University:
Florida Gateway College
Course:
MAC 2312 | Calculus II
Academic year:

2024
Views:

115

Pages:

2
Author:

Ruby Bates

Question 1 Explain what you know about the roots of polynomial functions. How can you determine all of the roots? Give examples, including sketches of graphs. Answer Roots of Polynomial Functions: The roots of a polynomial equation are the points at which 𝑦 = 0 y=0 for 𝑦 = 𝑓 ( 𝑥 ) = ( 𝑥 − 𝛼 ) ( 𝑥 − 𝛽 ) ( 𝑥 − 𝛾 ) y=f(x)=(x−α)(x−β)(x−γ), where 𝛼 , 𝛽 , 𝛾 ∈ 𝑅 α,β,γ∈R. Example: For 𝑓 ( 𝑥 ) = ( 𝑥 − 2 ) ( 𝑥 + 1 ) ( 𝑥 − 3 ) f(x)=(x−2)(x+1)(x−3), the roots are 𝑥 = 2 , − 1 , 3 x=2,−1,3. Graphically, the roots are the x-intercepts of the polynomial. Question 2 Every cubic polynomial can be categorized into one of four types: Three real, distinct zeros: 𝑃 ( 𝑥 ) = ( 𝑥 − 𝛼 ) ( 𝑥 − 𝛽 ) ( 𝑥 − 𝛾 ) , 𝛼 ≠ 𝛽 ≠ 𝛾 P(x)=(x−α)(x−β)(x−γ),α  =β  =γ. Two real zeros, one repeated: 𝑃 ( 𝑥 ) = 𝑎 ( 𝑥 − 𝛼 ) 2 ( 𝑥 − 𝛽 ) , 𝑎 ≠ 0 P(x)=a(x−α) 2 (x−β),a  =0. One real zero, repeated three times: 𝑃 ( 𝑥 ) = 𝑎 ( 𝑥 − 𝛼 ) 3 , 𝑎 ≠ 0 P(x)=a(x−α) 3 ,a  =0. One real and two imaginary zeros: 𝑃 ( 𝑥 ) = ( 𝑥 − 𝛼 ) ( 𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 ) , Δ = 𝑏 2 − 4 𝑎 𝑐 < 0 , 𝑎 ≠ 0 P(x)=(x−α)(ax 2 +bx+c),Δ=b 2 −4ac<0,a  =0. Experiment with the graphs of Type 1 cubics. Clearly state the effect of changing both the size and sign of 𝑎 a. What is the geometrical significance of 𝛼 , 𝛽 , 𝛾 , α,β,γ, and 𝑎 a? Answer 𝑎 a: The size of 𝑎 a controls the vertical stretch or shrink of the graph. Changing the sign of 𝑎 a reflects the graph about the x-axis. 𝛼 , 𝛽 , 𝛾 α,β,γ: The x-intercepts of the graph correspond to the roots 𝛼 , 𝛽 , 𝛾 α,β,γ. Question 3 For the following exercise, for each polynomial, a. find the degree, b. find the zeros, if any, c. find the 𝑦 y-intercept(s), if any, d. use the leading coefficient to determine the graph's end behavior, and e. determine algebraically whether the polynomial is even, odd, or neither. 𝑓 ( 𝑥 ) = 3 𝑥 − 𝑥 3 f(x)=3x−x 3 Answer a) Degree: 3. b) Zeros: 𝑥 = ± 3 , 𝑥 = 0 x=± 3 ,x=0. c) 𝑦 y-intercept: 0. d) Graph rises to the left and falls to the right. e) Function is odd. Question 4 How can proficiency in factoring help when graphing polynomial functions without using a calculator? Answer When we factor a polynomial partially or completely, we break that polynomial into low-degree factors. This allows us to: Identify roots and their multiplicities, Determine the graph’s x-intercepts, and Understand how the function behaves around these roots. Factoring simplifies analysis, making it easier to sketch the graph manually. Question 5 The graph of a polynomial function has the following characteristics: a) Its domain and range are the set of all real numbers. b) There are turning points at 𝑥 = − 2 , 0 , 3 x=−2,0,3. a) Draw the graphs of two different polynomial functions that have these three characteristics. b) What additional characteristics would ensure that only one graph could be drawn? Answer The other equation that works is: 𝑓 ( 𝑥 ) = 1 4 𝑥 4 − 1 3 𝑥 3 − 3 𝑥 2 − 3 𝑥 . f(x)= 4 1 x 4 − 3 1 x 3 −3x 2 −3x. This is one possible polynomial equation that satisfies the given conditions. a) There are two polynomials that work for this situation. b) To ensure only one graph could be drawn, we would need to specify additional information such as the leading coefficient or more precise behaviors (e.g., end behavior or exact values of 𝑓 ( 𝑥 ) f(x) at specific points).