GRAPHING POLYNOMIALS DAY 2 U N I T 1 1

Transcription

1 GRAPHING POLYNOMIALS DAY 2 U N I T 1 1

2 ODD/EVEN DEGREE POLYNOMIAL Odd degree polynomial: A polynomial whose largest power is an odd integer Even degree polynomial : A polynomial whose largest power is an even integer 2

3 LEADING COEFFICIENT TEST The end behavior of a polynomial can be determined by the sign of the leading coefficient (a) and the degree of the polynomial When n is odd AND a >0 then x -, f(x) - and x, f(x) When n is odd AND a<0 then x -, f(x) and x, f(x) - 3

4 LEADING COEFFICIENT TEST (CONTINUED) When n is even AND a>0 x -, f(x) and x, f(x) When n is even AND a<0 x -, f(x) - and x, f(x) - Describe the end behavior of 2 4 x -, f(x) - and x, f(x) x -, f(x) - and x, f(x) - 4

5 MULTIPLICITY (REPEATED ZEROS) Multiplicity is the number of times (k) a particular number is a zero for a given polynomial. When k is odd, the graph crosses over the x-axis at x=a When k is even, the graph touches or bounces off the x-axis at x=a As k increases the graph around x=a becomes flattened 5

6 MULTIPLICITY (CONT) For roots repeated an even number of times Notice The graph never crosses the x-axis The more the root is repeated, the curve becomes more flat 6

7 MULTIPLICITY (CONT) For roots repeated an odd number of times Notice The graph crosses the x-axis The more the root is repeated, the curve becomes more flat near x=a 7

8 TURNING POINTS Turning points on a graph are points where the graph changes direction Every polynomial function of degree n has at most n- 1 turning points. If the polynomial function has n number of distinct, real roots, the number of turning points is exactly n-1 8

9 EXAMPLES 1 & 2 Given the graphs below, determine the minimum degree, even or odd degree, the sign of the leading coefficient, factors, and multiplicity Degree even/odd: Minimum degree: Sign of leading coef: Factors: Multiplicity: even 4 positive (x+3)(x+1)(x-1)(x-3) none even 6 positive (x+3) 2 (x+1)(x-1) 3 Yes, 2@(-3,0); 3@(1,0) 9

10 YOU TRY! Given the graphs below, determine the minimum degree, even or odd degree, the sign of the leading coefficient, factors, and multiplicity Degree even/odd: Minimum degree: Sign of leading coef: Real factors: Multiplicity: odd 3 positive (x+0) 2 (x-4) Yes: 2@(0,0) 10

11 EXAMPLE 3 Given 2 2 List end behavior, roots with multiplicity, y-intercept. Sketch the graph End behavior Even degree and a<0 x -, f(x) - and x, f(x) - Roots and Multiplicity , 0, 0, 1 Multiplicity: yes 2@ (0,0) Y-intercept When x = 0, f(x) = 0 (0, 0) 11

12 EXAMPLE 3 (CONTINUED) Sketch 2 2 Let s think about the symmetry max number of turning points x y Charts must include intercepts

13 YOU TRY! Given 3 4 List end behavior, roots with multiplicity, y-intercept. Sketch the graph End behavior Even degree and a>0 x -, f(x) and x, f(x) Roots and Multiplicity 3 4 0, 0, 0, multiplicity=3 with the x=0 root Y-intercept When x = 0, f(x) = 0 (0, 0) 13

14 YOU TRY! (CONTINUED) Sketch 3 4 Symmetry: none Max turning points: 3 Actual:1 x y

15 EXAMPLE 4 Given List end behavior, roots with multiplicity, y-intercept. Sketch the graph End behavior Odd degree and a>0 x -, f(x) - and x, f(x) Roots and Multiplicity , 1.5, 2 Multiplicity: no Y-intercept When x = 0, f(x) = 24 (0, 24) 15

16 EXAMPLE 4 (CONTINUED) Sketch Let s think about the symmetry max number of turning points x y

17 YOU TRY! Given List end behavior, roots with multiplicity, y-intercept. Sketch the graph End behavior Even degree and a>0 x -, f(x) and x, f(x) Roots and Multiplicity , -.667, -.25, 1 Multiplicity: no Y-intercept When x = 0, f(x) = -6 (0, -6) 17

18 YOU TRY! (CONTINUED) Sketch symmetry max number of turning points x y