a n >0) Step 2: Determine the behavior of the graph.   Behavior of the graph: Since the degree of the polynomial is 3 and it is odd and the leading coefficient is greater than 0, using the Leading Coefficient Test the graph is FALLING to the extreme left and RISING to the extreme right.

23. Step 4: Determine the multiplicity of roots. Since there is no repeated root, there is no multiplicity of roots. Step 5: Determine the y-intercepts This can be done by simply solving for f(x) when x= o. y-intercept: f(x) = 2x 3 +6x 2 -2x -6 let x = 0 f(x) = 2(0) 3 +6(0) 2 -2(0) -6 y-intercept is -6 which is located at (0, -6) in the coordinate plane.   y-intercept is -6 which is located at (0, -6) in the coordinate plane. Step 6: Determine the number of turning points. Number of turning points: Since n is 3, the graph has at most 2 turning points. This means that there will be one or two turning points

24. DESCRIBE ME!!! Describe the properties of the graph of the polynomial function F(x) = (x-1) 2 (x+1) as to the following: Standard Form Leading Term x-intercepts y-intercepts number of turning points possible graph with end behavior (Refer to Learning Task 2, Letter B of PIVOT Learners Material for Grade 10, Quarter 2 page 10)  

25. DESCRIBE ME!!! Describe the properties of the graph of the polynomial function F(x) = (x-1) 2 (x+1) as to the following: Standard Form F(x) = x 3 - x 2 -x +1 Leading Term x 3 x-intercepts 1 with mult. of 2 and -1 y-intercepts 1 number of turning points 2  

26. f. graph with end behavior FALLING- RISING

27. THINK OF THIS Directions: Use the situation to answer the questions that follow. Karl Benedic, the president of Mathematics Club, proposed a project: to put up a rectangular Math Garden whose lot area is given by the equation f(x) = x 4 -7x 2 + 6x or in factored form f(x)= x ( x+3) (x-1)(x-2). Study the graph and answer the questions that follows. What is/are the x-intercepts? ____________________________________________ What is/are the y- intercept? ____________________________________________ How many turning points does the graph have? _____________________________ What are the end-behaviors of the graph? _____________________________ Suppose the width is x-2, what is the length? _______________________________ Show the possible sketch of the graph.

28. THINK OF THIS Directions: Use the situation to answer the questions that follow. Karl Benedic, the president of Mathematics Club, proposed a project: to put up a rectangular Math Garden whose lot area is given by the equation f(x) = x 4 -7x 2 + 6x or in factored form f(x)= x ( x+3) (x-1)(x-2). Study the graph and answer the questions that follows. What is/are the x-intercepts? 0, -3 , 1 , 2 What is/are the y- intercept? How many turning points does the graph have? 3 What are the end-behaviors of the graph? RISING- RISING Suppose the width is x-2, what is the length? x ( x+3) ( x -1)

29. 6. Show the possible sketch of the graph.

30. CARD MATCH Directions: Match the card that contains the POLYNOMIAL FUNCTION to the card that contains its corresponding GRAPH and determine the properties of the graph such as: x- intercepts y-intercepts Number of turning points End behavior of the graph

31. x- intercepts- -2, 1, 3 y-intercepts -6 Number of turning points 2 End behavior of the graph RISING- FALLING

32. x- intercepts- -2, -3 y-intercepts 6 Number of turning points 1 End behavior of the graph RISING- RISING

33. x- intercepts- -1, 1, -3, 2 y-intercepts 6 Number of turning points 3 End behavior of the graph RISING- RISING

34. What I have learned Answer the following questions based on their understanding 1. How do you determine the intercepts in the graph of a polynomial function? 2. How do you know the number of turning points of the graph of a polynomial function? 3. What is the use of the Leading Coefficient Test and how do we use it?

35. ASSESSMENT Directions: Choose the letter of the best answer. B C B A C

36. CLOSURE 3-2-1 Exit Card List 3 things you learned today.      List 2 things you’d like to learn more about.   List 1 question you have.