 STRATEGIC INTERVENTION MATERIAL IN MATHEMATICS 10 Least Learned Competency:The graph of a polynomial function My Behaviour, My Destiny. DUJALI NATIONAL HIGH SCHOOL MARVIN VICENTE ORETA LEAST MASTERED COMPETENCY FOCUS SKILLS: THE GRAPH OF POLYNOMIAL FUNCTIONS SUPPORT SKILLS: I. ILLUSTRATE A POLYNOMIAL FUNCTION II. DETERMINE THE INTERCEPTS AND PLOT THE POINTS III. GRAPH A POLYNOMIAL FUNCTION TO THE STUDENTS This learning package is intended to supplement your classroom learning while working independently . The activities and exercises will widen your understanding of the different concepts you should learn. OBJECTIVE: DEFINE A POLYNOMIAL FUNCTION  OBJECTIVE: DEFINE A POLYNOMIAL FUNCTION  A LEADING TERM is the term with the highest exponent. A LEADING COEFFICIENT is the coefficient of the term with the highest degree. A LEADING TERM is the term with the highest exponent. A LEADING COEFFICIENT is the coefficient of the term with the highest degree. The DEGREE is the greatest exponent of a polynomial . The Constant is a term with 0 as exponent A LEADING TERM is the term with the highest exponent. For example, the polynomial function f(x) = -2x4 + x3 – 5x2 – 10 For example, the polynomial function f(x) = -2x4 + x3 – 5x2 – 10 Leading term: -2x4 For example, the polynomial function f(x) = -2x4 + x3 – 5x2 – 10 Leading term: -2x4 Leading coefficient: -2 For example, the polynomial function f(x) = -2x4 + x3 – 5x2 – 10 Leading term: -2x4 Leading coefficient: -2 Degree: 4 For example, the polynomial function f(x) = -2x4 + x3 – 5x2 – 10 Leading term: -2x4 Leading coefficient: -2 Degree: 4 Constant: -10. For example, the polynomial function f(x) = -2x4 + x3 – 5x2 – 10 Leading term: -2x4 Leading coefficient: -2 Degree: 4 Constant: -10. What is the degree, leading coefficient and constant of P(x)=3x5 – 3x + 2 ? For example, the polynomial function f(x) = -2x4 + x3 – 5x2 – 10 Leading term: -2x4 Leading coefficient: -2 Degree: 4 Constant: -10. What is the degree, leading coefficient and constant of P(x)=3x5 – 3x + 2 ? Leading term: Constant: Leading coefficient: Degree: 3x5 ? 3 ? 2 ? 5 ?   Polynomial function f (x) = x3 +3x2 –x –3 Polynomial function f (x) = x3 +3x2 –x –3 f (x) = x3 +3x2 –x -3 Factors: (x + 3) (x + 1) (x – 1) = 0 Polynomial function f (x) = x3 +3x2 –x –3 f (x) = x3 +3x2 –x – 3 Factors: (x + 3) (x + 1) (x – 1) = 0 Solve for x: Polynomial function f (x) = x3 +3x2 –x –3 f (x) = x3 +3x2 –x – 3 Factors: (x + 3) (x + 1) (x – 1) = 0 Solve for x: x+3=0 x+1=0 x-1=0 x = -3 x = -1 x = 1
Polynomial function f (x) = x3 +3x2 –x –3 f (x) = x3 +3x2 –x – 3 Factors: (x + 3) (x + 1) (x – 1) = 0 Solve for x: x+3=0 x+1=0 x-1=0 x = -3 x = -1 x = 1
X INTERCEPTS(refer to the factors): -3, -1, 1 Polynomial function f (x) = x3 +3x2 –x –3 f (x) = x3 +3x2 –x – 3 Factors: (x + 3) (x + 1) (x – 1) = 0 Solve for x: x+3=0 x+1=0 x-1=0 x = -3 x = -1 x = 1
X INTERCEPTS(refer to the factors): -3, -1, 1 Y INTERCEPT (refer to the constant): -3  Polynomial function f (x) = x3 +3x2 –x –3 f (x) = x3 +3x2 –x – 3 Factors: (x + 3) (x + 1) (x – 1) = 0 Solve for x: x+3=0 x+1=0 x-1=0 x = -3 x = -1 x = 1
X INTERCEPTS(refer to the factors): -3, -1, 1 Y INTERCEPT (refer to the constant): -3  Polynomial function f (x) = x3 +3x2 –x –3 f (x) = x3 +3x2 –x – 3 Factors: (x + 3) (x + 1) (x – 1) = 0 Solve for x: x+3=0 x+1=0 x-1=0 x = -3 x = -1 x = 1
X INTERCEPTS(refer to the factors): -3, -1, 1 Y INTERCEPT (refer to the constant): -3    (-3, 0) -3 -2 -1 -1 -3 3 -2 2 1 -4 1 2 3 (-3, 0) (-1, 0) -3 -2 -1 -1 -3 3 -2 2 1 -4 1 2 3 (-3, 0) (-1, 0) -3 (1, 0) -2 -1 -1 -3 3 -2 2 1 -4 1 2 3 The X intercepts (-3, 0) (-1, 0) -3 (1, 0) -2 -1 -1 -3 3 -2 2 1 -4 1 2 3 The X intercepts (-3, 0) (0, -3) (-1, 0) -3 (1, 0) -2 -1 -1 -3 3 -2 2 1 -4 1 2 3 The X intercepts The Y intercept (-3, 0) (0, -3) (-1, 0) -3 (1, 0) -2 -1 -1 -3 3 -2 2 1 -4 1 2 3  OBJECTIVE: COMPARE THE BEHAVIOR OF THE GRAPH The destiny of our behavior is just congruent to the graph of a polynomial function  If we are positive thinker or optimistic, then we are always rising up OBJECTIVE: COMPARE THE BEHAVIOR OF THE GRAPH The destiny of our behavior is just congruent to the graph of a polynomial function If we are positive thinker or optimistic, then we are always rising up FALLING RISING    If we are positive thinker or optimistic, then we are always rising up LEADING COEFFICIENT POSITIVE DEGREE EVEN GRAPH RISING RISING BEHAVIOR OF GRAPH FALLING RISING  If we are positive thinker or optimistic, then we are always rising up LEADING COEFFICIENT POSITIVE DEGREE EVEN ODD GRAPH FALLING RISING RISING RISING BEHAVIOR OF GRAPH FALLING RISING  If we are negative thinker or pessimistic, then we are always falling down LEADING COEFFICIENT NEGATIVE DEGREE EVEN GRAPH FALLING FALLING BEHAVIOR OF GRAPH FALLING RISING  If we are negative thinker or pessimistic, then we are always falling down. LEADING COEFFICIENT NEGATIVE DEGREE EVEN ODD GRAPH FALLING FALLING RISING FALLING BEHAVIOR OF GRAPH FALLING RISING Therefore, the behavior of the graph of Polynomial function f (x) = x3 +3x2 –x –3 Therefore, the behavior of the graph of Polynomial function f (x) = x3 +3x2 –x –3 is FALLING RISING -3 -2 -1 -1 -3 3 2 1 -2 -4 1 2 3  Therefore, the behavior of the graph of Polynomial function f (x) = x3 +3x2 –x –3 is FALLING RISING -3 -2 -1 -1 -3 3 2 1 -2 -4 1 2 3   Move the answers given to the question mark.  P(x)= 4x3 – 3x2 –25x – 6 P(x)= x3 +5x2 – 9x – 45 P(x)= x4 +x3 –19x2 +11x+30 P(x)= –2x3 +3x2 +8x + 3 P(x)= x2 – 2x + 8 4x3 ? x3 ? x2 ? -2 ? 1 4 ? 1 ? 3 4 ? 2 ? 3 ? 30 ? -6 ? 8 ? 3    P(x)= –2x3 +3x2 +8x + 3 P(x)= 4x3 – 3x2 –25x – 6 P(x)= x4 +x3 –19x2 +11x+30 P(x)= x3 +5x2 – 9x – 45 POLYNOMIAL FUNCTION P(x)= x2 – 2x + 8 (x + 5)(x+1)(x-3)(x-2) (x + 5)(x + 3)(x - 3) (x - 3)(4x + 1)(x + 2) (x - 4) (x + 2) -(x +1)(2x + 1)(x -3) FACTORS CONSTANT -45 8 X INTERCEPTS Y INTERCEPT P(x)= –2x3 +3x2 +8x + 3 P(x)= 4x3 – 3x2 –25x – 6 P(x)= x4 +x3 –19x2 +11x+30 P(x)= x3 +5x2 – 9x – 45 POLYNOMIAL FUNCTION P(x)= x2 – 2x + 8 (x + 5)(x+1)(x-3)(x-2) (x + 5)(x + 3)(x - 3) (x - 3)(4x + 1)(x + 2) (x - 4) (x + 2) -(x +1)(2x + 1)(x -3) FACTORS CONSTANT (4,0),(-2,0) X INTERCEPTS (3,0),(-2,0), (-1/4,0) ? (-1,0),(3,0),(-1/2,0), ? (-5,0),(-3,0),(3,0) ? (-5,0),(-1,0),(3,0),(2,0) ? Y INTERCEPT P(x)= –2x3 +3x2 +8x + 3 P(x)= 4x3 – 3x2 –25x – 6 P(x)= x4 +x3 –19x2 +11x+30 P(x)= x3 +5x2 – 9x – 45 POLYNOMIAL FUNCTION P(x)= x2 – 2x + 8 (x + 5)(x+1)(x-3)(x-2) (x + 5)(x + 3)(x - 3) (x - 3)(4x + 1)(x + 2) (x - 4) (x + 2) -(x +1)(2x + 1)(x -3) FACTORS CONSTANT X INTERCEPTS Y INTERCEPT (0, -45) (0, 30) ? (0, -6) ? (0, 8) (0, 3) ?      Falling Rising Falling Falling Leading Coefficient + + ? - ? - ? Rising Rising Degree Even ? Odd ? Even Odd ? Rising Rising ? Rising Falling Rising Falling Falling Rising ? Falling Falling ? Behavior of the Graph   The graph of f (x) = x3+3x2 - 4x -12 STATION Note: use ^ for exponentExample: 3x2 for 3x2 STATION 1 f (x) = x3+3x2 - 4x -12
Note: use ^ for exponentExample: 3x2 for 3x2 STATION 1 f (x) = x3+3x2 - 4x -12 Leading Term: Leading Coefficient: Degree: Constant:
Note: use ^ for exponentExample: 3x2 for 3x2 STATION 1 f (x) = x3+3x2 - 4x -12 Leading Term: x3 Leading Coefficient: 1 Degree: 3 Constant: -12 FACTOR COMLETELY: f (x) = (x+3)(x+2)(x-2)
 STATION 2: DETERMINE THE X AND Y INTERCEPTS AND PLOT THE POINT. STATION What are the x intercepts of f (x) = x3+3x2 - 4x -12 ? (0, -3), (0, -2), (0, -3 ) (0, -3), (0, -2), (0, 2) (0, -3), (0, 2), (0, 3 ) (0, 3), (0, -2), (0, 3 ) What is the y intercept of f (x) = x3+3x2 - 4x -12? (-12, 0) (0, 10) (3, 0) (0, -12) Plot the points -3 -2 -1 -3 -9 9 -6 6 3 -12 1 2 3 -3 -2 -1 -3 -9 9 -6 6 3 -12 1 2 3 (-3, 0) ? (-2, 0) ? -3 -2 -1 -3 -9 9 -6 6 3 -12 1 (2, 0) ? 2 (0, -12) ? 3  STATION 3: DETERMINE THE BEHAVIOR AND CONNECT THE POINTS f (x) = x3+3x2 - 4x -12 STATION  Use the magnetic graphing boardin graphing your answer. STATION 3: DETERMINE THE BEHAVIOR AND CONNECT THE POINTS f (x) = x3+3x2 - 4x -12 What is the behavior of the graph? Rising Rising Falling Falling Rising Falling Falling Rising  The graph of f(x) = x4 - 5x3 +5x2 +5x - 6 STATION Note: use ^ for exponentExample: 3x2 for 3x2 STATION 1 f(x) = x4 - 5x3 +5x2 +5x - 6 Leading Term: Leading Coefficient: Degree: Constant: FACTOR COMLETELY: f (x) = (x+1)(x-1)(x-2)(x-3) STATION (0, -3), (0, -2), (0, 3 ) (-1, 0) (-1, 0), (1, 0), (2, 0), (3, 0) (0, -3), (0, -2), (0, 2 ) (-1, 0) (0, -3), (0, -2), (0, 3 ) (1,0) STATION 2: DETERMINE THE X AND Y INTERCEPTS AND PLOT THE POINT. f(x) = x4 - 5x3 +5x2 +5x - 6 X INTERCEPTS STATION 2: DETERMINE THE X AND Y INTERCEPTS AND PLOT THE POINT. f(x) = x4 - 5x3 +5x2 +5x - 6 What is the Y intercept? (0, -6) (-12, 0) (3, 0) (0, -12)  STATION 2: DETERMINE THE X AND Y INTERCEPTS AND PLOT THE POINT. f(x) = x4 - 5x3 +5x2 +5x - 6 (-1, 0) ? (1,0) ? (0, -6) ? (2, 0) ? (3, 0) ? STATION Use the magnetic graphing boardin graphing your answer. STATION 3: DETERMINE THE BEHAVIOR AND CONNECT THE POINTS f (x) = x4 - 5x3 +5x2 +5x - 6 What is the behavior of the graph? Falling Falling Rising Falling Falling Rising Rising Rising P(x)= 2x2 - 3x5 –1 P(x)= -3x4 - 2x –8 P(x)= -x3 + 4x3 –1 P(x)= x5 - 3x3 –2x Polynomial Function Leading Coefficient Positive P(x)= 2x2 - 3x5 –1 P(x)= -3x4 - 2x –8 P(x)= -x3 + 4x3 –1 P(x)= x5 - 3x3 –2x Polynomial Function Leading Coefficient Positive Positive Negative Negative Odd Degree Falling Rising P(x)= 2x2 - 3x5 –1 P(x)= -3x4 - 2x –8 P(x)= -x3 + 4x3 –1 P(x)= x5 - 3x3 –2x Polynomial Function A Falling Falling Leading Coefficient Positive Negative Positive Negative B Odd Even Odd Even Rising Rising Degree C Behavior of the Graph D ? C ? A ? B Rising Falling D  IF THE LEADING COEFFICIENT IS POSITIVE WITH ____________ DEGREE, THE GRAPH IS FALLING RISING. IF THE LEADING COEFFICIENT IS ______________ WITH EVEN DEGREE, THE GRAPH IS FALLING FALLING. IF THE LEADING COEFFICIENT IS POSITIVE WITH EVEN DEGREE, THE GRAPH IS ________________. IF THE LEADING COEFFICEINT IS NEGATIVE WITH ODD DEGREE, THE GRAPH IS ________________. RISING FALLING ? RISING RISING ? NEGATIVE ? ODD ? |