7-1: Polynomial Functions -Evaluate Polynomial Functions -Identify general shapes of graphs and of polynomial functions
Algebra 2 Review - Simplify
I. 7 13 11 2 8 4
II. 2 1
Terms: O Polynomial in One Variable – O Polynomial (ie. NO: , , 2 , xy) O Only ONE variable identified O Degree of a Polynomial – O Degree of the largest monomial (greatest exponent) O Leading Coefficient – O Coefficient of the term with the greatest exponent O Polynomial Function – O Function notation f(x); domain vs range Example 1: State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. ( pg. 346)
a) Yes, Deg. Of 4, L.C. of 7 b) NO, more than one variable c) NO, variable in the denominator d) Yes, Deg. Of 5, L.C. of 1 111 Your Turn: State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1) NO; variable in the radicand 2) Yes; Deg. Of 4; L.C. of 10 101010 3) NO; variable with a negative exponent Evaluating Polynomial Functions Ex. 22:: Evaluate each polynomial function a) Find ppp(3(3(3)(3 ) and ppp((( ( 1)1) if ppp(x(x(x)(x ) = x xxx3 + x+ x 222 –––x.x.x. Evaluating Polynomial Functions Ex. 22:: Evaluate each polynomial function b) Find mmm(a(a(a 222))) if mmm(x(x(x)(x ) = x 333 + 4x 222 –––5x. Evaluating Polynomial Functions Ex. 22:: Evaluate each polynomial function c) Find bbb(2x –––1) –––333 bbb(x) if bbb(m) = 2m 222 + m –––1.1.1. Evaluating Polynomial Functions Your Turn: Evaluate each polynomial function 1) Find ppp(8a) if 2) Find rrr(x + 2) if Graphs of Polynomial Functions:
Constant Function Linear Function Degree of 0 Degree of 1 Graphs of Polynomial Functions:
Quadratic Function Cubic Function Degree of 2 Degree of 3 Graphs of Polynomial Functions:
Quartic Function Quintic Function Degree of 4 Degree of 5 Graphs of Polynomial Functions What do you notice???? Roots/Zeros (real) – Degree stats MAX. number of real roots Graphs: Even vs Odd degree – Even – end in same direction Odd – end in opposite directions Will an odd function ALWAYS cross the x axis? YES – end in opposite directions Will an even function ALWAYS cross the x axis? NO – end in same direction Graphs of Polynomial Functions End Behavior :
Describes the behavior of the graph as it approaches infinity (∞).
→ ∞ → ∞ “the function fff of xxx is approaching positive infinity as x approaches negative infinity” (the y coord. is increasing as the x coord. is decreasing) Ex. 3: For each graph: 1) Describe the end behavior 2) Determine whether the graph represents an odd or even function 3) state the number of real zeros
a) b) Ex. 3: For each graph: 1) Describe the end behavior 2) Determine whether the graph represents an odd or even function 3) state the number of real zeros
c) Your Turn: For each graph: 1) Describe the end behavior 2) Determine whether the graph represents an odd or even function 3) state the number of real zeros
1) 2)
a) → ∞ → ∞ a) → ∞ → ∞ → ∞ → ∞ → ∞ → ∞ b) Even function b) Odd function c) 2 real zeros c) 5 real zeros Homework:
Pg. 350: 17 37 odds, 39 44 all, 57 & 58 Due Thursday